Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) [1] 3037192046, 9783037192047

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IRMA Lectures in Mathematics and Theoretical Physics 31

This is volume 1 of a 2-volume work comprising a total of 14 refereed research articles which stem from the CARMA Conference (Algebraic Combinatorics, Resurgence, Moulds and Applications), held at the Centre International de Rencontres Mathématiques in Luminy, France, from June 26 to 30, 2017. The conference did notably emphasise the role of Hopf algebraic techniques and related concepts (e.g. Rota–Baxter algebras, operads, Ecalle’s mould calculus) which have lately proved pervasive in combinatorics, but also in many other fields, from multiple zeta values to the algebraic study of control systems and the theory of rough paths. The volumes should be useful to researchers or graduate students in mathematics working in these domains and to theoretical physicists involved with resurgent functions and alien calculus.

ISBN 978-3-03719-204-7

www.ems-ph.org

Chapoton et al. | IRMA 31 | FONT: Rotis Sans | Farben: Pantone 287, Pantone 116 | 170 x 240 mm | RB: 23? mm

Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 1

Frédéric Chapoton, Frédéric Fauvet, Claudia Malvenuto and Jean-Yves Thibon, Editors

Frédéric Chapoton, Frédéric Fauvet, Claudia Malvenuto and Jean-Yves Thibon, Editors

Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA), Volume 1

Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 1 Frédéric Chapoton Frédéric Fauvet Claudia Malvenuto Jean-Yves Thibon Editors

IRMA Lectures in Mathematics and Theoretical Physics 31 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. For a complete listing see our homepage at www.ems-ph.org. 1 0 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) 23 Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Lizhen Ji and Athanase Papadopoulos (Eds.) 24 Free Loop Spaces in Geometry and Topology, Janko Latschev and Alexandru Oancea (Eds.) 25 Takashi Shioya, Metric Measure Geometry. Gromov‘s Theory of Convergence and Concentration of Metrics and Measures 26 Handbook of Teichmüller Theory, Volume V, Athanase Papadopoulos (Ed.) 27 Handbook of Teichmüller Theory, Volume VI, Athanase Papadopoulos (Ed.) 28 Yann Bugeaud, Linear Forms in Logarithms and Applications 29 Eighteen Essays in Non-Euclidean Geometry, Vincent Alberge and Athanase Papadopoulos (Eds.) 30 Handbook of Teichmüller Theory, Volume VII, Athanase Papadopoulos (Ed.)

Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 1 Frédéric Chapoton Frédéric Fauvet Claudia Malvenuto Jean-Yves Thibon Editors

Editors: Frédéric Chapoton IRMA, UFR Mathématiques et Informatique Université de Strasbourg 7, rue René Descartes 67084 Strasbourg CEDEX, France

Frédéric Fauvet IRMA, UFR Mathématiques et Informatique Université de Strasbourg 7, rue René Descartes 67084 Strasbourg CEDEX, France

e-mail: [email protected]

e-mail: [email protected]

Claudia Malvenuto Dipartimento di Matematica “Guido Castelnuovo” Università di Roma La Sapienza Piazzale A. Moro 5 00185 Roma, Italy

Jean-Yves Thibon Laboratoire d’Informatique Gaspard Monge Université Paris-Est Marne-la-Vallée 5, boulevard Descartes, Champs sur Marne 77454 Marne-la-Vallée CEDEX 2, France

e-mail: [email protected]

e-mail: [email protected]

2010 Mathematics Subject Classification: 05E, 81T15, 81T18, 81Q30, 34C20, 37C10, 18D50, 34M40, 34M60, 11M32, 30D60 Key words: Operad, Hopf algebra, algebraic combinatorics, moulds, renormalization, periods, multiple zeta values, resurgent functions, alien calculus, vector fields, diffeomorphisms

ISBN 978-3-03719-204-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2020 European Mathematical Society

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Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Contents of Volume 1

Shuffle quadri-algebras and concatenation by Mohamed Belhaj Mohamed, Dominique Manchon : : : : : : : : : : : : : : : :

1

Structure theorems for dendriform and tridendriform algebras by Emily Burgunder, Bérénice Delcroix-Oger : : : : : : : : : : : : : : : : : : : : : :

29

A group-theoretical approach to conditionally free cumulants by Kurusch Ebrahimi-Fard, Frédéric Patras : : : : : : : : : : : : : : : : : : : : : :

67

The Natural Growth Scale by Jean Ecalle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

95

Realizations of Hopf algebras of graphs by alphabets by Loïc Foissy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

225

Duplicial algebras, parking functions, and Lagrange inversion by Jean-Christophe Novelli, Jean-Yves Thibon : : : : : : : : : : : : : : : : : : : : :

263

The triduplicial operad is Koszul by Anthony Mansuy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

291

The Hopf algebra of integer binary relations by Vincent Pilaud and Viviane Pons : : : : : : : : : : : : : : : : : : : : : : : : : : : :

299

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

Combinatoire Algébrique, Résurgence, Moules et Applications (Algebraic Combinatorics, Resurgence, Moulds and Applications)

Conférence de clôture du 26 au 30 juin 2017 Ines Aniceto (Krakow) David Broadhurst (Open U.) Francis Brown (Oxford) Emily Burgunder (Toulouse) Gerald Dunne (U. Connecticut) Kurusch Ebrahimi-Fard (Trondheim) Jean Écalle (Orsay) Michael Hoffman (Annapolis) Carlos Mafra (Southampton) Dominique Manchon (Clermont-Ferrand) Marcos Marino (Genève) Frédéric Menous (Orsay) Jean-Christophe Novelli (Marne-la-Vallée) Erik Panzer (Oxford) Frédéric Patras (Nice) Viviane Pons (Orsay) David Sauzin (Pisa) Karen Yeats (Vancouver)

Scientific & Organizing Committee Frédéric Chapoton (CNRS/Strasbourg) Frédéric Fauvet (Strasbourg) Claudia Malvenuto (Roma) Jean-Yves Thibon (Marne-la-Vallée)

Contents of Volume 2 Special values of finite multiple harmonic q-series at roots of unity by Henrik Bachmann, Yoshihiro Takeyama and Koji Tasaka : : : : : : : : : : : :

1

Mould calculus: from primary to secondary mould symmetries by Olivier Bouillot : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

19

Renormalisation and locality: branched zeta values by Pierre Clavier, Li Guo, Sylvie Paycha and Bin Zhang : : : : : : : : : : : : : :

85

The scrambling operators applied to multizeta algebra and singular perturbation analysis by Jean Ecalle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

135

Quasi-shuffle algebras and applications by Michael E. Hoffman : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

327

Planar binary trees in scattering amplitudes by Carlos R. Mafra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

349

A study on prefixes of c2 invariants by Karen Yeats : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

367

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

385

Introduction This is the first of two volumes that constitute a follow-up to the CARMA Conference (Algebraic Combinatorics, Resurgence, Moulds and Applications), held at the Centre International de Rencontres Mathématiques in Luminy, France, from June 26 to 30, 2017. The CARMA project of the French National Research Agency devoted to Algebraic Combinatorics, Resurgence, Moulds and Applications was started in 2012. The conference CARMA 2017 aimed at taking stock of the numerous recent achievements and publications on these topics, from experts both outside and inside this research group. The conference did notably emphasise the role of Hopf algebraic techniques and related objects such as Rota Baxter algebras, operads and Ecalle s mould calculus. They have lately proved pervasive in combinatorics as well as in many other fields, from multiple zeta values to the algebraic study of control systems and the theory of rough paths. Moreover, recent years have seen a burst of articles in various areas of theoretical physics that enhance the role of resurgent functions beyond their original applications in dynamical systems, which were also among the topics of the conference. Some of the papers in these volumes correspond to talks given during the conference, others are original articles from participants. The contributions to the first volume essentially center about mould calculus and combinatorial Hopf algebras. The second volume mainly deals with multizetas and physical applications, although there is no clearcut separation between both.

Shuffle quadri-algebras and concatenation M. Belhaj Mohamed and D. Manchon Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A modified tensor product . . . . . . . . . . . . . . . . . . . . 3 Dendriform algebras . . . . . . . . . . . . . . . . . . . . . . . . 4 Quadri-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 From quadri-algebras to dendriform algebras . . . . . . . 6 The shuffle quadri-algebra . . . . . . . . . . . . . . . . . . . . 7 Module-algebra structures on the shuffle quadri-algebra References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 3 5 5 7 7 22 27

1 Introduction A dendriform algebra is a vector space equipped with an associative product which can be written as a sum of two operations  and  called left and right respectively, which satisfy the three following rules: .x  y/  z .x  y/  z .x  y/  z C .x  y/  z

D x  .y  z/ C x  .y  z/ D x  .y  z/ D x  .y  z/

They were introduced by Jean-Louis Loday [11, §5] in 1995 with motivation from algebraic K-theory and have been studied by other authors in different domains [1, 5, 6, 7, 12, 13, 17]. In 2004, Marcelo Aguiar and Jean-Louis Loday introduced the notion of quadrialgebra in [2]. A quadri-algebra is an associative algebra the multiplication of which can be decomposed as the sum of four operations &, %, - and . satisfying nine axioms. Two dendriform structures are attached to a quadri-algebra: the first dendriform structure is given by the two operations  and  such that: xy xy

WD x % y C x & y WD x - y C x . y;

and the second is given by the two operations _ and ^ where: x_y x^y

WD x & y C x . y WD x % y C x - y:

2

M. Belhaj Mohamed and D. Manchon

Quadri-algebras were studied by Loïc Foissy together with quadri-coalgebras and quadri-bialgebras [9]. In this article we revisit the canonical example of shuffle quadri-algebra [4, 10, 13, 16] treated by Marcelo Aguiar, Jean-Louis Loday and Loïc Foissy. We prove that there exist relations between the quadri-algebra laws, the concatenation product and the deconcatenation coproduct. We show that, for any elements u; v; w in this quadri-algebra H we have: X u % .vw/ D .u1 & v/.u2 ^ w/ uDu1 u2

D

X

.u1  v/.u2 % w/;

uDu1 u2

u & .vw/ D

X

.u1 & v/.u2 _ w/

uDu1 u2

D

X

.u1  v/.u2 & w/;

uDu1 u2

u . .vw/ D

X

.u1 . v/.u2 _ w/

uDu1 u2

D

X

.u1  v/.u2 & w/;

uDu1 u2

u - .vw/ D

X

.u1 . v/.u2 ^ w/

uDu1 u2

D

X

.u1  v/.u2 % w/

uDu1 u2

whenever these expressions make sense. We derive from these results a set of relations between the dendriform laws, the concatenation and the deconcatenation coproduct. We show that, for any u; v; w 2 H, we have: u ^ .vw/ D u % .vw/ C u - .vw/ X D .u1 _ v/.u2 ^ w/; uDu1 u2

u  .vw/ D u - .vw/ C u . .vw/ X D .u1  v/.u2  w/; uDu1 u2

Shuffle quadri-algebras and concatenation

3

u _ .vw/ D u . .vw/ C u & .vw/ X .u1 _ v/.u2 _ w/; D uDu1 u2

u  .vw/ D u & .vw/ C u % .vw/ X D .u1  v/.u2  w/; uDu1 u2

and consequently, two relations between the shuffle product, the concatenation and the deconcatenation coproduct. We show that, for any u; v; w 2 H, we have: X u qq .vw/ D .u1 _ v/.u2 qq w/ uDu1 u2

X

D

.u1 qq v/.u2  w/:

uDu1 u2

At the end of this article, we prove the existence of two module-algebra structures on H given by _ and , and further compatibility relations. All these results are best expressed in terms of commutative diagrams involving an extended version of the tensor product. Acknowledgements. We thank Loïc Foissy for his useful remarks, and the referee for a careful reading and numerous suggestions which greatly helped us to improve the presentation.

2 A modified tensor product Let V and W be two vector spaces over a field k, let "V W V ˚k ! k be the projection onto the second component, and let us consider "W similarly. We set: V ˝W WD Ker."V ˝ "W / ' .V ˝ W / ˚ V ˚ W:

(2.1)

This modified tensor product is symmetric and associative. We denote by v C 1 the generic element of V ˚ k. The generic element of V ˝W can be written as n X

vk ˝ wk C v ˝ 1 C 1 ˝ w;

kD1

where v; v1 ; : : : ; vn 2 V and w; w1 ; : : : ; wn 2 W. For any pair of linear maps f W V ! V 0 and g W W ! W 0 , there is a unique linear map f ˝g W V ˝W ! V 0 ˝W 0 such that for any v 2 V and w 2 W we get f ˝g.v ˝ w/ D f .v/ ˝ g.w/; f ˝g.1 ˝ w/ D 1 ˝ g.w/:

f ˝g.v ˝ 1/ D f .v/ ˝ 1;

4

M. Belhaj Mohamed and D. Manchon

These data turn the collection of k-vector spaces into a symmetric monoidal category. The unit for this tensor product ˝ is the zero-dimensional space f0g. Recall that any associative algebra V gives rise to a unital associative algebra V D V ˚k1. As a consequence, the product m W V ˝V ! V is extended to a product from V ˝ V into V , and its restriction to V ˝V takes value in V . Associativity of this extension can be described by the commutativity of the following diagram: m˝I

V ˝V ˝V I ˝m

/ V ˝V



m



V ˝V

/V

m

In the same line of thought, a left module structure ˆ W V ˝ M ! M yields an extension ˆ W V ˝M ! M via ˆ.1 ˝ m/ D m;

ˆ.v ˝ 1/ D 0;

making the following diagram commute: V ˝V ˝M I ˝ˆ

m˝I

/ V ˝M





V ˝M

ˆ

/M

ˆ

and similarly for right modules. Dually, a coassociative coproduct wt W V ! V ˝V can be modified to a co-unital coproduct  W V ! V ˝ V . Its restriction  W V ! V ˝V makes the following co-associativity diagram commute: V 



˝I



V ˝V

/ V ˝V

I ˝

 / V ˝V ˝V

Commutative diagrams for extended left and right comodule structures involving ˝ can be drawn accordingly.

5

Shuffle quadri-algebras and concatenation

3 Dendriform algebras A dendriform algebra is a vector space D together with two operations W D˝D ! D and W D ˝ D ! D, called left and right respectively, such that: .x  y/  z .x  y/  z .x  y/  z C .x  y/  z

D x  .y  z/ C x  .y  z/ D x  .y  z/ D x  .y  z/:

(3.1)

Dendriform algebras were introduced in [11, §5]. See also [1, 5, 6, 7, 12, 13, 17] for additional work on this subject. Defining a new operation ? by: x ? y WD x  y C x  y

(3.2)

permits us to rewrite axioms (3.1) as: .x  y/  z .x  y/  z .x ? y/  z

D x  .y ? z/ D x  .y  z/ D x  .y  z/:

(3.3)

By adding the three relations we see that the operation ? is associative. For this reason, a dendriform algebra may be regarded as an associative algebra .D; ?/ for which the multiplication ? can be decomposed as the sum of two coherent operations. It is standard to extend the dendriform operations  and  to D˝D by setting: a  1 D 0;

1  a D a;

1  a D 0;

a1Da

(3.4)

for any a 2 D. The space D WD D ˚ k1 is called somewhat incorrectly a unital dendriform algebra, although 1  1 and 1  1 are not defined. Let us however remark that ? D C  can be extended to D ˝ D , making D a unital associative algebra. The extension (3.4) is consistent with the dendriform axioms (3.3) in the sense that each of the three axioms makes sense provided both members of the equality are defined.

4 Quadri-algebras In this section, we use definitions and results on quadri-algebra structures given by Marcelo Aguiar and Jean-Louis Loday in [2] and Loïc Foissy in [9]. A quadri-algebra structure consists in splitting an associative product into four operations, which in turn gives rise to two distinct dendriform structures. Definition 4.1. A quadri-algebra is a vector space Q together with four operations: &; %; - and .W Q ˝ Q ! Q;

6

M. Belhaj Mohamed and D. Manchon

satisfying the nine axioms below. In order to state them, consider the following operations: xy xy x_y x^y

WD WD WD WD

x x x x

%y Cx -y Cx &y Cx %y Cx

&y .y .y -y

(4.1) (4.2) (4.3) (4.4)

and: x?y

WD x % y C x & y C x - y C x . y WD x  y C x  y WD x _ y C x ^ y:

(4.5)

The nine axioms, stated by Marcelo Aguiar and Jean-Louis Loday in [2] are: .x - y/ - z D x - .y ? z/; .x ^ y/ % z D x % .y  z/;

.x % y/ - z D x % .y  z/;

.x . y/ - z D x . .y ^ z/; .x _ y/ % z D x & .y % z/;

.x & y/ - z D x & .y - z/;

.x  y/ . z D x . .y _ z/; .x ? y/ & z D x & .y & z/:

.x  y/ . z D x & .y . z/;

The operations &; %; -; . are referred to as southeast, northeast, northwest, and southwest, respectively. Accordingly, ^; _;  and  are called north, south, west and east respectively. The axioms are displayed in the form of a 3  3 matrix. As in [2], we will make use of standard matrix terminology (entries, rows and columns) to refer to them. Let Q be a quadri-algebra. Following [9, Paragraph 3.1], we extend the four products to Q˝Q in the following way: if a 2 Q, a - 1 D a; a . 1 D 0;

a % 1 D 0; a & 1 D 0;

1 - a D 0; 1 . a D 0;

1 % a D 0; 1 & a D a:

It follows that we have for any a 2 Q: a ^ 1 D a; a  1 D 0;

1 ^ a D 0; 1  a D a;

1 _ a D a; 1  a D 0;

a _ 1 D 0; a  1 D a:

Shuffle quadri-algebras and concatenation

7

5 From quadri-algebras to dendriform algebras The three column sums in the matrix of quadri-algebra axioms yield: .x  y/  z D x  .y ? z/;

.x  y/  z D x  .y  z/

and .x ? y/  z D x  .y  z/: Thus, endowed with the operations west for left and east for right, Q is a dendriform algebra. We denote it by Qh and call it the horizontal dendriform algebra associated to Q. Considering instead the three row sums in the matrix of quadri algebra axioms yields: .x ^ y/ ^ z D x ^ .y ? z/;

.x _ y/ ^ z D x _ .y ^ z/

and .x ? y/ _ z D x _ .y _ z/: Thus, endowed with the operations north for left and south for right, Q is a dendriform algebra. We denote it by Qv and call it the vertical dendriform algebra associated to Q. The associative operations corresponding to the dendriform algebras Qh and Qv by means of (3.2) coincide, according to (4.5). In other words, ? D% C & C . C -D C D ^ C _:

(5.1)

6 The shuffle quadri-algebra L Let k be a field, and let V be a k-vector space. Let H D T .V / D n0 V ˝n be the tensor algebra of V , where we denote by  the deconcatenation coproduct and by m L ˝n the concatenation product. Let HC D T C .V / D be the augmentation n1 V ideal. For all u; v 2 H, we have: m.u ˝ v/ D uv;

(6.1)

and .u/ D

X

u1 ˝ u2 :

(6.2)

uDu1 u2

Here u1 (resp. u2 ) is the left (resp. right) part of the word u defined by the place where u is cut. This notation matches with Sweedler’s notation for a coproduct in a

8

M. Belhaj Mohamed and D. Manchon

coalgebra in general. The shuffle product qq is defined for any u D u1 u2 : : : up and v D upC1 upC2 : : : upCq with u1 ; : : : ; upCQ 2 V by: X u 1 .1/ u 1 .2/ : : : : : : u 1 .pCq/ ; (6.3) u qq v D 2Sh.p;q/

where Sh.p; q/ denotes the set of  2 SpCq verifying  .1/ <    <  .p/ and  .p C 1/ <    <  .p C q/. The triple .H; qq ; / becomes a commutative Hopf algebra called the shuffle Hopf algebra. The shuffle algebra of a vector space V provides an example of a commutative quadri-algebra (see Remark 6.1 below). Let us recall the two recursive formulas for the shuffle product: au qq bv D a.u qq bv/ C b.au qq v/; ua qq vb D .u qq vb/a C .ua qq v/b for any a; b 2 V and u; v 2 T .V /. The quadri-algebra laws on HC are defined by Marcelo Aguiar and Jean-Louis Loday in [2] recursively on the degrees of u and v, and can be extended to HC ˝HC as explained above. We recall here the construction. 1. If u D 1 and v 2 HC , we have: 1 qq v D v and: 1 % v D 0; v . 1 D 0;

1 & v D v; v - 1 D v;

1 . v D 0; v % 1 D 0;

1 - v D 0; v & 1 D 0;

which immediately gives: 1  v D v  1 D v; 1 ^ v D v _ 1 D 0;

1  v D v  1 D 0; 1 _ v D v ^ 1 D v:

2. If u; v 2 V , we have: u qq v D uv C vu and: u % v D vu; u . v D uv;

u & v D 0; u - v D 0;

which immediately gives: u  v D vu; u ^ v D vu;

u  v D uv; u _ v D uv:

Shuffle quadri-algebras and concatenation

9

3. If u 2 V , and v D cd with c; d 2 V and  2 V ˝.n2/ , we have: u qq v D u qq cd D ucd C c.u qq  /d C cdu C 0: The four quadri-algebra laws on H are given by: u % v D cdu; u . v D ucd;

u & v D c.u qq  /d; u - v D 0;

(6.4)

which immediately gives: u  v D ucd; u ^ v D cdu;

u  v D c.u qq  /d C cdu; u _ v D c.u qq  /d C ucd:

4. If u; v 2 H, such that u; v are pure tensors of of degree  2, i.e, u D awb and v D cd with a; b; c; d 2 V , we have: u qq v D a.wb qq c /d C c.awb qq  /d C a.w qq cd /b C c.aw qq d /b: The four quadri-algebra operations on H are defined by: u % v D c.aw qq d /b; u . v D a.wb qq c /d;

u & v D c.awb qq  /d; u - v D a.w qq cd /b:

The dendriform algebra operations on H are defined by: u  v D c.awb qq d /; u ^ v D .aw qq cd /b;

u  v D a.wb qq cd /; u _ v D .awb qq c /d:

We verify easily then: u qq v D u % v C u & v C u - v C u . v D u vCuv D u _ v C u ^ v:

(6.5)

The nine axioms of quadri-algebra laws can now be easily verified. Remark 6.1. By commutativity of the shuffle product, the quadri-algebra laws verify for any u; v of length  2: u%v D D D u&v D D D

c.aw qq d /b c.d qq aw/b v . u: c.awb qq  /d c. qq awb/d v - u:

10

M. Belhaj Mohamed and D. Manchon

The verification of these two commutativity statements for any u; v such that u ˝ v 2 HC ˝HC is straightforward and left to the reader. The shuffle quadri-algebra HC is said to be commutative. Remark 6.2. [9] The four quadri-algebra operations also admit a non-recursive definition in terms of shuffles: supposing that u (resp. v) is a word of length p (resp. q), X u&vD u 1 .1/ u 1 .2/ : : : : : : u 1 .pCq/ ; 2Sh.p;q/;  1 .1/DpC1 and  1 .pCq/DpCq

X

u%vD

u 1 .1/ u 1 .2/ : : : : : : u 1 .pCq/ ;

2Sh.p;q/;  1 .1/DpC1

and

2Sh.p;q/;  1 .1/D1

and

 1 .pCq/Dp

and

 1 .pCq/DpCq

X

u-vD

u 1 .1/ u 1 .2/ : : : : : : u 1 .pCq/ ;

X

u.vD

2Sh.p;q/;  1 .1/D1

 1 .pCq/Dp

u 1 .1/ u 1 .2/ : : : : : : u 1 .pCq/ :

Informally,  u & v is the sum of words obtained by shuffling u and v in such a manner that the first letter is the first letter of v and the last letter is the last letter of v,  u % v is the sum of words obtained by shuffling u and v in such a manner that the first letter is the first letter of v and the last letter is the last letter of u,  u - v is the sum of words obtained by shuffling u and v in such a manner that the first letter is the first letter of u and the last letter is the last letter of u,  u . v is the sum of words obtained by shuffling u and v in such a manner that the first letter is the first letter of u and the last letter is the last letter of v. Similar expressions for ; ; ^; _ and

qq

are straightforward and left to the reader.

We can now state the main result of this article. Theorem 6.3. Let m be the concatenation product of words. The eight following diagrams commute. H ˝ HC ˝ HC 23 ı˝I ˝I

I ˝m

/ H ˝ HC

I ˝m

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC &˝^

H ˝ HC ˝ HC

%



HC ˝ HC



m

/ HC

 H ˝ HC ˝ H ˝ HC

%

˝%

 HC ˝ HC

m

 / HC

11

Shuffle quadri-algebras and concatenation

H ˝ HC ˝ HC 23 ı˝I ˝I

I ˝m

/ H ˝ HC

I ˝m

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC &˝_

H ˝ HC ˝ HC 23 ı˝I ˝I

 H ˝ HC ˝ H ˝ HC

&



m

I ˝m

&

˝&



HC ˝ HC

/ HC

 HC ˝ HC

/ H ˝ HC

H ˝ HC ˝ HC

m

I ˝m

 / HC

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC .˝_

H ˝ HC ˝ HC 23 ı˝I ˝I

 H ˝ HC ˝ H ˝ HC

.



m

I ˝m

.

˝&



HC ˝ HC

/ HC

 HC ˝ HC

/ H ˝ HC

H ˝ HC ˝ HC

m

I ˝m

 / HC

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC .˝^

H ˝ HC ˝ HC

-



HC ˝ HC

 H ˝ HC ˝ H ˝ HC ˝%



m

-

 HC ˝ HC

/ HC

m

In other words, for any u 2 H and v; w 2 HC we have: (1) u % .vw/ D

X

.u1 & v/.u2 ^ w/

uDu1 u2

D

X

.u1  v/.u2 % w/;

uDu1 u2

(2) u & .vw/ D

X

.u1 & v/.u2 _ w/

uDu1 u2

D

X

uDu1 u2

.u1  v/.u2 & w/;

 / HC

12

M. Belhaj Mohamed and D. Manchon

(3) u . .vw/ D

X

.u1 . v/.u2 _ w/

uDu1 u2

D

X

.u1  v/.u2 & w/;

uDu1 u2

(4) u - .vw/ D

X

.u1 . v/.u2 ^ w/

uDu1 u2

D

X

.u1  v/.u2 % w/:

uDu1 u2

Proof. We will prove this theorem by induction on the length of u. Let us verify that the theorem is true for u D 1 and for u 2 V . For u D 1 and for v; w 2 HC we have: u % .vw/ D 1 % .vw/ D 0; and: X uDu1 u2

.u1 & v/.u2 ^ w/ D .1 & v/ .1 ^ w/ „ ƒ‚ … 0

D 0 D 1 % .vw/; X uDu1 u2

.u1  v/.u2 % w/ D .1  v/ .1 % w/ „ ƒ‚ … 0

D 0 D 1 % .vw/: Similarly: u & .vw/ D 1 & vw X uDu1 u2

D vw:

.u1 & v/.u2 _ w/ D .1 & v/.1 _ w/ D vw D 1 & .vw/:

13

Shuffle quadri-algebras and concatenation

X

.u1  v/.u2 & w/ D .1  v/.1 & w/

uDu1 u2

D vw D 1 & .vw/;

and by a similar computation, we prove that the two other assertions are true for u D 1. For u 2 V and for any v; w 2 HC , we have: u % .vw/ D vwu; and:

X uDu1 u2

.u1 & v/.u2 ^ w/ D .1 & v/.u ^ w/ C .u & v/ .1 ^ w/ „ ƒ‚ … 0

D vwu D u % .vw/; X uDu1 u2

.u1  v/.u2 % w/ D .1  v/.u % w/ C .u  v/ .1 % w/ „ ƒ‚ … 0

D vwu D u % .vw/: Similarly: u & .vw/ D vuw: X uDu1 u2

.u1 & v/.u2 _ w/ D .1 & v/.u _ w/ C .u & v/.1 _ w/ „ ƒ‚ … 0

D vuw D u & .vw/: X uDu1 u2

.u1  v/.u2 & w/ D .1  v/ .u & w/ C.u  v/.1 & w/ „ ƒ‚ … D vuw D u & .vw/:

0

By a similar computation, we prove that the two other assertions are true for u 2 V . We will now use the induction hypothesis to prove the theorem. Let u D ab, v and w be three elements of HC .

14

M. Belhaj Mohamed and D. Manchon

Proof of (1): X X .u1 & v/.u2 ^ w/ D .u & v/.1 ^ w/ C .u1 & v/.u2 ^ w/: ƒ‚ … „ uDu1 u2

uDu1 u2 u2 ¤1

0

The condition u2 ¤ 1 gives: u2 D u12 b where u1 u12 D a , hence: X X .u1 & v/.u2 ^ w/ D .u1 & v/.u12 b ^ w/: a Du1 u12

uDu1 u2

We distinguish here two cases, the first case where v is a single-letter word and the second case where v is a word of length  2, i.e. v D cd , where c; d 2 V and  2 V ˝n . 1 1 1 If v 2 V , by Remark 6.1 we have P u & v 1D v - u12 D 0 for all u ¤ 1 (see equation (6.4)). Hence the sum a Du1 u12 .u & v/.u b ^ w/ gives one term where u1 D 1, the other terms all vanish, and thus:

X

.u1 & v/.u2 ^ w/ D

X

.u1 & v/.u12 b ^ w/

a Du1 u12

uDu1 u2

D .1 & v/.u ^ w/ D v.u ^ w/ D u % .vw/: The last equality is valid because v is a single letter here. Now if v D cd we obtain the same result: X .u1 & v/.u2 ^ w/ uDu1 u2

D

X

.u1 & v/.u12 b ^ w/

a Du1 u12

D

X

.u1  c/d.u12 qq w/b

a Du1 u12

D

X

a Du1 u12

D

X

.u1  c/.u12  dw/b 

.u1  c/.u12 % dw/b C .u1  c/.u12 & dw/b

a Du1 u12

D .a % vw/b C .a & vw/b D .a  vw/b D .ab/ % .cdw/ D u % .vw/:

(induction hypothesis)



15

Shuffle quadri-algebras and concatenation

Similary, we have: X .u1  v/.u2 % w/ D .u  v/.1 % w/ C ƒ‚ … „ uDu1 u2

0

X

.u1  v/.u2 % w/:

uDu1 u2 u1 ¤ u;u2 ¤1

The condition u1 ¤ u and u2 ¤ 1 gives: u2 D u12 b where u1 u12 D a , hence: X .u1  v/.u2 % w/ uDu1 u2

X

D

.u1  v/.u12 b % w/

a Du1 u12

X

D

.u1  v/.u12  w/b

a Du1 u12

X

D

.u1  v/.u12 % w/b C

a Du1 u12

X

.u1  v/.u12 & w/b

a Du1 u12

D .a . vw/b C .a - vw/b D .a  vw/b D c.a qq dw/b D .ab/ % .cdw/ D u % .vw/:

(induction hypothesis)

Proof of (2): By a similar method we prove the second assertion. We distinguish here two cases, the first case where v is a single-letter word and the second case where v is a word of length  2, i.e v D cd , where c; d 2 V and  2 V ˝n . 1 1 1 If v 2 V , by Remark 6.1 we have P u & v1 D v - 2u D 0 for all u ¤ 1 (see equation (6.4)). Hence the sum uDu1 u2 .u & v/.u _ w/ gives one term where u1 D 1 and u2 D u, the other terms all vanish, we have:

X

.u1 & v/.u2 _ w/ D .1 & v/.u _ w/

uDu1 u2

D v.u _ w/ D u & .vw/; and if v D cd , we have: X .u1 & v/.u2 _ w/ uDu1 u2

D

X uDu1 u2

.u1 & cd /.u2 _ w/

16

M. Belhaj Mohamed and D. Manchon

D

X

X

.u11 & c/.u12 _ d /.u2 _ w/ (induction hypothesis)

uDu1 u2 u1 Du11 u12

D

X

.u11 & c/.u12 _ d /.u2 _ w/

uDu11 u12 u2

D

X

.u11 & c/

uDu11 u0

D

X

X

.u12 _ d /.u2 _ w/

u0 Du11 u12

.u

11

& c/.u0 _ dw/ (induction hypothesis)

uDu11 u0

D

X

.c - u11 /.u0 _ dw/:

uDu11 u0

The last sum contains one term because c - u11 D 0 if u11 ¤ 1, then we have: X .u1 & v/.u2 _ w/ D c.u _ dw/ uDu1 u2

D u & .cdw/ D u & .vw/: Similarly, we distinguish here two cases, the first case where w is a single-letter word and the second case where w is a word of length  2, i.e w D ef , where e; f 2 V and  2 V ˝n . P If w 2 V , the sum uDu1 u2 .u1  v/.u2 & w/ gives one term where u1 D u and u2 D 1, the other terms vanish, which gives: X .u1  v/.u2 & w/ D .u  v/.1 & w/ uDu1 u2

D .u  v/w D u & .vw/; and if w D ef , we have: X .u1  v/.u2 & w/ uDu1 u2

D

X

.u1  v/.u2 & ef /

uDu1 u2

D

X

X

.u1  v/.u21  e/.u22 & f / (induction hypothesis)

uDu1 u2 u2 Du21 u22

D

X

uDu1 u21 u22

.u1  v/.u21  e/.u22 & f /

17

Shuffle quadri-algebras and concatenation

D

X

X

.u1  v/.u21  e/.u22 & f /

uDu0 u22 u0 Du1 u21

D

X

.u0  ve/.u22 & f / (induction hypothesis)

uDu0 u22

D

X

.u0  ve/.f - u22 /:

uDu0 u22

The last sum contains one term because f - u22 D 0 if u22 ¤ 1, then we obtain: X .u1  v/.u2 & w/ D .u  ve/f uDu1 u2

D D D D

.u  cde/f c.u qq de/f u & .cdef / u & .vw/:

Proof of (3): X .u1 . v/.u2 _ w/ D .1 . v/.u _ w/ C ƒ‚ … „ uDu1 u2

X

.u1 . v/.u2 _ w/:

uDu1 u2 u1 ¤1;u2 ¤u

0

The condition u1 ¤ 1 gives: u1 D au11 where u11 u2 D b, hence: X .u1 . v/.u2 _ w/ uDu1 u2

D

X

.au11 . v/.u2 _ w/

uDau11 u2

D

X

a.u11 _ v/.u2 _ w/

uDau11 u2

D

X

a.u11 . v/.u2 _ w/ C

uDau11 u2

X uDau11 u2

D a.b . vw/ C a.b & vw/ D a.b _ vw/ D .ab/ . .vw/ D u . .vw/:

(induction hypothesis)

Similarly, we have: X .u1  v/.u2 & w/ D .1  v/.u & w/ C ƒ‚ … „ uDu1 u2

a.u11 & v/.u2 _ w/

0

X uDu1 u2 u1 ¤1;u2 ¤u

.u1  v/.u2 & w/:

18

M. Belhaj Mohamed and D. Manchon

The condition u1 ¤ 1 gives: u1 D au11 where u11 u2 D b, hence: X X .u1  v/.u2 & w/ D .au11  v/.u2 & w/: uDu1 u2

uDau11 u2

We distinguish here two cases, the first case where w is a single-letter word and the second case where w is a word of length  2, i.e w D ef , where e; f 2 V and  2 V ˝n . P If w 2 V , the sum uDau11 u2 .au11  v/.u2 & w/ gives one term where u2 D 1, the other terms all vanish, we have: X X .u1  v/.u2 & w/ D .au11  v/.u2 & w/ uDu1 u2

uDau11 u2

D .u  v/.1 & w/ D .u  v/w D u . .vw/: Now if w D ef , we have: X .u1  v/.u2 & w/ uDu1 u2

D

X

.au11  v/.u2 & w/

uDau11 u2

D

X

.au11  v/.u2 & ef /

uDau11 u2

D

X

a.u11 qq v/e.u2 qq /f

uDau11 u2

D

X

a.u11 _ ve/.u2 _ f /

uDau11 u2

D

X

a.u11 & ve/.u2 _ f / C

uDau11 u2

D a.b & vef / C a.b . vef / D a.b _ vw/ D u . .vw/; which proves the third assertion.

X

a.u11 . ve/.u2 _ f /

uDau11 u2

(induction hypothesis)

Shuffle quadri-algebras and concatenation

Proof of (4): X uDu1 u2

19

.u1 . v/.u2 ^ w/ D .1 . v/.u ^ w/ C .u . v/.1 ^ w/ ƒ‚ … „ C

0

X

.u . v/.u2 ^ w/: 1

uDu1 u2 u1 ;u2 ¤1;u

The condition u1 ; u2 ¤ 1; u gives: u1 D au11 and u2 D u12 b where u11 u12 D  , hence: X .u1 . v/.u2 ^ w/ uDu1 u2

X

D

.au11 . v/.u12 b qq w/b

 Du11 u12

X

D

.au11 . v/.u12 _ w/b C

 Du11 u12

X

.au11 . v/.u12 ^ w/b

 Du11 u12

D .a . vw/b C .a - vw/b D .a  vw/b D a. qq vw/b D .ab/ - .vw/ D u - .vw/:

(induction hypothesis)

Similarly we have: X .u1  v/.u2 % w/ D .1  v/.u % w/ C .u  v/.1 % w/ ƒ‚ … „ uDu1 u2

C

X

0

.u  v/.u2 % w/: 1

uDu1 u2 u1 ;u2 ¤1;u

The condition u1 ; u2 ¤ 1; u gives: u1 D au11 and u2 D u12 b where u11 u12 D  , hence: X .u1  v/.u2 % w/ uDu1 u2

D

X

.au11  v/.u12 b % w/

 Du11 u12

D

X

.au11  v/.u12  w/b

 Du11 u12

D

X

 Du11 u12

.au11  v/.u12 % w/b C

X  Du11 u12

.au11  v/.u12 & w/b

20

M. Belhaj Mohamed and D. Manchon

D .a . vw/b C .a - vw/b D .a  vw/b D a. qq vw/b D .ab/ - .vw/ D u - .vw/;

(induction hypothesis)



which proves the fourth assertion.

Remark 6.4. A non-recursive proof of Theorem 6.3 is available. Indeed, to prove the first assertion of (2) we note that u & .vw/ is obtained by summing all terms in the shuffle of u with vw so that the first letter belongs to v and the last letter belongs to w. We cut each of these terms just after the last letter of v. The left part is obtained by shuffling a prefix of u with v such that the first and last letters are in v. The right part is obtained by shuffling a suffix of u with w such that the last letter is in w. We proceed similarly for the second assertion, cutting just before the first letter of w. Items (1), (3) and (4) can be handled similarly. Corollary 6.5. The four following diagrams commute: H ˝ HC ˝ HC 23 ı˝I ˝I

I ˝m

/ H ˝ HC

I ˝m

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC _˝^

H ˝ HC ˝ HC 23 ı˝I ˝I

 H ˝ HC ˝ H ˝ HC

^



m

I ˝m



˝



HC ˝ HC

/ HC

 HC ˝ HC

/ H ˝ HC

H ˝ HC ˝ HC

m

I ˝m

 / HC

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC _˝_

H ˝ HC ˝ HC

_



HC ˝ HC

 H ˝ HC ˝ H ˝ HC ˝



m



 HC ˝ HC

/ HC

m

In other words, given u 2 H and v; w 2 HC we have: 1. u ^ .vw/ D u % .vw/ C u - .vw/ X D .u1 _ v/.u2 ^ w/; uDu1 u2

 / HC

21

Shuffle quadri-algebras and concatenation

2. u  .vw/ D u - .vw/ C u . .vw/ X D .u1  v/.u2  w/; uDu1 u2

3. u _ .vw/ D u . .vw/ C u & .vw/ X .u1 _ v/.u2 _ w/; D uDu1 u2

4. u  .vw/ D u & .vw/ C u % .vw/ X .u1  v/.u2  w/: D uDu1 u2

Proof. The diagram in position .1; 1/ of the 2  2-matrix of diagrams above is obtained by adding both diagrams .4; 1/ and .1; 1/ in the 4  2-matrix of diagrams of Theorem 6.3. Similarly, diagram .1; 2/ is obtained by adding both diagrams .3; 2/ and .4; 2/ of Theorem 6.3, diagram .2; 1/ is obtained by adding .2; 1/ and .3; 1/ of Theorem 6.3, and finally .2; 2/ is obtained as the sum of .1; 2/ and .2; 2/ thereof.  Corollary 6.6. Both following diagrams commute: H ˝ HC ˝ HC 23 ı˝I ˝I

I ˝m

/ H ˝ HC

I ˝m

/ H ˝ HC

23 ı˝I ˝I



H ˝ HC ˝ H ˝ HC _˝ qq

H ˝ HC ˝ HC

qq



HC ˝ HC

qq

qq ˝



m

 H ˝ HC ˝ H ˝ HC  HC ˝ HC

/ HC

In other words, for any u 2 H and v; w 2 HC , we have: X .u1 _ v/.u2 qq w/ u qq .vw/ D

m

 / HC

(6.6)

uDu1 u2

D

X

.u1 qq v/.u2  w/:

(6.7)

uDu1 u2

Proof. The first diagram is obtained by adding diagrams .1:1/ and .2:1/ of Corollary 6.5, the second is obtained by adding .1:2/ and .2:2/ thereof. 

22

M. Belhaj Mohamed and D. Manchon

Example 6.7. An example of computation for u D u1 u2 2 V ˝2 , v D v1 v2 2 V ˝2 and w 2 V . u qq .vw/ D .u1 u2 / qq .v1 v2 w/ D u1 u2 v1 v2 w C u1 v1 u2 v2 w C u1 v1 v2 u2 w C u1 u2 v1 v2 w C u1 v1 v2 wu2 C v1 v2 u1 u2 w C v1 u1 v2 u2 w C v1 u1 v2 wu2 C v1 v2 u1 wu2 C v1 v2 wu1 u2 : Also we have: X .u1 _ v/.u2 qq w/ uDu1 u2

D .1 _ v/.u qq w/ C .u1 _ v/.u2 qq w/ C .u1 u2 _ v/.1 qq w/ D v.u1 u2 w C u1 wu2 C wu1 u2 / C .u1 v1 v2 C v1 u1 v2 /.u2 w C wu2 / C .u1 u2 v1 v2 C u1 v1 u2 v2 C v1 u1 u2 v2 /w D v1 v2 u1 u2 w C v1 v2 u1 wu2 C v1 v2 wu1 u2 C u1 v1 v2 u2 w C v1 u1 v2 u2 w C u1 v1 v2 wu2 C v1 u1 v2 wu2 C u1 u2 v1 v2 w C u1 v1 u2 v2 w C v1 u1 u2 v2 w; and, X

.u1 qq v/.u2  w/

uDu1 u2

D .1 qq v/.u  w/ C .u1 qq v/.u2  w/ C .u1 u2 qq v/.1  w/ D v1 v2 wu1 u2 C .u1 v1 v2 C v1 u1 v2 C v1 v2 u1 /wu2 C .u1 u2 v1 v2 C u1 v1 u2 v2 C v1 u1 u2 v2 C u1 v1 v2 u2 C v1 u1 v2 u2 C v1 v2 u2 u2 /w D v1 v2 wu1 u2 C u1 v1 v2 wu2 C v1 u1 v2 wu2 C v1 v2 u1 wu2 C u1 u2 v1 v2 w C u1 v1 u2 v2 w C v1 u1 u2 v2 w C u1 v1 v2 u2 w C v1 u1 v2 u2 w C v1 v2 u2 u2 w: Then we have: u qq .vw/ D

X

.u1 _ v/.u2 qq w/

uDu1 u2

D

X

.u1 qq v/.u2  w/:

uDu1 u2

7 Module-algebra structures on the shuffle quadri-algebra We consider the bialgebra .H; qq ; / and the non-unitary infinitesimal bialgebra .HC ; m; /. The infinitesimal bialgebra compatibility relation is written as: .uv/ D .u/.1 ˝ v/  u ˝ v C .u ˝ 1/.v/:

(7.1)

23

Shuffle quadri-algebras and concatenation

Here we consider the restriction to HC of the full deconcatenation coproduct  W HC ! HC ˝HC : Proposition 7.1. Both maps _ and  are left actions of .H; qq / on HC , and both maps ^ and  are right actions of .H; qq / on HC . In other words, the four following diagrams are commutative: HC ˝HC ˝HC I ˝_

/ HC ˝HC



I ˝

HC ˝HC ˝HC



I ˝ qq

/ HC



/ HC ˝HC ^



/ HC ˝HC 

I ˝ qq



HC ˝HC



HC ˝HC

^˝I

/ HC

_

HC ˝HC ˝HC

HC ˝HC ˝HC

_



HC ˝HC

qq ˝I

qq ˝I

/ HC

^ ˝I

/ HC ˝HC





HC ˝HC





/ HC

That is to say: _ ı .I ˝_/ D _ ı . qq ˝I /;

^ ı .^˝I / D ^ ı .I ˝ qq /;

(7.2)

 ı .I ˝ / D  ı. qq ˝I /;

 ı . ˝I / D ı.I ˝ qq /:

(7.3) 

Proof. This is immediate from the dendriform axioms.

Theorem 7.2. The dendriform products _ and  define two .H; m/-module-algebra structures on HC , i.e. the two following diagrams are commutative: H ˝ HC ˝ HC I˝ m



H˝H _



C

H o

I˝ m

H

23



_˝ _

H ˝ H ˝ H ˝ HC m

H˝H C



C

 

/ H ˝ H ˝ HC ˝ HC C

C

H ˝ HC ˝ HC



˝I ˝I

o

˝I ˝I

C

H ˝ HC / H ˝ H ˝ HC ˝ HC 

23



˝ 

C

H ˝ H ˝ H ˝ HC C

m

H ˝ HC

24

M. Belhaj Mohamed and D. Manchon

That is to say: m ı ._ ˝ _/ ı 23 ı . ˝ I ˝ I / D _ ı .I ˝ m/; m ı . ˝ / ı 23 ı . ˝ I ˝ I / D  ı.I ˝ m/:

(7.4) (7.5)

Proof. For more details on module-algebras, see e.g. [15, Definition 4.1.1]. To prove the commutativity of these diagrams, we will use the results of Corollary 6.5. For u 2 H and v; w 2 HC we have: m ı ._ ˝ _/ ı 23 ı . ˝ I ˝ I /.u ˝ v ˝ w/ 1 0 X u1 ˝ u2 ˝ v ˝ w A D m ı ._ ˝ _/ ı 23 @ 0 D m ı ._ ˝ _/ @ D

uDu1 u2

X

1

u1 ˝ v ˝ u2 ˝ w A

uDu1 u2

X

.u1 _ v/.u2 _ w/;

uDu1 u2

whereas _ ı .I ˝ m/.u ˝ v ˝ w/ D _.u ˝ vw/ D u _ .vw/ X .u1 _ v/.u2 _ w/: D uDu1 u2

We also have: m ı . ˝ / ı 23 ı . ˝ I ˝ I /.u ˝ v ˝ w/ 1 0 X u1 ˝ u2 ˝ v ˝ w A D m ı . ˝ / ı 23 @ 0 D m ı . ˝ / ı @ D

X

uDu1 u2

X

1

u1 ˝ v ˝ u2 ˝ w A

uDu1 u2

.u1  v/.u2  w/;

uDu1 u2

whereas  ı.I ˝ m/.u ˝ v ˝ w/ D  .u ˝ vw/ D u  .vw/ X .u1  v/.u2  w/: D uDu1 u2



Shuffle quadri-algebras and concatenation

25

Proposition 7.3. The action _ makes the following diagram commute: HC ˝HKC KK KK KK _ KK KK  KK23 ı.0 ˝0 / KK HC KK KK KK KK 0 KK  % H ˝ HC o H ˝ H ˝ .HC˝HC / qq ˝_

where 0 .u/ WD .u/  u ˝ 1 for any u 2 HC . Proof. The compatibility of the deconcatenation with the shuffle product is written as follows: X .u qq v/ D .u1 qq v 1 / ˝ .u2 qq v 2 / (7.6) uDu1 u2 ; vDv 1 v 2

for any pure tensors u; v 2 H. Dropping the terms with 1 on the right side of the tensor product yields X 0 .u qq v/ D .u1 qq v 1 / ˝ .u2 qq v 2 /: (7.7) uDu1 u2 ; vDv 1 v 2 ; .u2 ;v 2 /¤.1;1/

Keeping only the terms of both sides of (7.7) with righmost letter in v gives: X .u1 qq v 1 / ˝ .u2 _ v 2 /; 0 .u _ v/ D

(7.8)

uDu1 u2 ; vDv 1 v 2 ; .u2 ;v 2 /¤.1;1/



which proves Proposition 7.3. Remark 7.4. The following diagram also commutes: HC ˝HKC KK KK KK  KK KK  K23 KKı.00 ˝00 / C H KK KK KK KK 00 KK K%  HC ˝ H o .HC ˝HC / ˝ H ˝ H ˝ qq

where 00 .u/ WD .u/  1 ˝ u for any u 2 HC . The proof is similar to the proof of Proposition 7.3.

26

M. Belhaj Mohamed and D. Manchon

Remark 7.5. The infinitesimal bialgebra structure on .H; m; / (in the category of vector spaces) does not give rise to an infinitesimal bialgebra structure in the category of .H; qq /-modules, because H is not a module-algebra on H. In other words the diagram below is not commutative: H˝H˝H I˝ m

˝I ˝I



H˝H˝H˝H



Ho

23



H˝H qq

/ H˝H˝H˝H



m

qq ˝ qq

H˝H

Let us finally make a simple restatement of Corollary 6.6. Proposition 7.6. The two following diagrams are commutative: H ˝ HC ˝ HC I˝ m

˝I ˝I



H ˝ HC ˝ H ˝ HC



HC o

H ˝ HC ˝ HC I˝ m

m

˝I ˝I



HC ˝ HC

/ H ˝ H ˝ HC ˝ HC 23



H ˝ HC ˝ H ˝ HC



HC o

_ ˝ qq



H ˝ HC qq

23



H ˝ HC qq

/ H ˝ H ˝ HC ˝ HC



m

qq ˝

HC ˝ HC

which means that the concatenation m is a morphism of left HC -modules, where H acts on HC by the shuffle product qq , and H acts on HC ˝ HC either by ._ ˝ qq / ı 23 ı . ˝ I ˝ I / or by . qq ˝ / ı 23 ı . ˝ I ˝ I /.

Shuffle quadri-algebras and concatenation

27

References [1] M. Aguiar, Infinitesimal bialgebras, pre-Lie and dendriform algebras, in Hopf algebras, Lecture Notes in Pure and Applied Mathematics 237, 1–33 (2004). [2] M. Aguiar and J. L. Loday, Quadri-algebras, J. Pure Appl. Algebra 191(3), 205–221 (2004). [3] D. Calaque, K. Ebrahimi-Fard, and D. Manchon, Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series, Adv. Appl. Math. 47(2) (2011), 282–308. [4] G. Duchamp, F. Hivert, and J. Thibon, Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Int. J. Algebra Comput. 12(5) (2002), 671–717. [5] K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math. 9(3) (2009), 295–316. [6] K. Ebrahimi-Fard and D. Manchon, Dendriform Equations, J. Algebra 322 (2009), 4053–4079. [7] K. Ebrahimi-Fard, D. Manchon, and F. Patras, New identities in dendriform algebras, J. Algebra 320 (2008), 708–727. [8] L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra 209(2) (2007), 439–459. [9] L. Foissy, Free quadri-algebras and dual quadri-algebras, arXiv: math.CO/1504.06056. [10] L. Foissy and F. Patras, Natural endomorphisms of shuffle algebras, Int. J. Algebra Comput. 23(4) (2013), 989–1009. [11] J. L. Loday, Dialgebras, Dialgebras and Related Operads, Lecture Notes in Mathematics 1763, Springer (2001), 7–66. [12] J. L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139(2) (1998), 293–309. [13] J. L. Loday and M. O. Ronco, Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Comb. 15 (2002), 253–270. [14] R. K. Molnar, Semi-Direct Products of Hopf Algebras, J. Algebra 47 (1977), 29–51. [15] S. Montgomery, Hopf algebras and their actions on rings, Regional Conference Series in Mathematics 82, CMBS/Amer. Math. Soc. (1993), 238. [16] C. Reutenauer, Free Lie Algebras, Oxford University Press, New York (1993). [17] M. O. Ronco, Primitive elements in a free dendriform algebra, New trends in Hopf algebra theory (La Falda, 1999), 245–263, Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000.

Structure theorems for dendriform and tridendriform algebras Emily Burgunder and Bérénice Delcroix-Oger Contents 1

Dual dendriform bialgebras and duplicial-dendriform bialgebras . . . . . . . . . . . . . . . 1.1 Confluence law et rigidity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definitions: dendriform and duplicial (co)algebras. . . . . . . . . . . . . . . . . . . . 1.3 Duplicial-dendriform bialgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Conilpotent dual dendriform bialgebras are free and cofree. . . . . . . . . . . . . . 2 Combinatorial description of the products and coproducts and the confluence law on PBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cutting paths and shuffle paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Confluence law with non-constant coefficients on PBT . . . . . . . . . . . . . . . . . 2.3 Dendriform operad and Tamari lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Skew-duplicial bialgebras, co-skew duplicial dendriform algebras . . . . . . . . . . . . . . 4 Application: freeness of algebras as dendriform algebras . . . . . . . . . . . . . . . . . . . . 4.1 The algebra of surjections : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Parking function algebra: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Terplicial and tridendriform operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definitions: tridendriform (co)algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Definitions: terplicial (co)algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Rigidity theorem for bialgebras endowed with terplicial and tridendriform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Application to the freeness of some tridendriform algebras . . . . . . . . . . . . . . . . . . . 6.1 Application to the freeness of the Solomon-Tits algebra as a tridendriform algebra. 7 Combinatorial description of the products, coproducts and confluence law on T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Path cutting and stuffled paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Combinatorial description of the terplicial structure on T1 via stuffle paths and trimming edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Combinatorial description of the tridendriform operations and cooperations via stuffle paths and cutting paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Confluence laws of the bialgebra structures on T1 . . . . . . . . . . . . . . . . . . . . 7.5 Combinatorial confluence laws for the co-tridendriform tridendriform bialgebra structure on T1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 35 36 38 38 40 43 44 46 46 47 48 48 50 52 55 55 57 57 58 58 59 63 64

30

E. Burgunder and B. Delcroix-Oger

Combinatorial objects gain in comprehension when seen as constructible from a few elements, for instance, when seen as a free algebra over a vector space. For a combinatorial object endowed with an associative algebraic operation with respect to an order, one may consider the dendriform and tridendriform algebra associated to it. Considering these finer structures reduces the dimension (rankwise) of the vector space when seen as a free dendriform or tridendriform algebra. But, proving that an object is free over one of these structure is not always easy. Some tools exist, mostly generalised Poincaré-Birkhoff-Witt Cartier-Milnor-Moore theorems or generalised Borel (called rigidity theorems). Namely, in the dendriform case with Foissy’s work [10] where he considers two different dendriform structures one as an algebra and one as a coalgebra which are not dual from each other; or the work of Loday and Ronco considering an associative coalgebra and a dendriform algebra [17]. One can also consider a different approach and find a duplicial structure (closely related to the dendriform structure) and consider a duplicial-duplicial rigidity theorem due to [16]. But such an analogue would not exist in the tridendriform case. In the tridendriform case, one structure theorem is due to Maria Ronco and the first author when considering an associative structure and a tridendriform algebra. We investigate different rigidity theorems, namely those arising from the dualisation of the dendriform or tridendriform operations. We provide explicit combinatorial expressions of the intertwining relations which permit rapidly to conclude to the nonfreeness of an object and provide a deeper knowledge of it. We introduce the terplicial algebra, and prove some rigidity theorems with a terplicial coalgebra structure, which plays an analogous role to the duplicial algebra in the dendriform case. The intertwining relations are short. Moreover, the free terplicial and the free tridendriform algebra are spanned by trees and the terplicial operation can be seen as a set operation related to the tridendriform one via an order, as defined in [27]. The paper has two parts one dedicated to the dendriform case and the second to the tridendriform case. It is organised as follows: we recall definitions of the dendriform and duplicial algebra and coalgebras. We introduce the duplicial dendriform bialgebra with their explicit intertwinning relations and state their associated rigidity theorem. To prove the intertwining relations, we give a precise description of the product and coproduct in the free algebra using paths. It introduces a whole new set of operations indexed by paths with their intertwining relations combinatorially explicited. The computation of the relations gives, as a byproduct, the number of elements of some intervals of the Tamari posets considered by Chatel and Pons [7]. The terplicial operad introduced later on is not exactly an analogous of the duplicial operad with regards to the dendriform operad but merely of a skew-symmetric version. We investigate this operad and rigidity theorems associated to it. In the next section, we illustrate on the Solomon-Tits algebra and the Parking functions these above structures and get as a result yet another proof of their freeness as dendriform algebra, see [2, 11, 10, 23, 27]. The second part of the paper is dedicated to the tridendriform case. After recalling the definition of the tridendriform algebra and coalgebra, we introduce the terplicial

Structure theorems for dendriform and tridendriform algebras

31

algebra and coalgebra. We then state the rigidity theorem for terplicial bialgebras, co-terplicial tridendriform bialgebras, and dual tridendriform bialgebras. The next section is devoted to a precise description of the tridendriform and the terplicial operations in the free algebra and cofree coalgebra through paths in order to prove the intertwining relations as in the dendriform case. It gives a new insight with their dual path indexed coproducts.

1 Dual dendriform bialgebras and duplicial-dendriform bialgebras This section is devoted to the dendriform algebra structure. We prove a new rigidity theorem for duplicial-dendriform bialgebras: a vector space with a duplicial coalgebra structure and a dendriform algebra structure which are moreover linked through intertwining relations. We also consider a dendriform bialgebras where the operations and the cooperations are obtained by dualisation. We investigate very precisely the relations between the product and coproduct to have a deeper knowledge of free dendriform algebra as dualising an operation is the most natural thing one can do on a given combinatorial object. 1.1 Confluence law et rigidity theorem We recall from [1] the definitions of a confluence law and a rigidity theorem, which extend the results presented by Loday in [16]. Note that we are considering non-symmetric quadratic operads here, which implies a simplification of definitions. We first recall the usual definitions on operads and algebras, for self-containedness. Definition 1.1. A (connected non-symmetric) operad A D ..A.n//n1; / is L  a graded vector space A D n1 A.n/, with A.1/ D K:1 and A.n/ finite dimensional for any n  1,  endowed with linear morphisms, called (partial) composition maps, ıi W Am ˝ An ! AmCn1 for any m; n  1, such that ıi and 1 satisfy associativity and unitality, for any elements x; z 2 A and y 2 Al : .x ıj z/ ıi y D .x ıi y/ ıj Cl1 z x ıi .y ıj z/ D .x ıi y/ ıi Cj 1 z 1 ı1 x D x D x ıi 1;

if i < j if 1 j l for any 1 i m

Let us first recall the definition of an algebra, a cooperad and a coalgebra over an operad. Definition 1.2. An algebra over an operad A is a vector space A equipped with a linear morphism mnA W A.n/ ˝ A˝n ! A. We denote the free algebra over an operad

32

E. Burgunder and B. Delcroix-Oger

A whose vector space of generators is V by 0 1 M M AD A.n/ ˝Sn @V A.n/ ˝Sn V ˝n : ˝ : : : ˝ V A.V / D „ ƒ‚ … n1

n

(1.1)

n1

Definition 1.3. A coalgebra over an operad C is a vector space C equipped with a linear morphism Cn W C.n/ ˝ C ! C ˝n . We denote the free (conilpotent) coalgebra over an operad C (or equivalently its associated dual cooperad C ) whose vector space of primitives is V by M C .n/ ˝Sn V ˝n : (1.2) Cc .V / D n1

Note that the coproducts considered in this article are reduced, i.e. obtained by removing from the coproduct terms in which the unit of the algebra appears, if it exists. We need moreover the following notion: Definition 1.4. The cofiltration Fn H can be defined on any Cc -coalgebra H: Fn H D fx 2 Hj8p > n; 8ı 2 C.p/; ı.x/ D 0g: The vector space F1 H is the vector space of primitive elements. Moreover, we denote by H W F1 H ! H the canonical inclusion. A Cc -coalgebra H is said to be conilpotent if H D [n1 Fn H. Definition 1.5. A confluence law ˛ between operads A and C is a family of maps M ˛m;n W C.m/ ˝ A.n/ ! A.n1 / ˝ : : : ˝ A.nm /; (1.3) n1 C:::Cnm Dn

such that ˛m;n is compatible with the structure of operad of C. Confluence laws are a generalisation of mixed distributive laws as defined by Fox and Markl in [12]. Example 1.6. It has been proven in [1] that a family satisfying a rigidity theorem (as defined below) is the family of preLie (product Ô) copreLie (coproduct ) bialgebras with confluence laws given on T 2 PreLie.n/ and S 2 PreLie.k/ by:  .T Ô S / D n  T ˝ S C .T Ô S1 / ˝ S2 C .T1 Ô S / ˝ T2 C T1 ˝ .T2 Ô S /; where .T / D T1 ˝ T2 and .S / D S1 ˝ S2 . Let us now define Cc ˛ A-bialgebras.

Structure theorems for dendriform and tridendriform algebras

33

Definition 1.7. A Cc ˛ A-bialgebra H is a K-vector space H endowed with a structure of A -algebra, a structure of Cc -coalgebra, satisfying a confluence law. Remark 1.8. Note that given a type of algebra and a type of coalgebra, there can be several choices of confluence law, none of them canonical. Theorem 1.9 ([1], Rigidity theorem). Let K be a field of characteristic 0 and let us consider two connected algebraic operads A and C, such that A.n/ and C.n/ are finite dimensional vector spaces. To any family of isomorphisms 'n W A.n/ ! C .n/ can be associated a confluence law .˛/ such that any conilpotent Cc ˛ A-bialgebras is free and cofree over the vector space of its primitive elements A.Prim H/ Š H Š Cc .Prim H/: 1.2 Definitions: dendriform and duplicial (co)algebras. We recall the definitions of dendriform algebras, dendriform coalgebras and describe the free dendriform algebra on the vector space spanned by the planar binary rooted trees PBT, and the conilpotent cofree codendriform coalgebra. We recall the definitions of duplicial algebras, coalgebras and describes their free algebra and cofree conilpotent coalgebra on PBT . Definition 1.10. A dendriform algebra (see [15]) structure on a vector space A is a pair of binary products W A ˝ A ! A and W A ˝ A ! A, satisfying that: .a  b/  c D a  .b  c C b  c/; .a  b/  c D a  .b  c/; .a  b C a  b/  c D a  .b  c/: Example 1.11. For any vector space V , the free dendriform algebra over V , denoted by PBT .V /, was defined in [17]. Its underlying vector space is spanned by labelled planar binary trees, where the vertices of planar rooted trees are labelled with the elements of a basis of V . For the sake of clarity, we recall that a planar rooted tree is a tree with a distinguished vertex called the root, such that all edges are oriented away from the root. A tree will be drawn with the root at the bottom. For an oriented edge from a vertex a to a vertex b, the vertex a will be called the source and b the target. The vertex a is then the parent of b and b a child of a. A leaf is a vertex which is note a source and an edge is said to be inner if its target is not a leaf. Vertices will be either leaves of sources of more than two edges (exactly two in the case of binary trees). We denote the vertices of a rooted tree t by V .t/, its root by r.t/ and its edges by E.t/. Definition 1.12. Given an integer ; tn on a root Sn n  2, the grafting of trees t1 ; : : :S n V .t / [ fg, whose edge set is is the tree whose vertex set is i i D1 i D1 E.ti / [ Sn i D1 f.; r.ti //g and whose root is . We denote it by _.t1 ; : : : ; tn /.

34

E. Burgunder and B. Delcroix-Oger

Any planar rooted tree T is the grafting of two trees of smaller degree, tl and tr ; that is T D _.tl ; tr / . The products ;  are defined recursively on any trees T and S by: T  ; D ;  T D T ; ;  T D T  ; D ;; T  S D _.tl ; .tr  S C tr  S // ; T  S D _..T  sl C T  sl / ; sr /:

(1.4) (1.5)

A codendriform coalgebra is a vector space C with two cooperations  and  satisfying equations obtained by dualizing the one in Definition 1.10. Example 1.13. As PBT .K/ is a dendriform algebra, its graded dual PBT .K/ is a codendriform coalgebra. The isomorphism between PBT .K/ and PBT .K/ given by the basis of planar binary trees on PBT .K/ induces a codendriform coalgebra structure on PBT cooperation is then given by P .K/. For any tree T , the codendriform P :  .T / D T.1/ ˝ T.2/ (resp.  D T.1/ ˝ T.2/ ), where the sum runs over all pairs of planar binary trees .T.1/ ; T.2/ / satisfying T  .T.1/  T.2/ / ¤ 0 (resp. T  .T.1/  T.2/ / ¤ 0). The definition of the operations gives a constructive way to define the cooperations on a tree T D _.tl ; tr /: X    .T / D tl ˝ ._.;; tr // C .tl /.1/ ˝ _ .tl /.2/ ; tr (1.6) X   _ tl ; .tr /.1/ ˝ .tr /.2/ (1.7)  .T / D _ .tl ; ;/ ˝ tr C P where  D  C ,  .t/ D t.1/ ˝t.1/ , and  ._.;; tr // D  ._.tl ; ;// D 0. This can be proven by direct inspection. The above coalgebra structure on the planar binary trees is the free conilpotent codendriform coalgebra. The proof is straightforward by dualisation. Definition 1.14. A duplicial algebra structure on A is a pair of binary products F W A ˝ A ! A and G W A ˝ A ! A, satisfying that: F and G are associative, .x F y/ G z D x F .y G z/, for any x; y; z in A. Example 1.15. The free duplicial structure on the planar binary rooted trees is given, for any trees T and S , by: T F; D;GT DT ; ;FT DT G; D; T F S D _..T F sl / ; sr / and T G S D _..tl ; .tr G S // A coduplicial coalgebra is a vector space C with two cooperations G and F satisfying equations obtained by dualizing the one in Definition 1.14. Note that coduplicial coalgebras appear in [14] under the name of L-coalgebras.

Structure theorems for dendriform and tridendriform algebras

35

Example 1.16. The free conilpotent coduplicial coalgebra is isomorphic to the vector space generated by planar binary rooted trees, endowed with the coproducts:   F .T / D tl ˝ _ .;; tr / C .tl/.F1/ ˝ _ .tl /.F2/ ; tr G .T / D _ .tl ; ;/ ˝ tr C _ tl ; .tr /.G1/ ˝ .tr /.G2/ , with G ._.tl ; ;// D F ._.;; tr // D 0. 1.3 Duplicial-dendriform bialgebras. Let H be a vector space with a dendriform algebra structure .H; ; / and a coduplicial coalgebra structure .H; F ; G /. Let us now determine the confluence law associated with the previously introduced products and coproducts on planar binary trees. Note that in [11], a confluence law is introduced for duplicial codendriform bialgebras. However, it differs from the dual of the one presented here as there is no term x ˝ y in  .x G y/. Definition 1.17. Let H be a vector space with a dendriform algebra structure .H; ; / and a coduplicial coalgebra structure .H; F ; G /. If it satisfies moreover 8x; y 2 H that:     (1.8) F .x  y/ D x ˝ y C x yF.1/ ˝ y.F2/ C x.F1/ ˝ x.F2/  y ;   (1.9) F .x  y/ D x.F1/ ˝ x.F2/  y ;   (1.10) G .x  y/ D x  y.G1/ ˝ y.G2/ ;     (1.11) G .x  y/ D x ˝ y C x  y.G1/ ˝ y.G2/ C x.G1/ ˝ x.G2/ y ; where D C , then H is said to be a coduplicial-dendriform bialgebra. Proposition 1.18. The relations introduced in the previous definition are a confluence law. Proof. The only property that has to be checked is the compatibility with the operadic structure. This comes from the fact that these relations are computed on planar binary trees. Equations (1.9) and (1.10) comes from the fact that the product is done on one of the two subtree of the root and the coproduct on the other. The two other equations are obtained by noting that in x  y (resp. y  x), the path from the root to the rightmost (resp. leftmost) leaf is first made of edges of x, and then of edges of y. The first term of the right side is obtained when the coproduct does a pruning of the tree at the leaf of x, the second, when it cuts an edge in y and the last when it cuts an edge in x.  Note that the confluence law obtained is in fact mixed distributive laws as studied in [16]. We can then apply the rigidity theorem in [16] to get: Proposition 1.19. Any conilpotent coduplicial-dendriform bialgebra is free and cofree over the vector space of its primitive elements.

36

E. Burgunder and B. Delcroix-Oger

1.4 Conilpotent dual dendriform bialgebras are free and cofree. In [10], Foissy considers dendriform bialgebras: bialgebras with a dendriform structure for the algebraic and the coalgebraic structure with a confluence law linking them both. But, the confluence law is not the one obtained when considering dendriform products and their dual coproducts. Indeed, it is obtained by considering dendriform products and the dualisations of other dendriform products (see [1] for more details). It seems natural to consider the case where the coalgebraic structure is the dual structure of the algebraic structure. In this section we focus on constructing the confluence law that links both structures. Let H be a vector space with a dendriform algebra structure .H; ; / and a codendriform coalgebra structure .H;  ;  /. We will introduce some notations: D .id ˝k ˝  / ı    ı .id ˝  / ı  kC1  kC1 D ı.id ˝ / ı    ı .id ˝k ˝ / X .1/k .id ˝ k / ı kC1 ˛ D  C  kD1

˛ D  C

X

.1/k .k ˝id / ı kC1 

kD1

where ˛ ; ˛ are called alternating convolution operations, from which we deduce the idempotents  ı˛  and  ı˛  which satisfy good properties (cf. [16, proposition 2.3.5]). Note that for any tree T D _.tl ; tr /, ˛ .T / is exactly tl ˝ _ .;; tr /. This is proven by induction while using the definitions (1.6), (1.7). Analogously, ˛ .T / is _ .tl ; ;/ ˝ tr . Definition 1.20. Let H be a vector space with a dendriform algebra structure .H; ; / and a codendriform coalgebra structure .H;  ;  /. If it satisfies moreover that for all x; y 2 H:  .x  y/ D ˛ .x  y/ C  .x ˛ .y/.1/ /.1/ ˝  .x ˛ .y/.1/ /.2/  ˛ .y/.2/ ;  .x  y/ D  .x/.1/ ˝  .x/.2/  y ;  .x  y/ D x   .y/.1/ ˝  .y/.2/ ;  .x  y/ D ˛ .x  y/ C ˛ .x/.1/   .˛ .x/.2/ y/.1/ ˝  .˛ .x/.2/ y/.2/ : then H is said to be a dual dendriform bialgebra. Note that this confluence law is not a finite sum of composition of tensors of operations and cooperations, but applied on an element x it is polynomial.

Structure theorems for dendriform and tridendriform algebras

37

Remark 1.21. One quick way to check that these relations differ from Foissy’s one is that in Foissy’s relations there is no term x ˝ y in  .x  y/, but a term y ˝ x instead. This means that on PBT for instance, we have in Foissy’s work: !    _  and here,   _ 

D

D 

˝_

(1.12)

!

 D 

D_˝

C

˝_

(1.13)

Proposition 1.22. The relations introduced in the previous definition are a confluence law. Proof. The only property that has to be checked is the compatibility with the operadic structure. This comes from the fact that these relations are computed on planar binary trees. PBT .V / endowed with the above codendriform and dendriform structure is a dual dendriform bialgebra. Indeed, the definition of coproducts of trees (1.6), (1.7) applied on the operations  and  of two trees T D _.tl ; tr /; S D _.sl ; sr / give the confluence laws when using the above note:    .T  S / D .T sl / ˝ _ .;; sr / C .T sl /.1/ ˝ _ .T sl /.2/ ; sr ; (1.14)    .T  S / D _ .tl ; ;/ ˝ .tr S / C _ tl ; .tr S /.1/ ˝ .tr S /.2/ : (1.15) The relations  .T  S / D T   .S /.1/ ˝  .S /.2/ , and  .T  S / D  .T /.1/ ˝  .T /.2/  S come from the fact that the product is done on one of the two subtree of the root and the coproduct on the other.  Note that this type of confluence law does not fit into the scope of Loday’s work in [16]. We then apply rigidity theorem from [1]: Proposition 1.23. Any conilpotent dual dendriform bialgebra is free and cofree over its primitive. Indeed, PBT endowed with its dual dendriform bialgebra is free and cofree on the vector space of its primitive elements. K  _. Remark 1.24. Let .H; ; ; ;  / be a conilpotent dual dendriform bialgebra whose primitive elements are explicitely known. We denote by proj1 W H ˝ H ! H the canonical projection of the first component and proj2 the canonical projection on the second component, we denote by 1F1 the application which is the identity on F1 H and null elsewhere, analogously for 1F1 , and by pi (resp. pi ) the composition pi WD proji ı.id ˝ 1F1 / ı  , (pi WD proji ı.1F1 ˝ id / ı  ). We then have ˛  D .1F1 ˝ id / ı  and ˛  D .id ˝ 1F1 / ı  on PBT .V /.

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E. Burgunder and B. Delcroix-Oger

2 Combinatorial description of the products and coproducts and the confluence law on PBT In the following section, we investigate combinatorically the confluence laws on bialgebras of the planar rooted trees. These descriptions are based on some shuffle paths and cutting paths in trees and are useful to give an explicit confluence law in terms of trees and coefficients, binomials, only. We then relate these to the work of Chatel and Pons on the Tamari Lattice. 2.1 Cutting paths and shuffle paths. The codendriform cooperations can be described in terms of cutting paths, and the dendriform operations in terms of shuffle paths. We describe more precisely these paths. 2.1.1 Cutting paths: Consider T D _.tl ; tr / a tree and q a path in the tree from the root to a leaf, denote ei its edges. Note that this path only depends on the choice of a leaf. The orientation of the tree defines for every vertex the notion of a right edge and a left edge. This path is referred to as a cutting path. We now construct a coproduct indexed by this cutting path as follows. If the path q is the leftmost path of T , denoted by lT , we will define lT .T / D ; ˝ T and the path is the rightmost path of T , denoted by rT , we will define rT .T / D T ˝ ; : For any other path, denoted .e1 ; : : : ; em /, define inductively the coproduct as follows:  _.tl ; .e2 ;:::;em / .T e1 /.1/ / ˝ .e2 ;:::;em / .T e1 /.2/ e1 is a right edge, .e1 ;:::;em / T D .e2 ;:::;em / .T e1 /.1/ ˝ _..e2 :::;em / .T e1 /.2/ ; tr / e1 is a left edge, (2.1) using Sweedler’s notation for the coproduct and where m is the total number of edges of q and T e1 is the right (resp. left) subtree of T if e1 is a right edge (resp. left). Note that this definition is the dual of the definition of the dendriform operations defined in [18]. Considering a cutting path q, the coproduct q of a tree can be understood as follows: duplicate the cutting path and consider the right-handside of the cutting path in the tree with the cutting path tensored with the cutting path and the left-handside of the cutting path. For a tree T we will denote Q.T / the set of all paths from root to a non-extremal leaf of T , by Ql (resp. Qr ) the subset of paths with their first edge being a left (resp. right) edge. Then, the co-operations defined in example 1.13 verify: X X X  .T / D q .T / ;  .T / D q .T / ;  .T / D q .T / : q2Q.T /

q2Ql .T /

q2Qr .T /

39

Structure theorems for dendriform and tridendriform algebras

, Q.T / D f

T D g .T / D

˝

C

˝

C

˝

,

,

,

˝

C

˝

, C

.

Figure 1.  in terms of cutting paths.

Indeed, it is immediate when considering the construction of the coproduct (1.7) and (1.6). An example is illustrated in figure 1. 2.1.2 Shuffle paths: We refer for example to [16, chapter 5] for the definition of the operations under n and over = on planar binary trees. Endowed with these operations the PBT is free as a duplicial algebra see [16, proposition 5.1.3]. Any tree T can be uniquely written as T D t 1 =t 2 = : : : =t m (resp. T D tn n : : : nt2 nt1 ) where ti , 1 i m, cannot be written as S=U (resp. tj , 1 j n, cannot be written as S nU ), for any S; U 2 PBT. Consider two trees T D t 1 =    =t m and S D smCn n    nsn , and a shuffle of the indices p 2 S h.m; n/ written as a list of its images .p.1/ : : : ; p.m C n//. We will define a product indexed by a shuffle p, denoted by T p S , inductively as follows: (2.2) T p S   1  2 m if p.1/ 2 1 I m t = .t = : : : =t / .p.2/;:::;p.mCn// S  D  T .p.2/;:::;p.mCn// .smCn n : : : nsmC2 / nsmC1 if p.1/ 2 m C 1 I n C m ; note that by definition p.1/ D 1 or m C 1, with the base case being : _ .1;2/ _ D

; _ .2;1/ _ D

:

(2.3)

For an example of a shuffle product between two trees, see example 2.1. For p 2 S h.m; n/ one can associate a path pQ in the product T p S which will be referred to as a shuffle path. In the following sequel we will consider the associated paths instead of the shuffles, as we will consider intersections of cutting paths and shuffle paths to describe the confluence laws. For two trees T; S , with T D t 1 =    =t m and S D smCn n    nsn , we will denote S h.n; m/ the set of n C m shuffles and by S h< .n; m/ (resp. S h> .n; m/) the shuffle p D .p.1/; : : : ; p.n C m// verifying p.1/ D 1 (resp. p.1/ D m C 1). Then, the

40

E. Burgunder and B. Delcroix-Oger

dendriform operations verify, see [18], X T S D T p S ; T  S D p2Sh.m;n/

X

T p S ; T  S D

p2Sh> .m;n/

X

T p S :

p2Sh< .m;n/

The proof is immediate when considering the constructive definition of the product (1.4) and (1.5). e b

d a and S D c , the set of right shuffle paths is Example 2.1. For T D P> .T; S / D ffc; d; e; a; bg; fc; a; b; d; eg; fc; d; a; b; eg; fc; a; d; e; bg; fc; d; a; e; bg; fc; a; d; b; egg then T p S , with p 2 P> .T; S /, are given respectively by:

2.2 Confluence law with non-constant coefficients on PBT . Understanding on PBT the confluence laws in a more combinatorial manner, with confluence laws with non-constant coefficients but depending on the cofiltration of the coalgebra is an efficient tool to determine either the structure "should" have a free structure on a given combinatorial object or to show rapidly that it is not free. Consider the free cofree conilpotent dual dendriform bialgebra on PBT with combinatorially defined products and coproducts. Understanding in a more combinatorial way the confluence law asks to compute the coefficients arising in front of the different elements appearing in it. Example 2.2. In the example 2.1, one gets that the number of elements of T ˝ S in the coproduct of  .T  S / is 6 which correspond to the cutting paths drawn with dots:

Note that the cutting paths and the shuffle paths coincide. From the above example, it becomes clear that elements in the coproduct  .T  S / will appear multiple times: it depends on the intersection of the cutting path and the shuffle path. Therefore, the first step is to understand the confluence law between the product associated to a shuffle path and the coproduct associated to a cutting path.

Structure theorems for dendriform and tridendriform algebras

41

2.2.1 Confluence laws between shuffle paths and cutting paths We will denote the set of edges of a path p as E.p/. Let T , S be two trees. Consider p a shuffle path of P .T; S /, and q cutting path of Q.T p S /. The intersection between the two paths p \ q is a (possibly empty) path in T p S with edges E.p/ \ E.q/. Consider also the remaining of the cutting path, denoted q c : it is the path in T p S composed of the edges E.q/ n .E.p/ \ E.q//. Lemma 2.3. This path, q c , is empty or a path with edges strictly in T or in S according to the reached leaf. We will denote that path qT (resp. qS ) if its edges are in T (resp. in S ). Proof. We will denote by rT the rightmost path of the tree T , and by lS the leftmost path of the tree S . The lemma is proven by induction on the number of leaves using the combinatorial definitions of the product (2.2) and coproduct (2.1). Indeed, in low dimensions it is clear. Suppose the property true for trees with the sum of their number of leaves inferior or equal to n. Consider two trees T D _.tl ; tr / and S D _.sl ; sr / such that the total number of their leaves is n C 1, with a shuffle path p denoted as a sequence of edges .pi /1i n and a cutting path q denoted as a sequence of edges .q i /1i m . The following cases can occur: if q 1 D p1 then consider the paths .p2 ; : : : ; pn / and .q 2 ; : : : ; q m / in the trees T 1 D tr , S 1 D S if p1 is the first edge of rT or T 1 D T , S 1 D sl if p1 is the first edge of lS and conclude by induction. If q 1 ¤ p1 , suppose moreover that p1 is an edge of rT (the symmetric being p1 is an edge of lS ). Then by construction q 1 is the left edge of the root of T . Moreover E.q/  E.T / n E.rT / as q .T p S / D .q 2 ;:::;q m / .tl /1 ˝  _..q 2 ;:::;q m / .tl /2 ; tr .p2 ;:::;pn / S /. As mentioned above the coproduct of a tree along a cutting path can be understood as thickening the cutting path and cutting it in two to give each side of the tensor. We therefore need to take more notations to describe the confluence law: When q c is a path of T we will denote by ŒqT the path in T defined by the sequence of edges E.q/ \ E.T / and ŒqS is defined analougously. Denote pc the remaining of the shuffle path in T p S , i.e. the path defined by the sequence of edges of E.p/n.E.p/\E.q//. The intersection path p\q has edges in S and in T . We will denote by pS the sequence of edges .E.p/\E.q/\E.S //[E.pc / which is the trace of the shuffle path in S , and analogously defined pT the sequence of edges .E.p/ \ E.q/ \ E.T // [ E.pc /. Lemma 2.4.

8 < T ˝S  T .T /1 ˝ ŒqT .T /2 pS S q .T p S / D : T Œq

pT ŒqS .S /1 ˝ ŒqS .S /2

if p D q else if q c D qT else if q c D qS

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E. Burgunder and B. Delcroix-Oger

Proof. A cutting path is determined by the leaf to which it leads, and by definition of T p S , this leaf is a former leaf of T or of S . Therefore, all the cases are considered. The lemma is proven by induction on the number of leaves, and consider the two cases whether or not the first edge of the shuffle path and the cutting path coincide and then by applying (2.1) to (2.2).  2.2.2 Confluence laws on co-dendriform dendriform bialgebras: Consider rT the rightmost path of T , denote RT its number of edges and .eiT /i its edges 1 i RT . Respectively denote LS D #lS to be the number of edges of the leftmost path lS of a tree S , and denote .eiS /i its edges. Corollary 2.5. of T ˝ S in  .T  S / (resp. in  .T  S /) RT CLS 1  The number RT CLSof1terms  is (resp. ). RT LS Proof. From the above lemma 2.4, the element T ˝ S will appear when the shuffle path and the cutting path coincide. It suffices to compute the cardinal of P< .T; S / which the RT CLis  collection of shuffles of two sequences of cardinal RT and LS  1, i.e. S 1 .  RT Proposition 2.6. For two planar binary trees T and S , the edges of the rightmost path of T will be denoted .eiT /i and the edges of the leftmost path of S will be denote .eiS /i . Denote by qiT (respectively qiS ) the cutting path such that i is the maximal integer such that the first i edges are the first i edges of rT (respectively of lS ), with 0 i RT , i.e. qiT D .e1T ; : : : ; eiT ; qi C1 ; : : : ; qjqj / with qi C1 an edge of S , denote pjT (respectively pjS ) the shuffle path such that j is the maximal integer such that the first j edges are edges of rT (resp. of lS ) with 0 j LS . The confluence law on PBT is given by:  .T  S/

D

! RT C LS  1 T ˝S RT C

C

 .T  S/

D

X

X

q 2 Q.T / q D qT i

eT p 2 P< .T i ; S/ p D pS j

X

X

q 2 Ql .S/ q D qS i

eS p 2 P .T; S i / p D pT i

 .T /1 ˝  .T /2  S ;

! i Cj 1 q .T /1 ˝ q .T /2 p S i ! i Cj 1 T p q .S/1 ˝ q .S/2 ; i

Structure theorems for dendriform and tridendriform algebras  .T  S/

D

 .T  S/

D

T   .S/1 ˝  .S/2 ; ! RT C LS  1 T ˝S C LS C

C

X

X

q 2 Qr .T / q D qT i

eT p 2 P .T i ; S/ p D pS j

X

X

q 2 Q.S/ q D qS i

eS p 2 P> .T; S i / T pDp j

43

! i Cj 1 q .T /1 ˝ q .T /2 p S j ! i Cj 1 T p q .S/1 ˝ q .S/2 i

Proof. The combinatorial description of the products and coproducts in term of shuffle paths and cutting paths gives  .T  S/ D

X

X

q .T p S/

p2P< .T;S/ q2Ql .T p S/

Then apply lemma 2.4 for the shuffle paths p and cutting paths q. The coefficients appear since for any shuffle path p0 obtained from p, and for any cutting path q 0 obtained from q, such that only the edges .E.p/\E.q//\E.T / and .E.p/\E.q//\ E.S / are shuffled, then, q 0 .T p0 S / will give the same element as q .T p S /  2.3 Dendriform operad and Tamari lattice. The set of planar binary trees can be endowed with the Tamari partial order, see [25], the obtained poset is a lattice known as the Tamari lattice. Chatel and Pons have shown its correspondence with intervalposets: Theorem 2.7 (Theorem 2.8 [7], [8]). Interval-posets are in one-to-one correspondence with intervals of the Tamari lattice. Moreover, the interval I1 is included in the interval I2 if and only if I1 is an extension of I2 (contains relations of I2 and possibly some other relations). We use this characterisation of intervals of the Tamari lattice. Loday introduced the notion of dendriform operad deeply linked with planar binary trees as explained above. He introduced with Ronco a structure of Hopf algebra on planar binary trees in [18]. The link between Tamari posets and dendriform operad, described by Chapoton, enables us to count elements of some intervals of the Tamari lattice. Proposition 2.8 ([6], prop 3.2). The dendriform and duplicial products enable the description of the following intervals: ŒT F S I T G S  D fX jX 2 T  S g; for any planar binary trees T and S .

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Let us remark that the interval-poset associated with interval ŒT F S I T G S  has two connected components, corresponding respectively to T and S . We can now enumerate elements in this type of interval, using Lemma 2.5. Corollary 2.9. The number of elements in the intervals described above is given by: ! RT C LS  1 jŒT F S I T G S j D LS where RT is the number of vertices on the rightmost path of the planar binary tree T and LS is the number of vertices on the leftmost path of the planar binary tree T , for any planar binary trees T and S .

3 Skew-duplicial bialgebras, co-skew duplicial dendriform algebras In the next section we will define a new type of algebras, which as an operad, has the same underlying graded vector space as the tridendriform operad and has the property of being a set-operad, see Mendez’s work [21]. The dendriform and duplicial operad share the same property. Moreover it will also verify the same compatibility towards composition as the dendriform and duplicial operads do, namely the operadic composition of the dendriform operad Dend and the duplicial composition Dup admit the existence of a map such that the diagram commute:

Dup

 Dup

De nd

/ lDend l l l l l l  ll vl l

Dup ı Dup D Dend ı Dend

In addition, any tridendriform algebra can be seen as a dendriform algebra, see Loday. Unfortunately, there exists no such thing as extending the duplicial algebra in a "terplicial" algebra, i.e. as operads a symmetric set operad sharing the same underlying graded vector space as the tridendriform operad and such that the right and the left products are both associative. The terplicial operad defined in the next section is then an extension of a new operad, named the Skew-duplicial operad, generated by two binary products J and jI satisfying for any elements x, y and z: .x jI y/ jI z D x jI .y jI z/ .x jI y/ J z D x jI .y J z/ .x J y/ J z D x J .y jI z/ Note that these algebras are associative algebras equipped with an extra binary operation satisfying a condition of L algebra introduced by P. Leroux and a condition of dipterous algebra introduced by J.-L. Loday and M. Ronco.

Structure theorems for dendriform and tridendriform algebras

45

Proposition 3.1. The free skew-duplicial operad have the same underlying module as duplicial and dendriform operads. The operations J and jI are defined on two planar binary trees S D _.Sl ; Sr / and T D _.Tl ; Tr / by: T jI S D _.T jI Sl ; Sr /; with T jI ; D T T J S D _.Tl ; Tr jI S /; with ; jI T D T Proof. The proof is done by checking that the defined product is a skew-duplicial product. The freeness of the defined product comes from the freeness of the dendriform product on planar binary trees: indeed the skew-duplicial product is equivalent to associating to the dendriform product on binary trees the leading term for the order S > T if jSl j > jTl j or jSl j D jTl j and jSl j > jTl j.  The dual co-Skew-duplicial coproduct satisfies the following relations: .jI ˝ id / ı jI D .id ˝ jI / ı jI .jI ˝ id / ı J D .id ˝ J / ı jI .J ˝ id / ı J D .id ˝ jI / ı J It is given on planar binary trees by: jI ._.Tl I Tr // D Tl ˝ _.;I Tr / C jI .Tl /1 ˝ _.jI .Tl /2 I Tr / J ._.Tl I Tr // D _.Tl I ;/ ˝ Tr C _.Tl I jI .Tr /1 / ˝ jI .Tr /2 with J ._.Tl I ;// D jI ._.;I Tr // D 0. This operad enables us to introduce a rigidity theorem for co-skew-duplicial dendriform bialgebras, proven using the recursive definitions of products and coproducts: Proposition 3.2 (Rigidity theorem for co-skew-duplicial dendriform bialgebras). Any connected dendriform co-skew-duplicial bialgebra satisfying the following confluence laws is free and cofree over its primitive elements: J .T J .T jI .T jI .T

 S / D T  J .S /1 ˝ J .S /2  S / D T ˝ S C T  jI .S /1 ˝ jI .S /2 C J .T /1 ˝ J .T /2 S  S / D T ˝ S C T jI .S /1 ˝ jI .S /2 C jI .T /1 ˝ jI .T /2  S  S / D jI .T /1 ˝ jI .T /2  S

Proposition 3.3 (Rigidity theorem for co-skew-duplicial duplicial bialgebras). Any connected duplicial co-skew-duplicial bialgebra satisfying the following confluence laws is free and cofree over its primitive elements: J .T J .T jI .T jI .T

F S / D T F J .S /1 ˝ J .S /2 G S / D J .T /1 ˝ J .T /2 G S F S / D T ˝ S C T F jI .S /1 ˝ jI .S /2 C jI .T /1 ˝ jI .T /2 F S G S / D jI .T /1 ˝ jI .T /2 G S

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E. Burgunder and B. Delcroix-Oger

Proposition 3.4 (Rigidity theorem for co-skew-duplicial skew-duplicial bialgebras). Any connected skew-duplicial co-skew-duplicial bialgebra satisfying the following confluence laws is free and cofree over its primitive elements: J .T J S / D T ˝ S C J .T /1 ˝ J .T /2 jI S C T J jI .S /1 ˝ jI .S /2 J .T jI S / D T jI J .S /1 ˝ J .S /2 jI .T J S / D jI .T /1 ˝ jI .T /2 J S jI .T jI S / D T ˝ S C jI .T /1 ˝ jI .T /2 jI S C T jI jI .S /1 ˝ jI .S /2

4 Application: freeness of algebras as dendriform algebras We will consider two well-known combinatorial Hopf algebras : the algebra of surjections and the algebra of Parking functions. To illustrate our above definitions, we will endow them with their usual dendriform structure and describe a co-duplicial structure or co-skew-duplicial structure verifying the confluence law 1.17 or 3.4 respectively. As a corollary it will reprove their freeness as a dendriform algebra, see [3, 10, 23, 27, 26] for previous proofs. 4.1 The algebra of surjections : Let us consider the vector space of surjection and recall its dendriform structure. We will denote it as ST the Solomon-Tits algebra. It is also denoted FQSym by some authors. Consider the set STrn WD fx W Œn ! Œr; x surjectiveg and the vector space ST D ˚nr1 KŒS Tnr . For x 2 STrn , we write x D .x.1/; : : : ; x.n//, listing its images, and r D maxfx.i /; 1 i ng. For x 2 STrn ; y 2 STsm denote the shifted concatenation by x  y D .x.1/; : : : ; x.n/; y.1/ C r; : : : ; y.m/ C r/. This vector space can be endowed with a dendriform structure .; /, see for example [23, 10, 2], as follows: let x 2 STrn , y 2 STsm define P 1. x  y WD f 2Sh .r;s/ f ı .x  y/, P 2. x  y WD f 2Sh .r;s/ f ı .x  y/, where, f 2 S h is a .r; s/-shuffle such that f .r/ > f .r C s/ and f 2 S h is a .r; s/-shuffle such that f .r/ < f .r C s/. Fix that x  1K WD 0 DW 1K  x and x  1K WD x DW 1K  x, for all x 2 ST. Given a map x W Œn ! Œr there exists a unique surjective map std.x/ 2 STrn such that x.i / < x.j / if, and only if, std.x/.i / < std.x/.j /, for 1 i; j n. A left-to-right maximum i is such that all the images that precede are smaller than x.i / : x.i  k/ < x.i /, i  k  0. A right-to left maximum i verifies x.i C k/ < x.i /, i C k n. We will denote by LR.x/ the list of the left-to-right maxima of x and by RL.x/ its list of right-to-left maxima. We can now define a coduplicial structure on ST: For x 2 STrn : ÍF .x/ D .x.1/; : : : ; x.lk  1// ˝ std.x.lk /; : : : ; x.n//

Structure theorems for dendriform and tridendriform algebras

47

with lk is the maximum element in LR.x/ D .l1 ; : : : ; lx / satisfying that x.lk / D x.lkC1 / C 1 and that the left hand-side of the above tensor product belongs to ST. For x 2 STrn : G .x/ D std.x.1/; : : : ; x.rk // ˝ .x.rk C 1/; : : : ; x.n// where rk is the minimum element in RL.x/ D .r1 ; : : : ; rx /, such that the right handside of the above tensor product is in ST and that x.rk / D x.rkC1 /  1. Proposition 4.1. ST endowed with the coproducts ÍF ; G and the dendriform structure .; / is a Dupc -Dend bialgebra. In [26, 3.2.1, proposition 4], Vong introduces on S T the following operations : x jI y D x  y; x J y D .x.1/; : : : ; x.n  1/; y.1/ C r; : : : ; y.m/ C r; x.n// with x 2 S Tnr ; y 2 S Tms . These operations verify the skew-duplicial relations. Dualising them give co-duplicial cooperations. Proposition 4.2. ST endowed with the coproducts .J ; jI / and the dendriform structure .; / is a co-skew-duplicial dendriform bialgebra. Corollary 4.3. [10, 3, 23, 27] ST is free as a dendriform algebra on its primitives. It is straightforward from theorem 1.19 with the co-duplicial dendriform structure or theorem 3.4 with the co-skew duplicial dendriform structure. Remark 4.4. The basis of the primitives, in the case of the coduplicial dendriform bialgebra structure, is not the same as in [3] as in dimension 3 the primitives are .1; 3; 2/; .2; 3; 1/ whereas in [3] it is .1; 2; 1/; .2; 3; 1/. The number of elements of ST for each dimension is given by the Fubini numbers [24, A00670]. Remark 4.5. In the case of the co-skew duplicial dendriform bialgebra structure, this corollary gives an algebraic rewriting of Vong’s proof [26, section 3], which uses reductions and Grobner basis arguments. 4.2 The Parking function algebra: The set of Parking functions can be endowed with a dendriform structure given by the work of Novelli-Thibon [23]. We follow [2]. Definition 4.6. A map f W Œn ! Œn is called a n-non-decreasing parking function if f .i / i for 1 i n. The set of n-non-decreasing parking functions is denoted by NDPFn .The composition f WD f " ı  of a non-decreasing parking function f " 2 NDPFn and a permutation  2 Sn is called a n-parking function. The set of n-parking functions is denoted by PFn . The subset of those such that maxff .i /; 1 i ng D r is denoted PFrn .

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S Let PQSym denote the vector space spanned by the set n1 PFn of parking functions. The binary operations  and  on PQSym are defined in a similar way that in the case of ST: P P f  g WD max.h/>max.k/ hk; f  g WD max.h/max.k/ hk; where the sums are taken over all pairs of maps .h; k/ verifying that hk [ is parking, S Fn ! Park.h/ D f and Park.k/ D g, for f; g 2 n1 PFn and the map Park W [

n1

PFn is the Parking counterpart of the standardisation.

n1

We will now define on the Parking function a structure of coduplicial coalgebra similarly to ST. For x 2 PFrn , ÍF .x/ D .x.1/; : : : ; x.lk  1// ˝ std.x.lk /; : : : ; x.n// with lk 2 LR.x/ D .l1 ; : : : ; lx / the maximum element such that x.lk / D x.lkC1 /C1 and that x1 belong to PF. For x 2 PFrn : G .x/ D std.x.1/; : : : ; x.rk // ˝ .x.rk C 1/; : : : ; x.n// where rk 2 RL.x/ D .r1 ; : : : ; rx / the minimum element such that x2 2 PF and that x.rk / D x.rkC1 /  1. Proposition 4.7. PQSym endowed with the coproducts ÍF ; G and the dendriform structure .; / is a Dupc -Dend bialgebra. Proof. The coduplicial structure is satisfied as the Parkisation of words preserve the maxima. The confluence law is verified as in ST by direct inspection.  As a corollary: Corollary 4.8. [23, 3] The dendriform algebra of Parking function is free as a dendriform algebra over the vector space of its primitive elements.

5 Terplicial and tridendriform operads 5.1 Definitions: tridendriform (co)algebras. Let us recall the relations governing a q-tridendriform algebra linking the tridendriform structure described in [18] for q D 1 and in [5] for q D 0: Definition 5.1. [2] A q-tridendriform algebra is a vector space A together with three operations W A ˝ A ! A,  W A ˝ A ! A and W A ˝ A ! A, satisfying the following relations: .a  b/  c a  .b  c/ .a  b/  c .a  b/  c .a  b/  c

D a  .b  c C b  c C q b  c/; D .a  b C a  b C q a  b/  c; D a  .b  c/; .a  b/  c D a  .b  c/; .a  b/  c D a  .b  c/:

D a  .b  c/; D a  .b  c/;

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Note that the operation WD  Cq  C  is associative. Moreover, given a q-tridendriform algebra .A; ; ; /, the space A equipped with the binary operations  and  WD q  C  is a dendriform algebra. Example 5.2. Let TS n be the set of all planar reduced rooted trees with n C 1 leaves. Denote by T D t 2 Tn may be written in a unique way as t D 1 n Tn . Any P W 1 .t ; : : : ; t r /, with t i 2 Tni and riD1 ni C r  1 D n. On the space KŒT1 , define operations ,  and  recursively as follows: t  j D t  j D j  t D j  t D 0; for all t 2 T1 ; j  t D t  j D t; for all t 2 T1 ; W t  w WD .t 1 ; : : : ; t r1 ; t r w/; W t  w WD .t 1 ; : : : ; t r1 ; t r w 1 ; w 2 ; : : : ; w l /; W t  w WD .t w 1 ; w 2 ; : : : ; w l /; W W for t D .t 1 ; : : : ; t r / and w D .w 1 ; : : : ; w l /, where is the associative product

D Cq  C  previously defined. Following [5] and [19], .KŒT1 ; ; ; / is the free q-tridendriform algebra spanned by the unique element of T1 . Definition 5.3. A q-tridendriform coalgebra, or co-tridendriform coalgebra, is a vector space V endowed with three coproducts  ;  ;  satisfying the following relations: . ˝ id / ı  . ˝ id / ı  . ˝ id / ı  .id ˝  / ı  . ˝ id / ı  . ˝ id / ı  . ˝ id / ı 

D .id ˝  / ı  ; D .id ˝  / ı  ; D .id ˝  / ı  ; D . ˝ id / ı  ; D .id ˝  / ı  ; D .id ˝  / ı  ; D .id ˝  / ı  :

where  D  C q C  . A co-augmented conilpotent tridendriform coalgebra C is a coalgebra verifying that: C D [n0 Fn C where F0 C D K, F1 C D fx 2 C j  .x/ D  .x/ D  .x/ D 0g, Fn C D fx 2 C j  .x/ 2 Fn1 C ˝2 ;  .x/ 2 Fn1 C ˝2 ;  .x/ 2 Fn1 C ˝2 g. Example 5.4. As in the dendriform framework, we will consider the coproducts obtained as duals of the above tridendriform products on the vector space generated by planar reduced rooted trees KŒT1 .The isomorphism between KŒT1  and its graded dual KŒT1  given by the basis of planar rooted trees induces a cotridendriform coalgebra structure on KŒT1  : for every T 2 T1 the cooperations are given by

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P  .T / D T. 1/ ˝ T. .2/ where the sum runs over all pairs .T. 1/ ; T. 2/ / such that T  .T. 1/  T. 2/ / ¤ 0, with  D;  or . The definition of the operations gives a constructive way to define the cooperations as:  ._.t1 ; : : : ; tn // D  .t1 /.1/ ˝ _. .t1 /.2/ ; t2 ; : : : ; tn / C t1 ˝ _.;; t2 ; : : : ; tn /  ._.t1 ; : : : ; tn // D

n1 X

_.t1 ; : : : ; ti 1 ;  .ti /.1/ / ˝ _. .ti /.2/ ; ti C1 ; : : : ; tn /

i D2

C _.t1 ; : : : ; ti 1 ; ;/ ˝ _.ti ; : : : ; tn / C _.t1 ; : : : ; ti / ˝ _.;; ti C1 ; : : : ; tn /  ._.t1 ; : : : ; tn // D _.t1 ; : : : ; tn1 ;  .tn /.1/ / ˝  .tn /.2/ C _.t1 ; : : : ; tn1 ; ;/ ˝ tn ; where  D  C q C  . This definition gives well-defined coproducts verifying the cotridendriform relations. Note that any free tridendriform algebra is naturally endowed with this dual coalgebra structure. Dualising the proof for the freeness of T1 as a tridendriform algebra, one gets that this structure on T1 is the cofree conilpotent tridendriform coalgebra. 5.2 Definitions: terplicial (co)algebras. From the tridendriform operad, we define a new set-operad called terplicial, on which the tridendriform operad is quasi-set (see [1]), by analogy with the pair (Dend, skew-dupl). It is to be noted that an analogue of the pair (Dend, Dupl) is not possible as the analogue of Dupl with three associative products cannot be defined. Definition 5.5. A terplicial algebra is a vector space V endowed with three binary products fG; O; Í Fg satisfying the following relations: Í F and O are associative, .x G y/ G z D x G .y Í Fz/ .x Í Fy/ G z D x Í F.y G z/ .xOy/ G z D xO.y G z/ .x Í Fy/Oz D x Í F.yOz/ .x G y/Oz D xO.y Í Fz/ All the equations but the second and the last coincide with relations satisfied by triduplicial algebra defined by J.-C. Novelli and J.-Y. Thibon in [22]. Consider the planar rooted trees, and denote by _.t1 ; : : : ; tn / a planar tree T 2 T1 whose root has arity n and such that the ti are the (possibly empty) subtrees of T rooted in the children of the root of T . We can now describe the free terplicial algebras. Theorem 5.6. The free terplicial algebra on a vector space V can be described as the algebra whose underlying vector space has a basis given by reduced planar rooted

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trees with leaves decorated by V : T er.V / D ˚KŒTn  ˝ V ˝n . Hence, the dimension of the space of operations of arity n in the triplicial operad is given by the Schroeder– Hipparchus number. The operations G, Í F and O on free terplicial algebras are described recursively as follows, for any tree T D _.t1 ; : : : ; tn / and S D _.s1 ; : : : ; sm /, and denoting by ; the empty tree: T G S D _.t1 ; : : : ; tn1 ; tn Í FS / T OS D _.t1 ; : : : ; tn1 ; tn Í Fs1 ; s2 ; : : : ; sm / T Í FS D _.T Í Fs1 ; s2 : : : ; sm /; given that, ; Í FT D T and T Í F; D T: Example 5.7. If T D T Í FS D

and S D

,T GS D

, the products are given by: and T OS D

.

Proof. T1 endowed with these operations satisfy the terplicial relations, see section 7.4.2. The universal property of terplicial algebras is verified: for any morphism from f W V ! A, with A a terplicial algebra, W V ! T er.V / the canonical injection, then there exists a unique terplicial morphism W T er.V / ! A defined as _.t1 ; : : : ; tn / ˝ v1 : : : vk1 vk1 C1 : : : vk1 C:::Ckn 7!

.t1 ˝ v1 : : : vk1 / Í F . ._.;; t2 / ˝ 1vk1 C1 : : : vk2 /O : : : O ._.;; tn1 / ˝ 1vkn2 C1 : : : vkn1 / O ._.;; ;/ ˝ 1  1// G .tn ˝ vkn1 C1:::kn / : The uniqueness of the morphism is obtained by construction.



Dualising the notion of terplicial algebras, into terplicial coalgebras gives: Definition 5.8. A terplicial coalgebra, or coterplicial coalgebra, is a vector space C endowed with three coproducts G ; O ; ÍF W C ! C ˝ C satisfying the following relations: ÍF and O are co-associative, .G ˝ id / ı G D .id ˝ ÍF / ı G ; .ÍF ˝ id / ı G D .id ˝ G / ı ÍF ; .O ˝ id / ı G D .id ˝ G / ı O ; .ÍF ˝ id / ı O D .id ˝ O / ı ÍF ; .G ˝ id / ı O D .id ˝ ÍF / ı O :

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A co-augmented conilpotent terplicial coalgebra C is a coalgebra verifying that: C D [n0 Fn C where F0 C D K, F1 C D fx 2 C j G .x/ D O .x/ D ÍF .x/ D 0g, Fn C D fx 2 C j G .x/ 2 Fn1 C ˝2 ; O .x/ 2 Fn1 C ˝2 ; ÍF .x/ 2 Fn1 C ˝2 g. Example 5.9. We introduce the dual coproduct associated to the products G, O and Í F. They are given inductively on T D _.t1 ; : : : ; tn / by:  if t1 D ;, ÍF .T / D 0  if tn D ;, G .T / D 0  if n 2, O .T / D 0 and, otherwise, ÍF .T / D t1 ˝ _.;; : : : ; tn / C ÍF .t1 /1 ˝ _.ÍF .t1 /2 ; : : : ; tn /; O .T / D

n1 X

_.t1 ; : : : ; ti 1 ; ;/ ˝ _.ti ; : : : ; tn / C _.t1 ; : : : ; ti / ˝ _.;; ti C1 ; : : : ; tn /

i D2

C _.t1 ; : : : ; ti 1 ; ÍF .ti /1 / ˝ _.ÍF .ti /2 ; ti C1 ; : : : ; tn / ; G .T / D _.t1 ; : : : ; tn1 ; ;/ ˝ tn C _.t1 ; : : : ; tn1 ; ÍF .tn /1 / ˝ ÍF .tn /2 : It is by duality, the cofree conilpotent terplicial coalgebra on one generator, which can be extended to the cofree conilpotent terplicial coalgebra on generators by decorating the leaves, in a similar manner as the free tridendriform algebras. 5.3 Rigidity theorem for bialgebras endowed with terplicial and tridendriform structures 5.3.1 Terplicial bialgebras Definition 5.10. A terplicial bialgebra is a vector space H endowed with a terplicial algebra structure .H; Í F; O; G/ and a co-terplicial coalgebra structure .H; ÍF ; O ; G / satisfying the following mixed ditributive laws: ÍF .T Í FS / D T ˝ S C ÍF .T /1 ˝ ÍF .T /2 Í FS C T Í FÍF .S /1 ˝ ÍF .S /2 ÍF .T OS / D ÍF .T /1 ˝ ÍF .T /2 OS ÍF .T G S / D ÍF .T /1 ˝ ÍF .T /2 G S O .T Í FS / D T Í FO .S /1 ˝ O .S /2 O .T OS / D T ˝ S C O .T /1 ˝ O .T /2 OS C T OO .S /1 ˝ O .S /2 C G .T / ˝ G .T /2 Í FS C T G ÍF .S /1 ˝ ÍF .S /2 O .T G S / D O .T /1 ˝ O .T /2 G S G .T Í FS / D T Í FG .S /1 ˝ G .S /2 G .T OS / D T OG .S /1 ˝ G .S /2 G .T G S / D T ˝ S C G .T /1 ˝ G .T /2 Í FS C T G ÍF .S /1 ˝ ÍF .S /2 :

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Proposition 5.11. These above relations are a confluence law. The only property that has to be checked is the compatibility with the operad structure. This comes from the fact that the relations are computed on planar rooted trees. The proof is postponed to the section 7.4.2 on the combinatorial description of the products and coproducts. Applying [1, corollary 2.1.4]: Proposition 5.12 (Rigidity theorem for coterplicial terplicial bialgebras). Any connected terplicial co-terplicial bialgebra is free and cofree over its primitive elements. The bialgebra T1 is then free as a terplicial algebra and cofree as a connected terplicial coalgebra over one element. 5.3.2 Co-terplicial tridendriform bialgebras Definition 5.13. A co-terplicial tridendriform bialgebra is a vector space H endowed with a terplicial coalgebra structure .H; ÍF; O ; G /, a tridendriform algebra structure .H; ; P; / satisfying the following mixed ditributive laws: ÍF .T  S / D .ÍF .T //1 ˝ .ÍF .T //2  S ÍF .T  S / D .ÍF .T //1 ˝ .ÍF .T //2  S ÍF .T  S / D T ˝ S C .ÍF .T //1 ˝ .ÍF .T //2  S C T .ÍF .S //1 ˝ .ÍF .S //2 O .T  S / D .O .T //1 ˝ .O .T //2  S O .T  S / D T ˝ S C .O .T //1 ˝ .O .T //2  S C T  .O .S //1 ˝ .O .S //2 C .G .T //1 ˝ .G .T //2  S C T  .ÍF .S //1 ˝ .ÍF .S //2 O .T  S / D T  .O .S //1 ˝ .O .S //2 G .T  S / D T ˝ S C .G .T //1 ˝ .G .T //2 S C T  .ÍF .S //1 ˝ .ÍF .S //2 G .T  S / D T  .G .S //1 ˝ .G .S //2 G .T  S / D T  .G .S //1 ˝ .G .S //2 where D C  C. Proposition 5.14. The relations introduced in the previous definition are a confluence law. The property that has to be checked is the compatibility with the operadic structure. This comes from the fact that the these relations are computed on rooted planar trees. The proof of these relations are then postponed to section 7.4.2 using the combinatorial description of the terplicial products and coproducts. We apply the result of [1, corollary 2.1.4]: Proposition 5.15 (Rigidity theorem for coterplicial tridendriform bialgebras). Any connected tridendriform co-terplicial bialgebra is free and cofree over its primitive elements.

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Therefore the bialgebra T1 is free as a tridendriform algebra on one element, and cofree as a connected co-terplicial algebra on one element. 5.3.3 Tridendriform bialgebras Tridendriform-tridendriform bialgebras: Let us denote 1 D  , kC1 D .id k ˝  / ı k , and 1 D , kC1 D k ı .id k ˝ /; where  stands for one of the symbols ;  or . P For the sake of readability, we will denote ˛ WD  C k1 .1/k .id ˝ / ı P k k kC1 kC1  , ˛ D  C k1 .1/ . ˝id / ı  . Definition 5.16. A co-tridendriform tridendriform bialgebra is a vector space H endowed with a co-tridendriform coalgebra structure .H;  ;  ;  / and a tridendriform algebra structure .H; ; ; / linked by the confluence laws given by: for any x; y 2 H  .x  y/ D  .x ˛ .y/1 /1 ˝  .x ˛ .y/1/2  ˛ .y/2 C x ˛ .y/1 ˝ ˛ .y/2  .x  y/ D  .x/1 ˝  .x/2  y  .x  y/ D  .x/1 ˝  .x/2  y  .x  y/ D x   .y/1 ˝  .y/  .x  y/ D  .x/ ˝  .x/2  y C x   .y/1 ˝  .y/2 C C ˛ .x/1  . .˛ .x/2 ˛ .y/1 /1 / ˝ . .˛ .x/2 ˛ .y/1 /2 /  ˛ .y/2  .x  y/ D  .x/1 ˝  .x/2  y  .x  y/ D x   .y/1 ˝  .y/2  .x  y/ D x   .y/1 ˝  .y/2  .x  y/ D ˛ .x/1   .˛ .x/2 y/1 ˝  .˛ .x/2 y/2 C ˛ .x/1 ˝ ˛ .x/2 y The definition is considered for q D 1 but can be extended for any q. Proposition 5.17. The above relations are a confluence law. Proof. The only property that has to be checked is the compatibility with the operadic structure. This comes from the fact that these relations are computed on planar rooted trees. Prove by induction that for any tree T D _.t1 ; : : : ; tn / the element ˛ .T / is equal to t1 ˝ _.;; t2 ; : : : ; tn /, respectively, ˛ .T / is equal to _.t1 ; t2 ; : : : ; tn1 ; ;/ ˝ tn . The induction is on the cofiltration with respect to  ;  respectively. Moreover P ˛ D   . ı˛ ˝  ı˛ / ı  applied to T is given by n1 i D2 _.t1 ; : : : ; ti / ˝  _.;; ti C1 ; : : : ; tn / C _.t1 ; : : : ; ti 1 ; ;/ ˝ _.ti ; t2 ; : : : ; tn /.

Structure theorems for dendriform and tridendriform algebras

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Applying [1, corollary 2.1.4]: Theorem 5.18 (Rigidity theorem for co-tridendriform-tridendriform bialgebras). Any connected co-tridendriform tridendriform bialgebra is free and cofree over its primitives.

6 Application to the freeness of some tridendriform algebras 6.1 Application to the freeness of the Solomon-Tits algebra as a tridendriform algebra. The Solomon-Tits algebra can be endowed with a tridendriform structure, see for example [23, 2]. We keep the notations taken in section 4.1. The concatenation product  W STrn ˝ STsm ! STrCs nCm is given by the formula: f  g WD .f .1/; : : : ; f .n/; g.1/ C r; : : : ; g.m/ C r/: Similarly, for K D fj1 <    < jl g f1; : : : ; rg, the co-restriction of x to K is denoted xjK WD std.x.s1 /; : : : ; x.sq //, for x 1 .K/ D fs1 <    < sq g. For an element x 2 STrn , we denote by .x/ the cardinal of x 1 .frg/. Suppose that x 1 .r/ D fj1 <    < j .x/ g, and let x 0 2 STr1 nk be the coQ 0 f1;:::;r1g 0 . We denote x as x D j1    > f .n/. 2. SH .n1 ; : : : ; np / the subset of all surjective maps f 2 SH.n1 ; : : : ; np / such that f .n1 / < f .n1 C n2 / <    < f .n/. 3. SH .n1 ; : : : ; np / the subset of all surjective maps f 2 SH.n1 ; : : : ; np / such that f .n1 / D f .n1 C n2 / D    D f .n/. 4. SH< .n1 ; : : : ; np / the subset of all surjective maps f 2 SH.n1 ; : : : ; np / such that f .n1 / f .n1 C2 /    f .n/. Let x 2 STrn ; y 2 STsm , the tridendriform structure on ST is defined as follows: X X f ı .x  y/; x  y WD f ı .x  y/; x  y WD f 2SH  .r;s/

x  y WD

X

f 2SH  .r;s/

f 2SH  .r;s/

f ı .x  y/:

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The work of Vong [27] can be understood as a construction of a terplicial algebraic structure on STsm . For x 2 STrn ; y 2 STsm define the operations Y Y x0  y xOy D x0  y0 xGy D x j1x ;:::;j.x/

y

y

x j1x ;:::;j.x/ ;j1 ;:::;j.y/

x Í Fy D x  y Q where x D j x ;:::;j x x 0 and y D j y ;:::;j y y 0 . 1 1 .x/ .x/ The relations are checked by direct inspection. In this section, we focus on proving the freeness of ST as a free tridendriform algebra and terplicial algebra by adding a co-terplicial structure to ST which is dual to the terplicial structure. The confluence laws are those introduced above. This viewpoint permits to give a way to understand the double application of Grœbner basis algorithm as seen in the work of Vong as terplicial-tridendriform or terplicialterplicial bialgebra structure. The rigidity theorem guarantees that the reductions provided will fit the bill which is one of the most tricky challenge when trying to find the “good” reductions. The co-terplicial structure on ST is combinatorially constructed as follows: for x 2 STrn there is a unique way to describe it as x1  : : :  xp such that every xi is irreducible that Q do not exists u; v 2 ST such that xi D u  v. Qis to say that there Suppose x D j1 ;:::;j.x/ x 0 D j1 ;:::;j.x/ u1  : : :  uq where u1  : : :  uq is the irreducible decomposition of x 0 , ui 2 ST smi i . Denote by U1 D u1  : : :  up1 the Q decomposition x D j1 ;:::;j.x/ U1  upC1  : : :  uq with m1 C : : : mp1 1 C .x/ < j .x/ m1 C : : : mp1 C .x/ Q

G .x/ D ÍF .x/ D

X

Y

i

j1 ;:::;j.x/

X

x1  : : :  xi ˝ xi C1  : : :  xp

i

O .x/ D

  U1  up1 C1  : : :  up1 Ci ˝ std.up1 Ci C1  : : :  uq /

X Y  i;l j1 ;:::;ji

 u1  : : :  ul ˝

Y

  ulC1  : : :  uq

ji C1 ;:::;j.x/

where the last sum runs over i; l such that m1 C : : : C ml1 < ji m1 C : : : C ml . The relations are checked as these cooperations are the dual of the terplicial operations. Proposition 6.1. The Solomon-Tits algebra is endowed with a terplicialc -terplicial bialgebra structure and a terplicialc -tridendriform bialgebra structure. Proof. The terplicialc -terplicial mixed distributive relations are verified. The proof decomposition of an element into x D ux1 : : : Q is based on the unique x uw.x/  j1 ;:::;j.x/ uw.x/C1  uq D ux1  : : :  uxq and the definition of the terplicial

Structure theorems for dendriform and tridendriform algebras

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operations which for two elements x and y will only modify the place of the maxima of x and y according to the operation considered. For example the relation: O .xOy/ D O .x/1 ˝ O .x/2 Oy C G .x/1 ˝ G .x/2 Í Fy C xOO .y/1 ˝ O .y/2 C x G ÍF .y/1 ˝ Í F.y/2 C x ˝ y is satisfied as the product O levels Q the maxima of x and y while keeping the overall structure of x 0  y 0 with x D x 0 . The terplicialc -tridendriform distributive relations Q are verified. The proof is based on the unique decomposition of an element x D j1 ;:::;j.x/ ux1  : : :  uxq and the definition of the tridendriform operations which keeps the overall order (in regards to y y ) of ux1  : : :  uxq in x and of u1  : : :  uq in y.  Theorems 5.12 and 5.15 give as a corolla that: Proposition 6.2. The algebra of ST is free as terplicial algebra and free as a tridendriform algebra.

7 Combinatorial description of the products, coproducts and confluence law on T1 7.1 Path cutting and stuffled paths 7.1.1 Coproducts indexed by a cutting path: Let T D _.t1 ; : : : ; tn / a tree and let q be a path in T from the root to a leaf, and denotes its edges .e1 ; : : : ; ek /. We will refer to q as a cutting path and define the cut of a tree through this path q. Intuitively we duplicate the cutting path and rearrange the left hand-side of the cutting path with the cutting path included by moding out unary edges in order to make it into a tree, and do the same on the right hand-side. Thus, giving us the both trees needed for the coproduct. It is defined as follows: If the path q is the leftmost path of T , namely lT , define lT .T / as: lT .T / D ; ˝ T : If the path q is the rightmost path of T , namely rT , define rT .T / as: rT .q/ D T ˝ ; : If q is neither the leftmost nor the rightmost path define q .T / as: 8 .e2 ;:::;ek / .t1 /1 ˝ _..e2 ;:::;ek / .t1 /2 ; t2 ; : : : ; tn / ˆ ˆ ˆ ˆ ˆ if e1 is the leftmost edge attached to the root < _.t1 ; : : : ; .e2 ; ;ek / .tn /1 / ˝ .e2 ; ;ek / .tn /2 q .T / D if e1 is the rightmost edge attached to the root ˆ ˆ ˆ _.t ; : : : ; t ;  ˆ 1 i 1 .e2 ;:::;ek / .ti /1 / ˝ _..e2 ;:::;ek / .ti /2 ; ti C1 ; : : : ; tn / ˆ : if e1 is the i th edge attached to the root from left to right.

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7.1.2 Products indexed by a stuffle path: Consider two trees T D _.t1 ; : : : ; tn /, S D _.s1 ; : : : ; sm /. Consider the rightmost path of T , number its edges .1; : : : ; n/ ordered from root to leaf, and the leftmost path of S , number its edges .nC1; : : : ; nC m/ ordered from root to leaf, with n; m integers. Consider the sequence of edges noted .epi / resulting in the inverse image of a SH.n; m/ stuffle – recalled in section 6.1– of the edges of the two paths. The sequence will start with 1 or n C 1 or the set f1; n C 1g, and will be referred to as a stuffle path of T and S . Define the product T ?p S of two trees accordingly to the stuffle path p D 1 .e ; : : : ; ek / of T and S inductively as follows: 8 if e1 D 1 < _.t1 ; : : : ; tn1 ; tn ?.e2 ;:::;ek / S / _.T ?.e2 ;:::;ek / s1 ; s2 ; : : : ; sm / if e1 D f1; n C 1g T ?p S D : _.t1 ; : : : ; tn1 ; tn ?.e2 ;:::;ek / s1 ; s2 ; : : : ; sm / if e1 D n C 1 Example 7.1. Consider T and S as represented below where ti are subtrees of T and si are subtrees of S . Denote the edges of the rightmost path of T  by .1; : : : 4/ and the edges of the leftmost path of S by .5; 6; 7/. Consider the product of T and S indexed by the stuffle path p D .1; 5; 6; f2; 7g; 3; 4/. Then, t4 t3

t4 t3 t2

T D

t1

1

:

:

:

t2

s3

4

7

:: :

,S D

s2

s1

s1 5

s3 s2

and T ?p S D

t1 .

7.2 Combinatorial description of the terplicial structure on T1 via stuffle paths and trimming edges. Consider two planar rooted trees T; S , with rT the rightmost path of T with edges denoted by .1; : : : ; k/ and lS the leftmost path of S with edges denoted .k C 1; : : : k C l/. The terplicial operations defined in Theorem 5.6 verify: T G S D T ?.1;kC1;:::;kCl;2;:::;k/ S ; T Í FS D T ?.kC1;:::;kCl;1;:::;k/ S

T OS D T ?.f1;kC1g;kC2;:::;kCl;2;:::;k/ S ;

The terplicial cooperations are dual to the above operations and defined by induction with pruning over the edges attached to the root. 7.3 Combinatorial description of the tridendriform operations and cooperations via stuffle paths and cutting paths. For two trees T; S with the edges of the rightmost path of T numbered from 1 to m and the edges of the leftmost path of S numbered from mC1 to mCn, from root to leaf. We will denote the set of stuffle P.T; S /, the subset of sequences starting with 1 will be denoted P< .T; S / , the subset starting with m C 1 will be denoted P> .T; S /, and the subset of sequences starting with

Structure theorems for dendriform and tridendriform algebras

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f1; m C 1g will be denoted PD .T; S /. The tridendriform operations defined in 5.2 are described as follows: Let T and S be two rooted planar trees then X X T S D T ?p S ; T  S D T ?p S ; p2P.T;S/

X

T S D

p2P< .T;S/

T ?p S ; T  S D

p2PD .T;S/

X

T ?p S

p2P> .T;S/

It is immediate by induction through description of T ?p S and the constructive way to define the operation in 5.2. For p 2 SH.n; m/ one can associate a path pQ in the product T ?p S that will be referred to as a stuffle path. In the following sequel, we will consider the associated paths instead of the stuffle as we will consider intersections of cutting paths and shuffle paths to describe the confluence laws. For a tree T denote the set of cutting paths Q.T /, the subset of sequences which start with the leftmost edge of T will be denoted Q> .T /, the subset of sequence which start with the rightmost edge of T will denoted Q< .T /, and the subset of sequences which start with neither the rightmost nor the leftmost edge of T will be denoted QD .T /. The co-tridendriform cooperations defined in 5.4 are described as follows: X X  .T / D q .T / ;  .T / D q .T / q2Q.T /

 .T / D

X

q2QD .T /

q2Q< .T /

q .T / ;  .T / D

X

q .T /

q2Q> .T /

It is immediate by induction considering the description of q .T / for a cutting path q and the constructive description of the coproducts in 5.4. Example 7.2. Consider Figure 2. For a given tree T , we describe the set of cutting path Q.T / with the colours to indicate their belonging to the subsets Q> .T / if the cutting path is in red, QD .T / if the cutting path is in green, and Q< .T / if the cutting path is in blue. Then we give the associated coproducts as sum of q for a cutting path following the left to right order given in the description of Q. 7.4 Confluence laws of the bialgebra structures on T1 . In this section, we will investigate the confluence laws on the tridendriform-tridendriform bialgebra structure on T1 combinatorically and prove that the coefficients respect the Delannoy series. As in the section 2.2, the coefficients will depend on the cofiltration. The first step is to compute the confluence laws for the products indexed by a stuffle path with the coproducts indexed by a cutting path.

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T D

Q.T / D f

,

,

,

,

˝

,

,

,

g:

,  .T / D

„

˝

C

˝

C

˝

C

.

 .T / D  D

˝

C ˝

C

˝

C ˝

˝

C

. ˝

.

Figure 2.  ,  et  in terms of thickened paths.

7.4.1 Confluence law for a product indexed by a stuffle path and a coproduct indexed by a cutting path: Let T and S be two trees. Consider p a stuffle path in P.T; S /, and q a cutting path of Q.T ?p S /. The intersection between the two paths, p \ q, is a (possibly empty) path in T ?p T with edges E.p/ \ E.q/. Consider also the remaining edges of the cutting path composed of the edges E.q/ n .E.p/ \ E.q// and denoted q c . Lemma 7.3. q c is a path, with edges strictly in T or in S or empty. We will denote qT (resp. qS ) when q c is a path of T (resp. S ). Proof. The proof is analogous to the proof of lemma 2.3: it is proven by induction on the number of leaves. It is clear in low dimensions. Suppose the property true for trees such that the sum of their leaves is equal to n. Consider two trees T D _.t1 ; : : : ; tn / and S D _.s1 ; : : : ; sm / with total number of trees n C 1. Let the edges of the stuffle path be denoted .pi /1i k and the edges of the cutting path .q i /1i l . Two cases can occur: q1 D p1 then consider the stuffle path .p2 ; : : : ; pk / and the cutting path .q 2 ; : : : ; ql / in the trees T 1 D t n and S 1 D S if p1 2 E.T / n E.S /, in the trees T 1 D T and S 1 D s1 if p1 2 E.S / n E.T / and in the trees T 1 D tn and S 1 D s1 if p1 2 E.T / \ E.S / and conclude by induction. If q 1 ¤ p1 suppose moreover that p1 is an edge of the rightmost path of T , denoted rT , but not an edge of the leftmost path of S , denoted lS . The symmetric case where p1 is an edge of lS but not an edge of rT is analogous. Then by construction,

Structure theorems for dendriform and tridendriform algebras

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q 1 is an edge attached to the root of T which is not the rightmost edge. Let say it is the edge such that to grows from it. So E.q/  E.T / n E.rT / as q .T ?p S / D _.t1 ; : : : ; to1 ; .q 2 ;:::;q l / .to /1 //˝_..q 2 ;:::;q l / .to /1 /; toC1 ; : : : ; tn ?p2 ;:::;pk S / proving that q is a a path with edges strictly in T . Suppose that p1 is the first edge of rT identified to the first edge of lS , then one concludes by induction on the trees T 1 D tn , S 1 D s1 , with the stuffle path .p2 ; : : : ; pk / and cutting path .q 2 ; : : : q l /.  As mentioned above the coproduct of a tree along a cutting path can be understood as duplicating the cutting path to give each side of the tensor. We therefore need to take more notations to describe the confluence law: When q c is a path of T we will denote by ŒqT the path in T defined by the sequence of edges E.q/ \ E.T / and ŒqS is defined analougously. Denote pc the remaining of the shuffle path in T ?p S , i.e. the path defined by the sequence of edges of E.p/n.E.p/\E.q//. The intersection path p\q has edges in S and in T . We will denote by pS the sequence of edges .E.p/\E.q/\E.S //[E.pc / which is the trace of the shuffle path in S , and analogously defined pT the sequence of edges .E.p/ \ E.q/ \ E.T // [ E.pc /. Lemma 7.4.

8
.S/ q D qS i

D.i; j  1/ q .T /1 ˝ q .T /2 ?p S

eT p 2 P> .T i ; S/ p D pS j

X

eT p 2 P.T; S i / p D pT j

D.i; j  1/ T ?p q .S/1 ˝ q .S/2

64

E. Burgunder and B. Delcroix-Oger  .T  S/ D D.RT  1; LS  1/T ˝ S C  .T /1 ˝  .T /2  S C T   .S/1 ˝  .S/2 X X C D.i  1; j  1/ q .T /1 ˝ q .T /2 ?p S q 2 Q< .T / q D qT i

C

X

eT p 2 P.T i ; S/ p D pS j

X

q 2 Q> .S/ q D qT i

D.i  1; j  1/ T ?p q .S/1 ˝ q .S/2

eT p 2 P.T; S i / p D pS j

 .T  S/ D D.RT  1; LS / T ˝ S X X C q 2 Q< .T / q D qT i

C

X q 2 Q.S/ q D qS i

D.i  1; j / q .T /1 ˝ q .T /2 ?p S

eT p 2 P.T i ; S/ p D pT j

X

D.i  1; j / T ?p q .S/1 ˝ q .S/2

eT p 2 P> .T; S i / p D pT j

 .T  S / D  .T /1 ˝  .T /2  S  .T  S / D  .T /1 ˝  .T /2  S  .T  S / D T   .S /1 ˝  .S /2  .T  S / D  .T /1 ˝  .T /2  S  .T  S / D T   .S /1 ˝  .S /2  .T  S / D T   .S /1 ˝  .S /2 Proof. The description of the tridendriform products through stuffle paths and the co-tridendriform coproducts through cutting paths in Section 7.3 gives X X  .T  S / D q .T ?p S / p2P> .T;S/ q2Q> .T ?p S/

Then apply lemma 7.4. The coefficients appear as for a stuffle path p and a cutting path q, consider any stuffle of the edges .E.p/ \ E.q// \ E.T / and .E.p/ \ E.q// \ E.S /. Consider any path p0 obtained from p, and any cutting path q 0 obtained from q, such that only the edges .E.p/ \ E.q// \ E.T / and .E.p/ \ E.q// \ E.S / are stuffled, then, q 0 .T ?p0 S / will give the same element as q .Tp S /. 

References [1] E. Burgunder and B. Delcroix-Oger, Rigidity theorem, freeness of algebras and applications, arXiv: 1701.01323.

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[2] E. Burgunder and M. O. Ronco, Tridendriform structure on combinatorial Hopf algebras, J. Algebra 324(10) (2010), 2860–2883 . [3] E. Burgunder, P.-L. Curien, and M. O. Ronco, Free algebraic structures on the permutohedra, J. Algebra 487, 20–59. [4] C. Brouder and A. Frabetti, QED Hopf algebras on planar binary trees, J. Algebra 267(1) (2003), 298–322. [5] F. Chapoton, Opérades différentielles graduées sur les simplexes et les permutoèdres, Bull. Soc. Math. Fr. 130 (2002), 233–251. [6] F. Chapoton, On the Coxeter transformations for Tamari posets, Canad. Math. Bull. 50.2 (2007), 182–190. issn:0008-4395. [7] G. Châtel G. and V. Pons, Counting smaller elements in the Tamari and m-Tamari lattices, J. Combin. Theory Ser. A 134 (2015), 58–97. [8] G. Châtel G., Algebraic combinatorics on order of trees, Theses, Université Paris-Est, (2015). [9] A. Chouria and J.-G. Luque, r-Bell polynomials in combinatorial Hopf algebras, CRM (2016). [10] L. Foissy, Bidendriform bialgebras, trees and free quasi-symmetric functions, J. Pure Appl. Algebra 209.2 (2007), 439–459. [11] L. Foissy, Ordered forests and parking functions, IMRN 7, (2012), 1603–1633. [12] T. Fox and M. Markl, Distributive laws, bialgebras, and cohomology, Operads: Proceedings of Renaissance Conferences, Contemp. Math. 202 (1997), 167–205. [13] T. Lam and P. Pylyavskyy, Combinatorial Hopf algebras and K-Homology of Grassmannians, Int. Math. Res. Notices Vol. 2007. [14] Ph. Leroux An algebraic framework of weighted directed graphs, Int. J. Math. Math. Sci. 58 (2003). [15] J.-L. Loday, Dialgebras, Dialgebras and related operads, Springer Lecture Notes in Math. 1763 (2001), 7–66. [16] J.-L. Loday, Generalized bialgebras and triples of operads, Astérisque 320 (2008), x+116 pp. ISBN: 978-2-85629-257-0. [17] J.-L. Loday and M. O. Ronco, On the structure of cofree Hopf algebras, J. Reine Angew. Math. 595 (2006), 123–155. [18] J.-L. Loday and M. O. Ronco., Hopf algebra of the planar binary trees, Adv. Math. 139.2 (1998), 293–309. [19] J.-L. Loday and M. O. Ronco, Trialgebras and families of polytopes, Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Contemp. Math. 346, (2004), AMS. [20] A. Mansuy, Algèbres de greffes, Bull. Sci. Math. 136 (2012). [21] M. A. Mendez, Set operads in combinatorics and computer science, Springer Briefs in Mathematics, Springer, Cham, 2015, 20. [22] J.-C. Novelli and J.-Y. Thibon, Duplicial algebras and Lagrange inversion, arXiv:1209.5959. [23] J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fund. Math. 193 (2007), 189–241. [24] N. J. A. Sloane, The on-line encyclopedia of integer sequences, (2016), https://oeis.org/.

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[25] D. Tamari, The algebra of bracketings and their enumeration., Nieuw Arch. Wisk. 10(3) (1962) [26] V. Vong Combinatorial proofs of freeness of some P-algebras, DMTCS proc., FPSAC 2015 (2015). [27] V. Vong, On (non-) freeness of some tridendriform algebras, DMTCS proc. FPSAC 2016 (2016).

A group-theoretical approach to conditionally free cumulants Kurusch Ebrahimi-Fard and Frédéric Patras Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conditional freeness . . . . . . . . . . . . . . . . . . . . . . 3 Shuffle algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Shuffle and half-shuffle exponentials and logarithms 5 Monotone, free and boolean cumulants . . . . . . . . . 6 Conditionally free cumulants revisited . . . . . . . . . . 7 Conditionally free convolution . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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67 70 71 76 81 86 88 90

1 Introduction Voiculescu [46, 47] introduced in the 1980s the theory of free probability. It is based on the notion of free independence, or freeness, that is, the absence of algebraic relations. Free cumulants encode the notion of free independence. Speicher [40] uncovered a combinatorial approach to free cumulants based on the lattice of noncrossing set partitions and its Möbius calculus. The reader is referred to [34, 35, 42] for introductions and reviews. The free moment-cumulant relation for the n-th univariate moment mn is given by X Y kji j ; (1.1) mn D 2NCn i 2

where NCn is the lattice of non-crossing set partitions WD f 1 ; : : : ; j g of Œn WD f1; : : : ; ng and j i j denotes the number of elements in the block i 2 . Here kl denotes the l-th free cumulant. Analogous statements hold for monotone [26] and boolean cumulants [39]. Indeed, for monotone cumulants hl one has the monotone moment-cumulant relation X 1 Y mn D hj j : (1.2)  . /Š  2 i 2NCn

i

The so-called tree (forest) factorial  . /Š corresponds to the forest  . / of rooted trees associated to the nesting of the blocks of the non-crossing partition 2 NCn .

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K. Ebrahimi-Fard and F. Patras

See [3] for details. Boolean cumulants rl satisfy the boolean moment-cumulant relation X Y mn D rjli j ; (1.3) I 2Bn li 2I

where Bn is the boolean lattice of interval partitions. The multivariate generalisations of these moment-cumulant relations will be given further below in the shuffle algebra setting. Relations between the different cumulants have been studied in great detail by Arizmendi et al. in the recent article [3]. The framework for our group-theoretical approach to free, boolean and monotone moment-cumulant relations has been developed in a series of recent works [15, 16, 17, 18, 19]. It is based on a particular graded, connected, non-commutative, noncocommutative word Hopf algebra H defined on the double tensor algebra over a non-commutative probability space .A; '/ with linear unital map 'W A ! C. One may characterise H as a non-cocommutative generalisation of the classical cocommutative unshuffle Hopf algebra [36]. In [15] we first defined the coproduct of H . It is right-sided (right-handed in Turaev’s original terminology) [28, 33, 44] and splits into left and right half-coproducts, which define a structure of unshuffle bialgebra on H [21]. This implies, on the other hand, a splitting of the associative convolution product on the graded dual H  into two non-associative half-shuffles, which define a non-commutative shuffle algebra (aka dendriform algebra) structure on H  . It follows from the existence of these structures that, besides the classical exponential exp , two other exponential-type maps, denoted E and E , can be defined in terms of the two half-shuffles. This allows for a refinement of the classical correspondence between a group and its Lie algebra, by means of the exponential map since all three maps define bijections from the Lie algebra g  H  of infinitesimal characters to the group G  H  of algebra characters on H . Once a particular character ˆ 2 G has been defined in terms of the natural extension of ' from A to H , monotone, free, and boolean cumulants can be considered as infinitesimal characters , , ˇ in g, respectively, and are defined in terms of the identities ˆ D exp./ D E ./ D E .ˇ/:

(1.4)

In [19] we showed that, indeed, the left and right half-shuffle exponentials, E ./ respectively E .ˇ/, give rise to free respectively boolean multivariate moment-cumulant relations [39, 40]. The shuffle exponential exp./ describes instead the monotone moment-cumulant relations [26]. This yields a novel and unifying approach to moment-cumulant relations in non-commutative probability. In [19] we also showed how three logarithm-type maps (R-transformations) corresponding to these exponentials together with a particular shuffle group-theoretical adjoint operation permit to recover relations between cumulants, which were described explicitly in [3] using classical Möbius calculus on non-crossing set partitions. Regarding the latter an important remark is in order. In our group-theoretical approach non-crossing partitions enter the picture only through the closed formulas for the evaluation of the exponential-type maps exp , E and E on words from H . For instance, the right-hand side of the

A group-theoretical approach to conditionally free cumulants

69

univariate free moment-cumulant relation (1.1) results from calculating E ./.w/ for a word w D a˝n 2 H of length n in a single letter, i.e., random variable, a 2 A. There is a well-known connection between moment-cumulant relations in classical probability and relations between Green’s respectively connected Green’s functions (e.g., in perturbative quantum field theory (QFT)). It extends to free momentcumulant relations and relations between Green’s respectively connected Green’s functions in planar QFT (see, e.g., [17]). Still in perturbative QFT, replacing the vacuum by other, non-trivial, ground states gives rise to interesting phenomena meaningful to applications, see, e.g. [6, 7, 13]. The notion of conditionally free probability, introduced in [5] by Boz˙ ejko et al., shares with perturbative QFT over non-trivial vacua a key feature (the analogy stops there at the moment but deserves to be further analysed): the idea to consider a theory of free probabilities relative to a given arbitrary state or, equivalently, to consider the behaviour of a pair of states (where, however, the two states have different roles). This conditional extension of Voiculescu’s theory allows, among others, for the definition of a conditionally (or c-)free convolution product and R-transform, for the explicit calculation of distributions of conditionally free Gaussian and free Poisson distributions and other similar key behavioural properties one expects for a generalised free probability theory. The paper at hand shows how the combinatorial side of c-free cumulants is naturally captured by the group-theoretical picture sketched above. In this respect the aforementioned shuffle group-theoretical adjoint action, which permits to express monotone, free, and boolean cumulants in terms of each other, is the central object. Indeed, we will show how it allows to relate c-free cumulants with free and boolean, and therefore also with monotone cumulants. Finally, we remark that several works have appeared in recent years, applying Hopf algebra techniques in the context of free probability [22, 23, 31, 32]. Moreover, non-commutative shuffle algebras appeared in the work by Belinschi et al. [4] in relation to the problem of the infinite divisibility of the normal distribution with respect to additive convolution in free probability. However, our approach is rather different, and potential connections have to be explored in the future. The paper is organised as follows. In Section 2 we survey briefly the foundations of the theory of c-free probability. In Section 3 we present the necessary background on non-commutative shuffle algebras together with the particular combinatorial Hopf algebra (denoted H in the Introduction) as main example. This Hopf algebra will provide the underlying framework for our shuffle group-theoretical approach to momentcumulant relations. Section 4 introduces the three exponential bijections, which provide the group-theoretical setting for free, boolean and monotone moment-cumulant relations. The next section recalls the shuffle algebra approach to the latter. Section 6 contains the main result of the paper. It describes conditionally free cumulants and convolution using the group-theoretical machine introduced in Sections 4 and 5. Acknowledgements: We would like to thank the organisers of the CARMA 2017 workshop at CIRM in Luminy and the CNRS PICS project: Algèbres de Hopf combinatoires et probabilités non commutatives for its support.

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2 Conditional freeness We first fix some notation. Let NCn denote the lattice of non-crossing set partitions of order n and Bn the boolean lattice of interval partitions of order n. Recall that a partition D 1 t    t k of Œn is non-crossing if and only if there are no .i; l/ 2 a  a and .j; m/ 2 b  b , 1 a; b k with a 6D b and i < j < l < m. A partition is boolean if each of its blocks i is an interval, i.e., if a; b 2 i and a < c < b, then c 2 i . We let j i j denote the number of elements in the block i 2 . The elements in each block i 2 are naturally ordered, i WD fj1 <    < jji j g, and we set ai WD aj1    ajji j . See [35] for details. Conditionally free probability generalizes the fundamental notion of Voiculescu’s free probability theory in the context of two states. We introduce here briefly the main constructions relevant for our later purposes and refer the reader to the original article [5] for details. We work in the framework of unital algebras A and a state on A simply means a unital linear form ' W A ! C (unital meaning that '.1/ D 1). Given now unital algebras Ai ; i 2 I , each equipped with a pair of states .'i ; i /, their free product reads 1 M M AC ˝    ˝ AC ;

i 2I Ai Š C  1 ˚ i.1/ i.n/ nD1 i.1/DDi.n/

where AC j WD ker and

j.

A new state ' is then defined on i 2I Ai by requiring '.1/ D 1 '.a1    an / D 'i.1/ .a1 /    'i.n/ .an /;

and i.1/ D    D i.n/: Notice that the role of the 'i and that of the when aj 2 AC i.j / are not symmetrical. i Suppose that A D ChX i. The (free) moments of the states ' and on A are defined by m'n WD '.X n /; mn WD .X n /: Speicher’s free moment-cumulant relations (which define implicitly the free cumulants kn ) are then given by mn D

n X

X

pD1

l.1/;:::;l.p/0 l.1/CCl.p/Dnp

kp ml.1/    ml.p/

or, in terms of non-crossing partitions: mn D

X

Y

2NCn l 2

kjl j :

A group-theoretical approach to conditionally free cumulants .'; /

Conditionally (or c-)free cumulants Rk m'n D

n X

X

j D1

l.1/;:::;l.j /0 l.1/CCl.j /Dnj

for the pair .'; / are defined instead by: .'; /

Rj

'

ml.1/    ml.j 1/ ml.j / ;

or, in terms of non-crossing partitions: Y .'; / X Y kjl j Rjl j : m'n D 2NCn

71

l 2 l inner

(2.1)

l 2 l outer

Here, the terms “outer” and “inner” refer to the structure of non-crossing partitions. A block i of 2 NCn is inner if there exists a j and a; b 2 j such that a < c < b for all c 2 i . A block which is not inner is outer. As in classical free probability, cumulants characterise free convolution in the sense that the distribution of the c-free convolution .'; / D .'1 ;

1/

 .'2 ;

2/

of two pairs of states on ChX1 i respectively ChX2 i is characterised by kn D kn 1 C kn 2 ; Rn.'; /

D

Rn.'1 ;

1/

C

Rn.'2 ;

(2.2) 2/

:

(2.3)

3 Shuffle algebra Card shufflings appeared already in Poincaré’s classical treatise on probability. Later, so-called perfect shuffles led to the definition of commutative shuffle products. They were axiomatised independently by Eilenberg and MacLane in 1953 [20] and Schützenberger in 1958 [37], in relation to the homology of commutative algebras, respectively combinatorics and classical Lie algebra theory. Eilenberg and MacLane studied also non-commutative shuffle products using the idea of splitting such products into left and right half-shuffle products. The resulting algebraic relations between those half-shuffles (Eqns (3.1) (3.2) (3.3) below) allowed them to demonstrate abstractly the associativity of shuffle products familiar in topology. The commutative shuffle product is essential in many fields of pure and applied mathematics. In Chen’s work for example [11, 12], it encodes algebraically the product of iterated integrals. The splitting into half-shuffles reflects instead the integration by parts rule for Riemann integrals of ordinary functions. Reutenauer’s classic monograph [36] embedded Chen’s fundamental work into the setting of connected graded cocommutative Hopf algebras. Together with their origins in topology, combinatorics and Lie theory, these phenomena explain the ubiquity of shuffles. From 2001 onwards non-commutative

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shuffle products were axiomatised and explored systematically by Chapoton, Foissy, Loday, Ronco, and others [27], who called them dendriform products. We prefer, for historical and conceptual reasons, the classical name shuffle products. These works paved the way to many theoretical developments and the discovery of new structures. Indeed, anti-symmetrising non-commutative shuffle products yields Lie brackets. However, for the half-shuffle products the picture is more subtle. Indeed, they combine to Lie admissible pre-Lie products [8, 9, 10, 30], which have been discovered independently in geometry, algebraic deformation theory and control theory [1, 2, 24, 45]. Regarding the following definitions and statements we refer the reader, for instance, to Manchon’s survey [30]. In what follows, K denotes a field of characteristic 0. Definition 3.1. A shuffle algebra .D; ; / consists of a K-vector space D together with products  and  called respectively the left and right half-shuffle products, satisfying the shuffle relations .a  b/  c D a  .b  c C b  c/ .a  b/  c D a  .b  c/ a  .b  c/ D .a  b C a  b/  c;

(3.1) (3.2) (3.3)

for a; b; c 2 D. A commutative shuffle algebra is defined by including the extra relation a  b  b  a D 0: (3.4) A shuffle algebra is, in general, non-commutative and we will use from now on the terminology “non-commutative shuffle algebra” only to emphasise explicitly noncommutativity, i.e., the absence in a given shuffle algebra of relation (3.4). Proposition 3.2. Let .D; ; / be a shuffle algebra. The shuffle product m W D ˝ D ! D, m .a ˝ b/ DW a b, defined in terms of the two half-shuffles a b WD a  b C a  b;

(3.5)

for a; b 2 D, is associative. In a commutative shuffle algebra the product m is commutative. Definition 3.3. A left pre-Lie algebra .P; B/ consists of a K-vector space P with a binary product BW P ˝ P ! P satisfying the left pre-Lie identity .a B b/ B c  a B .b B c/ D .b B a/ B c  b B .a B c/;

(3.6)

for a; b; c 2 P . An analogous notion of right pre-Lie algebra exists. Proposition 3.4. Let .P; B/ be a left pre-Lie algebra. For a; b 2 P the commutator bracket Œa; b WD a B b  b B a defines a Lie algebra structure on P .

A group-theoretical approach to conditionally free cumulants

73

Proposition 3.5. Let .D; ; / be a shuffle algebra. For a; b 2 D the product a B b WD a  b  b  a

(3.7)

defines a left pre-Lie algebra structure on D. Moreover, one verifies quickly that Œa; b D a B b  b B a D a b  b a in .D; ; /. Observe that in a commutative shuffle algebra the pre-Lie product (3.7) becomes trivial. We define the left and right multiplication maps, La .b/ WD a  b, La .b/ WD a  b and Ra .b/ WD b  a, Ra .b/ WD b  a. Combining them, we can write LaB .b/ WD a B b D .La  Ra /.b/. The maps La and Rb commute thanks to relation (3.2). A consistent definition of unital shuffle algebras demands some caution – due to the fact that it is a priori difficult to split the identity 1 1 D 1 via half-shuffles. This said, the augmentation of the shuffle algebra .D; ; / by a unit 1 to D WD D ˚ K:1 is defined by requiring for any a 2 D, that a 1 WD 1 a WD a. Moreover, concerning the half-shuffles we define 1  a WD a DW a  1; and 1  a WD 0 DW a  1. This is further extended to include the pre-Lie product, i.e., a B 1 WD a DW 1 B a. However, it is important to note that the separate cases of 1  1 and 1  1 must be excluded as they can not be defined consistently. Two examples of (unital) shuffle algebras are given next. Example 3.6. The non-unital tensor algebra over a K-vector space A is defined by M TC .A/ WD A˝n : n>0

Elements in TC .A/ are denoted by words w D ai1    aim 2 A˝m . The number of letters of L a word w 2 TC .A/ defines its length jwj. The unital tensor algebra T .A/ WD n0 A˝n is defined by adding the empty word 1 in A˝0  T .A/. The commutative and associative shuffle product on words is defined iteratively on T .A/ by w 1 D w D 1 w and ai 1    ai m

aj1    ajl WD ai1 .ai2    aim aj1    ajl / C aj1 .ai1    aim aj2    ajl /;

(3.8)

for any words ai1    aim ; aj1    ajl 2 TC .A/. The two terms on the righthand side of (3.8) define respectively the left and right half-shuffles satisfying (3.1)-(3.3) and (3.4). Note that there exists a natural grading on T .A/ given by the length of words, where j1j D 0. Next we present an example of a non-commutative shuffle algebra. It provides the framework for our approach to non-commutative probability and consists of the double tensor algebra over a K-vector space A. Further below, the latter is supposed to be a unital K-algebra, which together with the linear unital map 'W A ! K defines a non-commutative probability space .A; '/.

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Example 3.7. [15] The non-unital double tensor algebra over a K-vector space A is defined by M TC .TC .A// WD TC .A/˝n : n>0

We use the bar-notation to denote elements w1 j    jwn 2 TC .TC .A//, where wi 2 TC .A/, i D 1; : : : ; n. The space TC .TC .A// is equipped with the concatenation prod0 in TC .TC .A// by wjw 0 WD uct, defined for w D w1 j    jwn and w 0 D w10 j    jwm 0 0 w1 j    jwn jw1 j    jwm . This non-commutative algebra is multigraded, that is, TC .TC .A//n1;:::;nk L WD Tn1 .A/ ˝    ˝ Tnk .A/, as well as graded. The degree n part is TC .TC .A//n WD n1 CCnk Dn TC .TC .A//n1;:::;nk . Similar observations hold for the unital case, that is, T .T .A// D ˚n0 T .A/˝n , and we will identify without further comments a bar symbol such as w1 j1jw2 with w1 jw2 . The empty word, which is the unit for the bar-product, is denoted 1 2 T .TC.A//0 . Given two (canonically ordered) subsets S U of the set of integers N, we call connected component of S relative to U a maximal sequence s1 ; : : : ; sn in S , such that there are no 1 i < n and u 2 U , such that si < u < si C1 . In particular, a connected component of S in N is simply a maximal sequence of successive elements s; s C 1; : : : ; s C n in S . Consider a word a1    an 2 TC .A/. Recall that for the (canonically ordered) non-empty set S WD fs1 ; : : : ; sp g Œn, we define aS WD as1    asp ;

(3.9)

and a; WD 1. Denoting by J1 ; : : : ; Jk the connected components of Œn  S , we then set aJ S WD aJ1 j    jaJk : (3.10) Œn

More generally, for S U Œn, set aJ S WD aJ1 j    jaJk , where the aJi are U now the connected components of U  S in U . We remark that the bar-notation in (3.10) respectively in aJ S may be interpreted as marking the places where sequences U of consecutive letters have been extracted from a word. Using (3.9) and (3.10) we define a coproduct on T .TC .A//. Definition 3.8. The coproduct  W T .A/ ! T .A/˝T .TC.A// is defined by .1/ WD 1 ˝ 1 and X .a1    an / WD aS ˝ aJ S : (3.11) S Œn

Œn

It is extended multiplicatively to all of T .TC .A//, i.e., .w1 j    jwm / WD .w1 /    .wm /. For example, the coproduct of a single letter is .a/ D a ˝ 1 C 1 ˝ a. For a word a1 a2 of length two it is .a1 a2 / D a1 a2 ˝ 1 C 1 ˝ a1 a2 C a1 ˝ a2 C a2 ˝ a1 :

A group-theoretical approach to conditionally free cumulants

75

For the word a1 a2 a3 2 A˝3 we calculate .a1 a2 a3 / D a1 a2 a3 ˝ 1 C 1 ˝ a1 a2 a3 C a1 ˝ a2 a3 C a2 ˝ a1 ja3 C a3 ˝ a1 a2 C a1 a2 ˝ a3 C a1 a3 ˝ a2 C a2 a3 ˝ a1 : The coproduct .a1    a6 / includes among others the sum a1 a3 a5 ˝ a2 ja4 ja6 C a3 a4 a5 ˝ a1 a2 ja6 C a3 a6 ˝ a1 a2 ja4 a5 C a4 a5 a6 ˝ a1 a2 a3 : The proofs of the following two theorems appeared in [15]. See also [16]. Theorem 3.9. [15] The graded algebra H WD T .TC.A// equipped with the coproduct (3.11) is a connected graded non-commutative and non-cocommutative Hopf algebra. The central observation in [15] consist of the splitting of the coproduct (3.11) into two parts  D  C  : The corresponding left respectively right half-coproducts are defined on H by X  .a1    an / WD aS ˝ aJ S DW C (3.12)  .a1    an / C a1    an ˝ 1 12S Œn

and  .a1    an / WD

X 1…SŒn

Œn

aS ˝ aJ S DW C  .a1    an / C 1 ˝ a1    an : Œn

(3.13)

Note that for w 2 H the reduced coproduct C .w/ WD .w/  w ˝ 1  1 ˝ w splits into C C D C  C  : For instance, the coproduct .a1 a2 a3 / is the sum of the left half-coproduct  .a1 a2 a3 / D a1 a2 a3 ˝ 1 C a1 ˝ a2 a3 C a1 a2 ˝ a3 C a1 a3 ˝ a2 and right half-coproduct  .a1 a2 a3 / D 1 ˝ a1 a2 a3 C a2 ˝ a1 ja3 C a3 ˝ a1 a2 C a2 a3 ˝ a1 : The two half-coproducts are extended to H by defining them on w1 j    jwm  .w1 j    jwm / WD  .w1 /.w2 /    .wm /  .w1 j    jwm / WD  .w1 /.w2 /    .wm /: Theorem 3.10. [15] The algebra H equipped with  and  is a unital unshuffle bialgebra.

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For details on the notion of unshuffle bialgebra we refer the reader to Foissy’s article [21] and, in the present context, to our previous articles, e.g., [15, 18]. Recall that the space of linear maps, Lin.H; K/, is (as for all Hopf algebras) an associative and untial K-algebra with respect to the non-commutative convolution product defined for ; ‰ 2 Lin.H; K/ in terms of the coproduct (3.11)  ‰ WD mK . ˝ ‰/; where mK stands for the product map in K. The augmentation map e W H ! K, defined by e.1/ WD 1 and zero on the so-called augmentation ideal HC WD TC .TC .A//, C is the unit for this convolution product. In light of the splitting C D C  C  we define accordingly the left and right convolution half-products on Lin.HC ; K/:   ‰ WD mK . ˝ ‰/C 

  ‰ WD mK . ˝ ‰/C :

These operations are extended by setting, for ‰ 2 Lin.HC ; K/, e  ‰ WD 0, e  ‰ WD ‰, ‰  e WD ‰; ‰  e WD 0: As a result we obtain the next proposition. Proposition 3.11. [15] The space Lin.H; K/ equipped with .; / is a unital shuffle algebra.

4 Shuffle and half-shuffle exponentials and logarithms Recall the formal, i.e., purely algebraic component of the relations between a group and its Lie algebra as encoded in the Baker–Campbell–Hausdorff formula [36]. The natural framework to understand these phenomena is provided by complete connected cocommutative Hopf algebras and, in particular, classical commutative shuffle Hopf algebras [36]. In this case the exponential and logarithm maps relate the Lie algebra of primitive elements bijectively to the group of group-like elements. In [18] we started to explore how the classical correspondence between groups and Lie algebras, and related properties and identities, translate in the setting of the non-(co)commutative shuffle bialgebra H in Theorem 3.9. It turns out that in this case one has to consider not only the usual exponential-logarithm correspondence but also two shuffle-type counterparts defined in terms of the two half-shuffle products. In this section we recall from [18] the shuffle and half-shuffle exponentials and logarithms and introduce the group-theoretical shuffle adjoint actions. A preliminary remark is in order regarding convergence issues. They are left aside in the present paper since we deal implicitly with formal series expansions over free shuffle algebras (insuring the convergence in the formal sense), or with graded algebras (in which case formal power series expansions restrict to finite expansions in each degree). In practice, “let D be a shuffle algebra” means therefore till the end of the present section, “let D be a free or a graded connected (i.e. with no degree zero component) shuffle algebra”.

77

A group-theoretical approach to conditionally free cumulants

Let .D; ; / be a unital shuffle algebra. For any element a 2 D we define the usual exponential and logarithm in terms of the associative shuffle product (3.5) exp.a/ WD 1 C

X an n>0

log .1 C a/ WD 

nŠ

X

.1/n

n>0

an : n

(4.1)

For a 2 D we define a0 WD 1 DW a0 and for n > 0, an WD a  .an1/, and an WD .an1 /  a. Then the left and right half-shuffle exponentials are defined for a 2 D X X E .a/ WD 1 C an E .a/ WD 1 C an : n>0

n>0

They are respectively the formal solutions of the two half-shuffle fixed point equations X D1Ca X

Y D 1 C Y  a:

(4.2)

Lemma 4.1. Let D be a shuffle algebra, and D D D ˚ K:1 its unital augmentation. 1) For a 2 D, the product of X WD E .a/ and Y WD E .a/ is Y X D X Y D 1, so that E1 .a/ D E .a/. We have therefore X .1/n .E .a/  1/n : (4.3) Y D X 1 D n0

2) For a 2 D and X 0 D X 0 .a/ WD E .a/  1, we have X n  a D X0  .1/nX 0 :

(4.4)

n0

Analogously, for Y 0 D Y 0 .a/ WD E .a/  1, we have X n  aD .1/nY 0  Y 0:

(4.5)

n0

Proof. For proofs and more details see, for instance, [15, 16, 19].



Definition 4.2. Let D be a shuffle algebra, and D its unital augmentation. For x 2 D define the left half-shuffle logarithm X  L .1 C x/ WD x  .1/n x n ; (4.6) n0

and the right half-shuffle logarithm L .1 C x/ WD

X

 .1/n x n  x:

n0

(4.7)

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For the following theorem we define the pre-Lie Magnus expansion [14] in terms of the recursion X bm .m/ X bm .m/ 1 L 0 B .a/ D a  a B a C L 0 .a/; mŠ 2 mŠ B id m0 m2 (4.8) where the bl ’s are the Bernoulli numbers. For a 2 D we define the map 0 .a/ WD

L 0 B L e 0 B 

W 0 .a/ WD

.a/ D

eLaB  id 1 1 .a/ D a C a B a C a B .a B a/ C    : LaB 2 6

(4.9)

The bijection W 0 is the compositional inverse of 0 , i.e., W 0 ı 0 D id D 0 ı W 0 . Theorem 4.3. The left and right half-shuffle exponentials, E .a/ respectively E .a/, satisfy     E .a/ D exp 0 .a/ E .a/ D exp  0 .a/ : (4.10) In the commutative case, i.e., when La D Ra for a 2 D, the map 0 reduces to the identity map. Hence, in a commutative shuffle algebra the two fixed point equations in (4.2) coincide and the solution is given by X D exp.a/: From (4.10) we deduce a key identity in shuffle algebra connecting the three exponentials. Lemma 4.4. Let D be a shuffle algebra, and D its augmentation by the unit 1. For a 2 D the following identity holds     (4.11) E W 0 .a/ D exp.a/ D E  W 0 .a/ : The interplay between the pre-Lie Magnus expansion and its inverse becomes most intriguing when combining it with the Baker–Campbell–Hausdorff expansion. Indeed, one can show that [19, 30]   (4.12) BCH 0 .a/; 0 .b/ D 0 .a#b/; where

   a#b D W 0 BCH 0 .a/; 0 .b/ :

(4.13)

From W 0 .a/ D eLaB 1  1 it follows that   W 0 .a/#W 0 .b/ D W 0 BCH.a; b/ D eLaB eLbB 1  1: Hence W 0 .a/#W 0 .b/ D W 0 .a/ C eLaB W 0 .b/. This yields the formula a#b D a C eL0 .a/B b:

(4.14)

A group-theoretical approach to conditionally free cumulants

79

We can use LaB .b/ D a B b D .La  Ra /.b/ to rewrite (4.14) a#b D a C eL0 .a/ eR0 .a/ b     D a C exp 0 .a/  b  exp  0 .a/ D a C E .a/  b  E1 .a/:

(4.15)

We used shuffle relation (3.1) which implies for n  0, that .   ..b  0 .a//  0 .a//     /  0 .a/ D b  .0 .a/n/. Likewise, from (3.3) it follows that 0 .a/  .    .0 .a/  .0 .a/  b//    / D .0 .a/n/  b. Using (4.12) we note for a; b 2 D that     E .a/ E .b/ D exp 0 .a/ exp 0 .b/    D exp BCH 0 .a/; 0 .b/   D exp 0 .a#b/ D E .a#b/: Similarly, we have that E .a/ E .b/ D E ..b#  a//, where we used a classical property of the Baker–Campbell–Hausdorff series, BCH.a; b/ D BCH.b; a/. Definition 4.5. [14, 19] Let D be a shuffle algebra, and D its unital augmentation. For x; y 2 D we define y x WD Ad x .y/ WD E1 .x/  y  E .x/

(4.16)

E1 .x/:

(4.17)

x

yx WD Ad .y/ WD E .x/  y 

First, we note that from (4.11) in Theorem 4.4 we see that the left half-shuffle exponentials in (4.16) and (4.17) can be expressed in terms of exp as well as the right half-shuffle exponential y x D exp.0 .x//  y  exp.0 .x// D E1 .W 0 .0 .x//  y  E .W 0 ..x//: Next, we show that the identity E1 .x/ D E .x/ implies that y x D Ad x .y/ D E1 .x/  y  E .x/ D E .x/  y  E1 .x/ f x .y/: DW Ad Proposition 4.6. Let D be a shuffle algebra, and D its unital augmentation. For x; y; z 2 D we have Ad x Ad y .z/ D Ad y#x .z/: (4.18)

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Proof. We calculate   Ad x Ad y .z/ D E1 .x/  E1 .y/  z  E .y/  E .x/ 1     z  E .y/ E .x/ D E .y/ E .x/ D E1 .y#x/  z  E .y#x/ D Ad y#x .z/:

(4.19) 

Corollary 4.7. Let D be a shuffle algebra, and D its unital augmentation. For a; b 2 D we have that a#b a D a C b D ab #b: (4.20) Proof. First, we observe for the product (4.14), that a#b D a C Ad a .b/, and that Ad a Ad a .b/ D .b a /a D b D .ba /a D Ad a Ad a .b/:

(4.21)

From this we deduce that a#b a D a C Ad a Ad a .b/ D a C b: Note that we will show the second equality in (4.20) explicitly in the following theorem.  Theorem 4.8 ([18, 19]). Let D be a shuffle algebra, and D its unital augmentation. For x; y 2 D we have the following factorisations

and

E .x C y/ D E .x/ E .y x /

(4.22)

E .x C y/ D E .x y / E .y/:

(4.23)

Proof. The proofs of both (4.22) and (4.23) follow from (4.12) together with (4.15). We verify (4.23) explicitly.     E .x y / E .y/ D exp  0 .x y / exp  0 .y/    D exp BCH  0 .x y /; 0 .y/    D exp  BCH 0 .y/; 0 .x y /   D exp  0 .y#  x y /   D exp  0 .y  .x y /y /   D exp  0 .y  x/ D E .x C y/: 

A group-theoretical approach to conditionally free cumulants

81

For later use, let us also include two technical identities that will prove relevant in the context of conditionally free cumulants. Lemma 4.9. For x; y as above, we have y/

.x y /..xy/

D xx :

Proof. Indeed, we have, using the relations between half-shuffles:       y .x y /..xy/ / D E1 .x  y/y  E1 .y/  x  E .y/  E .x  y/y        D E1 .x  y/y E1 .y/  x  E .y/ E .x  y/y D E1 .x/  x  E .x/ D xx :



Corollary 4.10. For x; y as above, we have:

  E .y/ D E .x/ E .x C y/y ;

or, equivalently,

  E .x/ E .y/ D E .x C y/y :

Proof. Indeed, we have, using the identity E .y  y/ D 1 D E .y y / E .y/ and the previous lemma:   E .y/ D E1 .y/ D E .y y / D E  x y C .x C y/y    y  D E .x y /..xy/ / E .x C y/y     D E .x/x E .x C y/y   D E1 .x/ E .x C y/y    D E .x/ E .x C y/y

5 Monotone, free and boolean cumulants Let us return to Example 3.7 and the Hopf algebra H D T .TC.A// in Theorem 3.9. Recall that it is connected, graded, non-cocommutative, and non-commutative. Its antipode S 2 EndK .H; H /, i.e., the inverse of the identity id 2 EndK .H; H / with respect to the convolution product defined on EndK .H; H / in terms of the coproduct (3.11) on H , is given by X SD .1/i P i : (5.1) i 0

The linear map P WD id  e is the augmentation ideal projector, that is, P .1/ D 0 and P D id on the kernel of the counit, HC D TC .TC .A//. Definition 5.1. A character ˆ 2 Lin.H; K/ is a unital multiplicative map, i.e., ˆ.1/ D 1 and ˆ.wjw 0 / D ˆ.w/ˆ.w 0 /, for w; w 0 2 HC . An infinitesimal character  2 Lin.H; K/ is a map such that .1/ D 0 and .wjw 0 / D 0; for w; w 0 2 HC .

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Recall that the set G  Lin.H; K/ of characters forms a group with respect to the convolution product [25, 29]. The convolution inverse of a character ˆ 2 G is ˆ1 WD ˆ ı S . The space g  Lin.H; K/ of infinitesimal characters forms a Lie algebra for the Lie bracket Œ˛; ˇ WD ˛ ˇ  ˇ ˛. The exponential and logarithm maps, exp and log , are set isomorphisms between the group G and its Lie algebra g. Recall that for any infinitesimal character ˛ 2 g and any word w 2 TC .A/ of finite length jwj, the exponential reduces to a finite sum, i.e., exp.˛/.w/ D Pjwj 1 j Pjwj .1/l1  j D1 j Š ˛ .w/. The same holds for the logarithm, log .eC˛/.w/ D lD1 l ˛ l .w/. For any ˛ 2 g, the left and right half-shuffle exponentials, E .˛/ respectively E .˛/, also reduce to finite sums when applied to a word w 2 TC .A/ of finite P j .w/, and similarly for E .˛/.w/. length, i.e., E .˛/.w/ D jwj j D1 ˛ Both half-shuffle exponentials provide as well natural bijections between G and g [18, 19]. It follows that for ˆ 2 G there exist unique infinitesimal characters ˛, ˇ, in g such that ˆ D exp. / D E .˛/ D E .ˇ/: (5.2) From this identity together with Theorem 4.3 the following relations between ˛, ˇ, in g can be deduced ˛ D W 0 . /; ˇ D W 0 . / (5.3) from which ˛ D W 0 .0 .ˇ// follows (see [18] for details). We now consider H D T .TC .A// where .A; '/ is supposed to be a non-commutative probability space, i.e., a unital K-algebra A with map 'W A ! K, and '.1A / D 1. See [35] for details. First, ' is extended to a linear map from TC .A/ to K by defining .a1    an / WD '.a1 A    A an /. Then is extended to a character ˆ on H . For a word w D a1    an 2 TC .A/, the n-th order multivariate moment is defined by mn .a1 ; : : : ; an / WD '.a1 A    A an / D ˆ.w/: From [15, 16, 19] the next theorem follows. Theorem 5.2 ([19]). Let .A; '/ be a non-commutative probability space with unital map 'W A ! K and ˆ its extension to H as a character. Let , , ˇ in g be infinitesimal characters defined in terms of the shuffle algebra identity ˆ D exp./ D E ./ D E .ˇ/:

(5.4)

For the word w D a1    an 2 TC .A/ we set kn .a1 ; : : : ; an / D .w/, rn .a1 ; : : : ; an / D ˇ.w/, and hn .a1 ; : : : ; an / D .w/. The maps kn , rn , and hn identify respectively with multivariate free, boolean and monotone cumulants, and we obtain the following multivariate moment-cumulant relations i) Free moment-cumulant relation [35]W E ./.w/ D

n X

 j .w/ D

j D1

where k .a1 ; : : : ; an / WD

Q

i 2

X 2NCn

.ai /:

k .a1 ; : : : ; an /;

(5.5)

A group-theoretical approach to conditionally free cumulants

83

ii) Boolean moment-cumulant relation [39]W E .ˇ/.w/ D

n X j D1

where rI .a1 ; : : : ; an / WD

X

ˇ j .w/ D

Q

lk 2I

rI .a1 ; : : : ; an /;

(5.6)

I 2Bn

ˇ.alk /:

iii) Monotone moment-cumulant relation [26]W n X X j .w/ 1 exp ./.w/ D D h .a1 ; : : : ; an /: jŠ  . /Š 

j D1

(5.7)

 2NCn

The tree factorial  . /Š corresponds to the forest  . / of rooted trees encoding the nesting structure of the non-crossing partition 2 NCn [3], and Q h .a1 ; : : : ; an / WD i 2 .ai /. We call the Lie algebra elements , , ˇ 2 g the monotone, free and boolean infinitesimal cumulant characters, respectively. Note that in all three cases, (5.5)-(5.7), the second equality follows from evaluating the lefthand sides on a word of finite length. The next result will be useful. Proposition 5.3. Let ;  2 g be the free and boolean infinitesimal characters of the state ‰ D E ./ D E . / 2 G. Following (4.18) we deduce from ‰ 1 D E1 . / D E . / that  D Ad  ./ D ‰    ‰ 1 D E1 . /    E . / D Ad  ./ D  : From (5.4) and (4.11) it follows that monotone, free and boolean cumulants are related. This implies that one can express monotone, free, and boolean cumulants in terms of each other. See [3] for details. We consider the following lemma, which will be useful in describing these relations. From [3] we recall that an irreducible non-crossing partition is a non-crossing partition of the set Œn with 1 and n being in the same block. The set of irreducible non-crossing partitions is denoted by NCni rr . Lemma 5.4. Let ; ;  be infinitesimal characters in g and ‰ D E ./ D E . / 2 G (so that  and  are the infinitesimal free respectively boolean cumulant characters associated to ‰). The following formula holds for the infinitesimal character  D Ad  ./ D ‰ 1    ‰ evaluated on a word w D a1    an 2 TC .A/ of length n X  .w/ D .aS /‰.aJ S / (5.8) 1;n2S Œn

D

X

Œn

Y

outer 2NCni rr  2 1

j1 j .a1 /

Y inner i 2

ji j .ai /:

(5.9)

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On the other hand, from Proposition 5.3 it follows that  D  evaluated on a word of length n, w D a1    an 2 TC .A/, gives X  .w/ D  .w/ D .aS /‰ 1 .aJ S / (5.10) Œn

1;n2S Œn

X

D

.1/jj1

Y outer 1 2

2NCni rr

Y

j1 j .a1 /

ji j .ai /:

(5.11)

inner i 2

Notice that we used for notational convenience the symbol product is trivial and involves only one block.

Q

outer 1 2

although the

Proof. We follow the proof given in [19] by using induction on the length of words. Let w D a1    an 2 TC .A/ and let  2 g be an infinitesimal character. The expression  D Ad  ./ D ‰ 1    ‰ is equivalent to ‰   D   ‰, such that ‰   .w/ D  .w/ C

n1 X

‰.aj C1    an / .a1    aj / D

j D1

X

.aS /‰.aJ S /:

12S Œn

Œn

(5.12) This implies X

 .w/ D

.aS /‰.aJ S / 

12S Œn

X

D

1;n2S Œn



n1 X

Œn

n1 X j D1

.aS /‰.aJ S / C Œn

‰.aj C1    an / .a1    aj / X

(5.13)

.aS /‰.aJ S /

12SŒn n…S

Œn

‰.aj C1    an / .a1    aj /:

(5.14)

j D1

A simple calculation for a single letter a 2 A shows that  .a/ D .a/. For a word of length n D 2 we find  .a1 a2 / D .‰ 1    ‰/.a1a2 / D ‰ 1 .a2 /.  ‰/.a1 / C .  ‰/.a1 a2 / D ‰.a2 /.a1 / C .a1 a2 / C .a1 /‰.a2 / D .a1 a2 /:

85

A group-theoretical approach to conditionally free cumulants

Using induction we write  .a1    aj / D X

 .w/ D

n1 X

Œn

X

X

Œn

.aT /‰.aJ T / Œj 

1;j 2T Œj 

.aS /‰.aJ S /:

(5.15)

Œn

1;n2S Œn

Then

.aS /‰.aJ S /

12SŒn n…S

X

‰.aj C1    an /

j D1

D

S / in (5.14). 1;j 2S Œj  .aS /‰.aJŒj 

.aS /‰.aJ S / C

1;n2S Œn



P

Here we used that X

.aS /‰.aJ S / D Œn

12SŒn n…S

n1 X j D1

.aT /‰.aJ T / ‰.aj C1    an /:

X

Œj 

1;j 2T Œj 

The formulation in terms of irreducible non-crossing partitions in (5.9) follows from the fact that the sum on the righthand side of (5.15) ranges over subsets S Œn which always contain both the elements 1 and n. For  D Ad  ./ D ‰    ‰ 1 we recall that, as  is the infinitesimal boolean cumulant character of ‰ 2 G, we have ‰ 1 D E1 . / D E . /. Therefore,  D  D Ad  ./ D E1 . /    E . /. Following the same argument as before this yields X X  .w/ D .aS /‰ 1 .aJ S / D .aS /E . /.aJ S /: Œn

1;n2S Œn

1;n2S Œn

Œn

Therefore, we have for any word w 2 TC .A/ of length n that ‰ 1 .w/ D E . /.w/ D P jj jj1 on the righthand side 2NCn .1/  .w/, which implies the coefficient .1/ in (5.11).  For instance, from this lemma and (5.4) we deduce immediately the relation between boolean and free cumulants [19]. Indeed, E ./ D E .ˇ/ implies that ˆ  ˇ D   ˆ, which yields ˇ D ˆ1    ˆ and  D ˆ  ˇ  ˆ1 . Therefore X X ˇ.w/ D .aS /ˆ.aJ S / D k .a1 ; : : : ; an /; Œn

1;n2S Œn

and .w/ D

X 1;n2S Œn

ˇ.aS /ˆ1 .aJ S / D Œn

2NCni rr

X

.1/jj1 r .a1 ; : : : ; an /:

2NCni rr

In the last equation we used that from ˆ D E .ˇ/ it follows that ˆ1 D E1 .ˇ/ D E .ˇ/.

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6 Conditionally free cumulants revisited Let ', be two states on the non-commutative probability space A. We denote by ˆ; ‰ their extensions to elements of G, i.e., characters on the Hopf algebra H D T .TC .A//. By ˇ; ˇ 0 2 g and ;  0 2 g we denote the corresponding infinitesimal boolean respectively free cumulant characters, i.e., ˆ D E .ˇ/ D E ./ and ‰ D E .ˇ 0 / D E . 0 /. Recall the relation between infinitesimal boolean and free cumulant characters ˇ D ˆ1    ˆ

ˇ 0 D ‰ 1   0  ‰:

(6.1)

In the following we would like to determine the infinitesimal cumulant character R 2 g such that 0 ˇ D R D ‰ 1  R  ‰; (6.2) and show that it is related to the c-free cumulants Rn.'; / as the infinitesimal characters ˇ, ˇ 0 and ,  0 are related to the corresponding boolean and free cumulants. Lemma 5.4 implies immediately that X R.aS /‰.aJ S / ˇ.w/ D 1;n2S Œn

X

D

Œn

Y

R.a1 /

outer 2NCni rr  2 1

Inverting (6.2) gives

Y

 0 .ai /:

inner i 2

R D ‰  ˇ  ‰ 1 ;

which evaluates to R.w/ D

X

ˇ.aS /‰ 1 .aJ S / Œn

1;n2S Œn

D

X

(6.3)

.1/jj1

Y

ˇ.a1 /

outer 1 2

2NCni rr

Y

ˇ 0 .ai /:

inner i 2

As ˇ D L .ˆ/ D ˆ1  .ˆ  e/, we can express R in terms of the characters ˆ and ‰   R D ‰  ˆ1  .ˆ  e/  ‰ 1 : (6.4) For instance, with the following notation for moments m'n .a1 ; : : : ; an / D ˆ.w/; mn .a1 ; : : : ; an / D ‰.w/; and using Lemma 5.4 together with some shuffle algebra we find quickly that R.a1 a2 a3 / D .ˆ1  .ˆ  e//.a1 a2 a3 / C .ˆ1  .ˆ  e//.a1 a3 /‰ 1 .a2 / D m'3 .a1 a2 a3 /  m'1 .a3 /m'2 .a1 a2 /  m'2 .a2 a3 /m'1 .a1 / C m'1 .a1 /m'1 .a2 /m'1 .a3 / '

'

'

 m2 .a1 a3 /m1 .a2 / C m1 .a1 /m1 .a3 /m1 .a2 /:

A group-theoretical approach to conditionally free cumulants

87

A closer look at (6.3) respectively (6.4) reveals that from the relation between boolean and free cumulants, expressed on the level of infinitesimal characters by (6.1), it follows that   R D .ˆ ‰ 1 /1  .ˆ  e/  ˆ1  .ˆ ‰ 1 /; (6.5) where  D L .ˆ/ D .ˆ  e/  ˆ1 2 g. This yields R D .ˆ ‰ 1 /1    .ˆ ‰ 1 /: In terms of half-shuffle exponentials this gives  0 ˆ D E R   D E ‰ 1  R  ‰   D E .ˆ ‰ 1 /  R  .ˆ ‰ 1 /1 :

(6.6)

(6.7) (6.8) (6.9)

Observe the change from left half-shuffle exponential to right half-shuffle exponential between equations (6.8) and (6.9). These half-shuffle exponentials solve the corresponding fixed point equations   ˆ D e C ˆ  ‰ 1  R  ‰ (6.10) respectively   ˆ D e C .ˆ ‰ 1 /  R  .ˆ ‰ 1 /1  ˆ:

(6.11)

Again, (6.10) reflects the boolean character of the picture, whereas (6.11) is in the free setting. Remark 6.1. Observe that for ‰ D 1, the half-shuffle fixed point equation (6.10) reduces to ˆ D e C ˆ  R, which implies that R D ˆ1    ˆ D ˇ. For ˆ D ‰ we deduce that ˆ D e C R  ˆ, such that R D  is the free infinitesimal cumulant character. Proposition 6.1. For a word w D a1    an 2 TC .A/ of length n ˆ.w/ D m'n .a1 ; : : : ; an / D

n X

X

R.aS /‰.aJ S /ˆ.aj C1 : : : an / Œj 

j D1 1;j 2S Œj 

D

X

Y

2NCn outer j 2

R.aj /

Y inner i 2

 0 .ai /:

(6.12)

(6.13)

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Proof. From (6.10) and Lemma 5.4 it follows for a word w D a1    an 2 TC .A/ that    (6.14) ˆ.w/ D ˆ  ‰ 1  R  ‰ .a1    an / n X   D ˆ.ak    an / ‰ 1  R  ‰ .a1    ak1 / (6.15) D

kD2 n X kD2

D

n  X

kD2

X

R.aS /‰.aJ S

Œk1

1;k12S Œk1

Y

X

Y

R.a1 /

outer i rr 2NCk1 1 2

/ ˆ.ak    an /

(6.16)



0

 .ai / ˆ.ak    an /;

(6.17)

inner i 2



which gives by iteration the expression in (6.13).

Corollary 6.2. The infinitesimal cumulant character R 2 g defined in (6.2) computes the multivariate c-free cumulant: R.a1    an / D Rn.'; / .a1 ; : : : ; an /; using the notation of Section 2. Proof. Equation (6.12) and (6.13) identify indeed with the multivariate generalisation of the equations defining c-free cumulants (see Section 2 and reference [5]).  Equation (6.4) then says that c-free cumulants are given by a shuffle adjoint action on the boolean logarithm (R-transformation) R D ‰  L .ˆ/  ‰ 1 :

7 Conditionally free convolution Recall from Section 2 that the c-free convolution .'; / of two c-free states .'1 ; and .'2 ; 2 /, is given on the associated free and c-free cumulants by

1/

kn D kn 1 C kn 2 ; Rn.';

/

D Rn.'1 ;

1/

C Rn.'2 ;

2/

:

Let us write ˆ; ‰ (respectively ˆ1 ; ‰1 , ˆ2 ; ‰2 ) for the associated characters, respectively ; ˇ;  0 ; ˇ 0 (and so on) for the associated infinitesimal free and boolean cumulant characters.

A group-theoretical approach to conditionally free cumulants

89

 0  In shuffle group theoretical terms, the c-free convolution of ˆ1 D E R11 D    0    E ‰11  R1  ‰1 and ˆ2 D E R22 D E ‰21  R2  ‰2 is defined by the resulting state   0  ˆ D E R D E ‰ 1  R  ‰ ; where  0 D 10 C 20 (7.1) and (7.2) R D R1 C R2 : Condition (7.1) implies that ‰ D E .10 C 20 /, that is 0

‰ D E .10 / E .20 1 /; which is free additive convolution of ‰1 with ‰2 , i.e., ‰ D ‰1  ‰2 . The second condition (7.2) is more involved and shows that c-free convolution is different from free additive convolution in general. We shall consider three particular cases to illustrate how shuffle group calculus can be developped. First, we assume that 10 D 20 D 0, i.e., ‰1 and ‰2 are just the shuffle unit, and  0 D 0. In light of (7.2) c-free convolution turns out to be just boolean additive convolution       ˆ D E R1 C R2 D E R1R2 E R2 ; that is, ˆ D ˆ1 ] ˆ2 . Next we consider the case when 1 D 10 and 2 D 20 . This means that ˆ1 D ‰1 and ˆ2 D ‰2 . Then ˆ1 D E ˆ1  R1  ˆ1 D E .R1 / and ˆ2 D 1  1 E ˆ2  R2  ˆ2 D E .R2 /. Now, conditions (7.1) and (7.2) imply that the c-free convolution coincides with the free additive convolution such that   ˆ D E .R1 C R1 / D E ˆ1  .R1 C R2 /  ˆ ; and ˆ D ‰. Last we consider the case, where 10 D 0 and 2 D 20 corresponding to ‰1 D e and ˆ2 D ‰2 . First notice that ‰ D ‰2 D ˆ2 D E .2 / D E .20 /, whereas ‰1 D e implies ˇ1 D R1 (so that ˆ1 D E .R1 /). Similarly, ˆ2 D ‰2 implies R2 D 2 and ˆ2 D E .R2 /. Using Lemma 4.10, we have:     ˆ1 ˆ2 D E R1 E R2 D E ..R1 C R2 /R2 / E .ˆ1  .R1 C R2 /  ˆ2 / D E .‰ 1  .R1 C R2 /  ‰/: 2 We get finally ‰ D ‰1  ‰2 D ‰2 as ‰1 D e and ˆ D E .‰ 1  .R1 C R2 /  ‰/ D ˆ1 ˆ2 : That is, in this case c-free convolution coincides with the convolution, i.e., shuffle product in H  . According to [18], this amounts to saying that in this case c-free convolution coincides with monotone convolution.

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The Natural Growth Scale Jean Ecalle Contents 1

2

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Program: exploring/completing the natural growth scale . . . . . . . . . . 1.1 Groups of real germs. Successive extensions. . . . . . . . . . . . . 1.2 Non-oscillation and comparability. . . . . . . . . . . . . . . . . . . . 1.3 Divergence: resurgent or/and cohesive. . . . . . . . . . . . . . . . . 1.4 The ‘display’ and its many uses. . . . . . . . . . . . . . . . . . . . . . 1.5 Extensions: exponential or ultra-exponential. . . . . . . . . . . . . 1.6 Functional incarnation of transfinite arithmetics. . . . . . . . . . . 1.7 Iso-differential operators and convexity. . . . . . . . . . . . . . . . . 1.8 Universal asymptotics of fast/slow germs. . . . . . . . . . . . . . . . 1.9 Stubborn indeterminacy in the realisation of ultraexponentials. 1.10 Spirit of this paper: exploratory rather than systematic. . . . . . . Tools: resurgence, acceleration, cohesiveness, analysability . . . . . . . 2.1 Resurgent functions. The three models. . . . . . . . . . . . . . . . . 2.2 Convolution preserving averages. . . . . . . . . . . . . . . . . . . . . 2.3 Alien derivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Pseudovariables and displays. . . . . . . . . . . . . . . . . . . . . . . . 2.5 Multicritical resurgence and accelero-summation. . . . . . . . . . 2.6 Acceleration and deceleration transforms. . . . . . . . . . . . . . . . 2.7 Pseudo-acceleration and -deceleration transforms. . . . . . . . . . 2.8 Cohesiveness and the Regularity Scale. . . . . . . . . . . . . . . . . 2.9 Time changes and the Great Divide. . . . . . . . . . . . . . . . . . . . 2.10 Transseries and transmonomials. . . . . . . . . . . . . . . . . . . . . . 2.11 Analysable germs. The complexity hierarchy. . . . . . . . . . . . . 2.12 Multicritical displays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Ultraseries and ultramonomials. . . . . . . . . . . . . . . . . . . . . . 2.14 Main germ groups and main extensions. . . . . . . . . . . . . . . . Groups of analysable germs. Complexity hierarchy . . . . . . . . . . . . . 3.1 Degrees of divergence. Rising complexity. . . . . . . . . . . . . . . 3.2 Groups of convergent analysable germs. . . . . . . . . . . . . . . . . 3.3 Groups of seriable analysable germs. . . . . . . . . . . . . . . . . . . 3.4 Groups of non-polarised analysable germs. . . . . . . . . . . . . . . 3.5 Groups of polarised analysable germs. . . . . . . . . . . . . . . . . . 3.6 Infinite exponential depth. . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Accelero-summation commutes with composition. . . . . . . . . . Conjugation=iteration of zero-exponentiality germs . . . . . . . . . . . . . 4.1 The three steps of conjugation. . . . . . . . . . . . . . . . . . . . . . .

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95 95 96 96 97 97 98 98 99 99 99 100 100 102 103 104 105 105 106 107 108 109 111 112 113 114 115 115 116 118 119 120 121 121 121 121

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J. Ecalle 4.2 The general transserial difference equation. . . . . . . . 4.3 The general transserial conjugation equation. . . . . . . 4.4 Some examples. . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugation/iteration of nonzero-exponentiality germs . . . . . 5.1 Conjugation of germs with the same exponentiality. . . 5.2 Graded examples. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Resurgence and displays. . . . . . . . . . . . . . . . . . . . . Universal asymptotics of ultra-slow germs . . . . . . . . . . . . . 6.1 The bialgebra of iso-differentiations. . . . . . . . . . . . . 6.2 The first two main bases Dnf g and Dsf g of ISO. . . . . 6.3 Universal asymptotics. The algebras Isolog  ]Isolog. 6.4 The bialgebra ]Iso and its two bases Deh i and Dah i . . 6.5 Action of ]Iso on the group G . . . . . . . . . . . .

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125 128 132 133 133 134 137 137 137 138 141 143 145

Iso-convexity and the extremal basis Daf g . . . . . . . . . . . . . . . . . . . . . . 7.1 The positive cones ]ISOC  ]ISO. . . . . . . . . . . . . . . . . . . . . . . . 7.2 The extremal basis. Main statements. . . . . . . . . . . . . . . . . . . . . . 7.3 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Up to ! ! : the ultra-exponential scale . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Towers of ultraexponentials and ultralogarithms. . . . . . . . . . . . . . 8.2 Central indeterminacy. Growth types. . . . . . . . . . . . . . . . . . . . . 8.3 Geometric incarnation of the semi-ring Œ1; ! ! Œ. . . . . . . . . . . . . . . 8.4 Some useful notations. Iterators and connectors. . . . . . . . . . . . . . 8.5 Construction of analytic ultraexponential towers. . . . . . . . . . . . . . 8.6 Action of the periodic towers on ultraexponential towers. . . . . . . . 8.7 Kneser’s analytic iteration of exp. . . . . . . . . . . . . . . . . . . . . . . . 8.8 Analytic ultra-quasiexponential towers. . . . . . . . . . . . . . . . . . . . 8.9 Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond ! ! : the meta-exponential scale . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Iterates of order ˛  ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 No all-inclusive quasi-analyticity class. . . . . . . . . . . . . . . . . . . . 9.3 Non-oscillating extensions beyond the ultraexponential range. . . . . Ultraseries and their all-round completeness . . . . . . . . . . . . . . . . . . . . . 10.1 Ultraseries and ultramonomials. . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Simplification rules for ultramonomials. . . . . . . . . . . . . . . . . . . . 10.3 Several competing presentations, but one well order on ultraseries. . 10.4 Integration of ultramonomials. . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Further remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition equations: resurgence and displays . . . . . . . . . . . . . . . . . . 11.1 Composition equations: alternance. . . . . . . . . . . . . . . . . . . . . . . 11.2 Composition equations: resurgence and displays. . . . . . . . . . . . . . 11.3 Some remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Iteration and conjugation equations: what is so special about them.

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146 146 146 149 152 152 153 154 154 156 156 159 159 161 161 161 163 163 163 163 164 166 167 168 168 168 169 175 176

The Natural Growth Scale 11.5 Stokes constants and coefficient asymptotics. . . . . . . . . . . . . . . . . . . . . . . . 12 Some examples of composition equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Example of non-polarising composition equation. . . . . . . . . . . . . . . . . . . . . 12.2 Example of polarising composition equations. . . . . . . . . . . . . . . . . . . . . . . . 12.3 Example of polarising composition equation with additional symmetry. . . . . . 12.4 Parity separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 More examples: twins and continued conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Reminders about formal, identity-tangent twins. . . . . . . . . . . . . . . . . . . . . . 13.2 Simplest instance of rigid twins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Removable and non-removable invariants. . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Simplest instance of non-rigid twins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 An analogue of continued fractions: continued conjugation. . . . . . . . . . . . . . 14 Tables: iso-derivations and iso-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The three bases Dnf g , Dsf g , Daf g of ISO. . . . . . . . . . . . . . . . . . . . . . . . . . e in the three bases of ISO. . . . . . . . . . . . . . . . . . . . . 14.2 The involution D 7! D 14.3 The co-product D 7! .D/ in the three bases of ISO. . . . . . . . . . . . . . . . . . 14.4 Embedding of ISO into ]ISO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Tables: how construction-sensitive is E1 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The c .e x  1/-based ultraexponentials for c near 1. . . . . . . . . . . . . . . . . . . 15.2 The c .e x  1/-based ultraexponentials for c moderate. . . . . . . . . . . . . . . . . 15.3 The c .e x  1/-based ultraexponentials for c large and the tardy onset of the staircase regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Ultraexponentials based on c .e x  1/ ; c x e x ; 2c sinh.x/. . . . . . . . . . . . . . x 15.5 The c .e e  1/-based ultraexponentials. . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Conclusion: central facts, main questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Central facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Some open questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 179 180 181 182 183 184 185 185 186 189 190 191 192 192 194 196 199 207 207 208 209 214 215 216 216 216 219 222

1 Program: exploring/completing the natural growth scale 1.1 Groups of real germs. Successive extensions. The present paper purports to investigate the various groups Gext of one-dimensional real germ mappings (for technical convenience, near C1 rather than C0) that can be obtained by starting from some elementary germ group G and then imposing closure under the resolution of various types of composition equations or systems – mainly the following four types

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Ti of increasingly general equations (where f denotes the unknown): f ıq D f0ıp

.p=q 2 Q/ .fractional iteration/

f ı f1 D f2 ı f

.conjugation/

.T1 / .T2 /

id D f ınr ı fr ı    ı f ın1 ı f1

.ni 2 N/

.positive composition/ .T3 /

id D f ınr ı fr ı    ı f ın1 ı f1

.ni 2 Z/

.general composition/

.T4 /

As it happens, neither the general shape of the ultimate group extensions Gext nor the sort of difficulties to arise along the way, significantly depend on the initial group G, whether P that be the group Gana of all invertible real analytic germs f W x 7! c0 x C cn x 1n .c0 > 0/ at C1, or the group G WD< T; E > generated by the unit shift T WD x 7! x C 1 and the exponential E WD exp, or even the group G WD< T > generated by the sole unit shift and its twins (as defined in §13)! All these constructions result in kindred groups Gext , each of which can serve as a fairly satisfactory model for what we may call the natural growth scale. On the other hand, complicating all these constructions but also providing for excitement and surprises, two main difficulties will keep arising: the omnipresence of divergence and the unavoidability of very fast-growing germs.1 1.2 Non-oscillation and comparability. We shall be working in a setting completely ‘free of oscillations’, in the sense that our initial groups G as well as their extensions Gext shall contain only pair-wise comparable germs. The corresponding (strict) order will systematically be noted  7! < Gext 1 ; G2 >  < G1 ; G2 >

(1.2)

in which the order  < still holds.2 1.3 Divergence: resurgent or/and cohesive. Divergence, in this context, can only be of two sorts, resurgent or cohesive, and it has the saving grace of being always resummable: it complicates but does not destroy the connexion between our germs f e as power series - or series of a as geometric objects, and their formal counterparts f far more general nature3 . Resurgence, whether mono- or polycritical (i.e. forcing us to go through one or several intermediary models to perform resummation), always results in germs 1

and of course of the very slow-growing reciprocal germs. the sequel will make clear, this is only a small part of what we mean when saying that the extensions do not depend too much on the initial groups. 3 namely, transseries or ultraseries – see below. 2 As

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f that are real-analytic on some tapering complex neighbourhood of :::; C1Œ. In resurgence’s wake come the so-called alien derivations and, dual to them, the pseudovariables, which together generate a rich, flexible, and very useful algebraic-analytic apparatus. Cohesiveness4, on the other hand, is closely related with the frequent occurrence e , which of finitely (resp. transfinitely) iterated exponentials in the formal objects f e then assume the form of transseries (resp. ultraseries). These generalised series f converge absolutely5 on some strictly real neighbourhood of C1, and their sums f belong to a remarkable class of quasi-analytic functions – the so-called ‘cohesive’ class COHES. 1.4 The ‘display’ and its many uses. To each resurgent germ f there corresponds an object, noted Dpl f (‘display’ of f ), that involves the variable proper, but also a huge number of so-called pseudo-variables. The ‘display’ has many uses. Firstly, it carries all the local information about f , including its Stokes contants or holomorphic invariants. Secondly, it is the key to a complete understanding of the relation (upset by resurgence) between the formal and the geometric side, i.e. between the e and the germs f . Thirdly, any relation R.f1 ; :::; fn / D id between (trans-)series f germs immediately extends to an identity R.Dpl f1 ; :::; Dpl fn / D id between their displays, unchanged in outward form but implying a much stronger set of constraints. This is hugely useful for establishing all sorts of transcendence and independence theorems. 1.5 Extensions: exponential or ultra-exponential. We cannot have stability under T1 ; T2 (see §1.1), let alone under T3 ; T4 , without introducing very fast or slow growing germs. There are actually two steps here. (i) In the first step, we are content with introducing finite iterates of the exponential and logarithm: En WD E ın ; Ln D Lın

.E WD exp ; L WD log/

(1.3)

On the formal side, this leads to so-called transseries, and on the geometric side to analysable germs. (ii) The second step has us introduce even more exotic newcomers, namely the transfinite iterates E˛ and L˛ , with an iteration order ˛ running through the semi-open transfinite interval Œ!; ! ! Œ, where ! stands for the first inaccessible ordinal. It is in fact enough to define the ultraexponentials En WD E! n and their reciprocals, the ultralogarithms Ln WD L! n . They are required to verify E1 WD E D expI L1 WD L D logI 4 cohesiveness

En .x C 1/ exp.En1 .x//  1 C Ln .x/ Ln1 .log.x//

(1.4) (1.5)

stricto sensu, i.e. non-analytic cohesiveness. directly, if they carry only convergent power series, or indirectly, after the divergent-resurgent power series they carry (tucked away within the exponential towers) have been separately resummed. 5 either

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These relations, though not fully determining En and Ln , yet suffice to rigidly constrain their growth regimen. Even on the formal side, this leads to serious complications. It forces us to consider so-called ultraseries, which, unlike the more manageable transseries, admit not one but several competing canonical forms (yet remain pairwise comparable). While transseries suffice for most purposes of non-oscillating asymptotics, in particular in differential calculus, ultraseries cannot be avoided if we demand closure under all composition equations T3 ; T4 . 1.6 Functional incarnation of transfinite arithmetics. Having got hold of a system – any system – of ultraexponentials and ultralogarithms, we easily define the corresponding general transfinite iterates E˛ and L˛ (˛ < ! ! ). We can then replace the slow-growing L˛ by suitable equivalence classes ŒL˛  so defined as to remove the indeterminacy inherent in the construction of the ultralogarithms. Next, we find that composition naturally carries over to the classes ŒL˛ , giving rise to a semi-group ŒL, with transfinite iteration itself smoothly extending to ŒL. As it turns out, this double structure on ŒL exactly reflects the semi-ring structure of the transfinite interval Œ1; ! ! Œ, with its non-commutative addition, non-commutative multiplication, and semi-distributivity.6 1.7 Iso-differential operators and convexity. We shall require a special class of operators, the so-called iso-differential operators Dnfng : Y Dnfn1 ;:::;nr g f WD .Dnfni g f / with Dnfni g f WD .1/ni @ ni log.1=f 0 / (1.6) i

They are indexed by non-ordered sequences of positive integers fng and span a bialgebra ISO which is far better suited to germ composition and to the description of fast/slow germs than the larger bialgebra DIFF spanned by the ordinary differential operators Dfng : Y Dfn1 ;:::;nr g f WD f .ni / .ni 2 N / (1.7) i

DIFF and ISO both possess non-cocommutative co-products, respectively  and , that reflect their action on germ composition ı. They also possess (quite distinct) commutative products, respectively  and . The bialgebra ISO owes its name to the fact thatPits operators7 have a double homogeneousness, measured by an ‘isodegree’ jnj WD ni simultaneously stable under  and . It is also useful to embed ISO into a vaster bialgebra ]ISO spanned by operators hni De which are no longer strictly differential and whose indices are now ordered 6 Thus, a logical-mathematical structure, which when first introduced met with fierce resistance on account of its supposedly ethereal character, reveals itself to be isomorphic to a very natural structure, firmly anchored in concrete, down-to-earth analysis. 7 unlike those of DIFF.

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integer sequences. On ]ISO, both product and co-product assume much simpler expressions. Moreover, ISO and ]ISO possess, as co-algebras, positive cones ISOC and ] ISOC with bases Dafng and Dahni rich in unexpected algebraic-combinatorial properties and leading to a new notion of iso-convexity better adapted to germ compostion than ordinary convexity. To sum up, we have these four structures: ISO



]

ISOI

ISOC



ISOC

]

(1.8)

1.8 Universal asymptotics of fast/slow germs. With their natural adequation to germ composition, the iso-differential operators enlarge the circle of operations and equations at our disposal for carrying out group extensions G 7! Gext . But their main utility lies in this: any iso-differential operator D acting on any ultra-slow germ L (say, on any transfinite iterate of L) produces a germ D:L whose natural asymptotic expansion depends on D alone, not on L.8 1.9 Stubborn indeterminacy in the realisation of ultraexponentials. The system (1.4)-(1.5) determines each pair .Ln ; En / in terms of .Ln1 ; En1 /, but only up to preresp. post-composition by a 1-periodic germ P . 9 That, plus the fact, just mentioned, of all ultra slow/fast germs sharing a universal asymptotics, dashes all hope of selecting a privileged solution .Ln ; En / based purely on real-asymptotic criteria. On the other hand, the possibility, however remote, cannot be dismissed off hand that one of these systems might possess extensions to the complex domain so regular or so distinctive as to single it (that system) out as clearly ‘optimal’. To further complicate the picture, we shall find that all the ‘reasonable’ candidates for the first non-elementary pair .L1 ; E1 /10 are extremely close to one another. So the question is still open, and likely to remain so for quite a while. 1.10 Spirit of this paper: exploratory rather than systematic. The present investigation is unapologetically exploratory in spirit and method. We isolate each of the main difficulties, describe in detail the methods for overcoming them (they involve a lot of fancy machinery), outline the unexpected features (there are quite a few of them), and illustrate everything on a series of select examples. But we do not attempt an exhaustive description of all possible extensions Gext of all possible germ groups G, especially where so doing would force us to grapple with the most general transseries or ultraseries. One excuse for this caution or restraint is that we are handling here an inflatable subject-matter and venturing into almost limitless territory, where exhaustive all too easily rhymes with exhausting, and thorough implies unreadable11. But there is another reason, which is the danger of diminishing returns. Indeed, the extensions Gext that we get by imposing full closure under T1 -T4 (and under isodifferential equations for good measure), though huge, are also in a sense sparsely is only the trans-asymptotic part of D:L that depends on L. precisely, a germ P that commutes with the unit shift T . 10 derived from the pair .L ; E / D .L; E/. 0 0 11 Cf Voltaire:“The secret of being a bore is to tell everything". 8 It

9 More

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populated. They do not seem, for the moment at least, to contain all that many native germs of intrinsic interest, by which we mean remarkable germs arising naturally and directly within the new framework, as opposed to germs obtained by solving composition equations with external, pre-extension data. To put it bluntly: these extensions, though huge, have a wasteland quality about them. They exhibit low biodiversity, compared with, say, classical complex analysis with its wealth of ‘special functions’. This applies in particular to the rarefied ultraexponential range, which would be hardest and most unrewarding to map out down to the last details and which for that reason shall receive here only a sketchy treatment.

2 Tools: resurgence, acceleration, cohesiveness, analysability This section presents – mainly for perspective and to settle notations – a very cursory survey of resurgence theory and its basic tools. 2.1 Resurgent functions. The three models. Resurgent ‘functions’ live simultaneously in three models: (i) in the formal model, as formal power series e ' .z/ or series of a more general type (here the tilda always stands for ‘formal’), (ii) in the convolutive model, as analytic germs b ' ./ defined near the origin 0 of C WD C  f0g; admitting an endless analytic continuation (usually highly ramified) laterally along any finite, finitely punctured broken line; possessing at most a discrete configuration of singular points !; and growing at most exponentially when  goes to 1 radially or ultimately radially,12 (iii) in the geometric model(s), as analytic germs ' .z/ defined in certain sectorial ' .z/ as asymptotic neighbourhoods j arg.z 1 /   j <  C =2 and admitting there e series. e ' .z/ .... ' .z/ z-plane .multiplication/ Fig: 2:1 B & % L b ' ./ -plane .convolution/

B

Despite its auxiliary character, the convolutive model or ‘Borel plane’ 13 is where most obstacles to resummation assume tangible form in the shape of singular points ! ultimately responsible for the divergence of e ' .z/, and where these obstacles can be overcome. The product there is the finite-path convolution (2.1), unambiguously defined for small values of , and then extended in the large by analytic continuation: Z .b '1 b ' 2 /./ WD b ' 1 .1 / b ' 2 .  1 / d 1 (2.1) 0

12 i.e.

following a broken line whose last segment is infinite. 13 ‘plane’ is here something of a misnomer, since the functions b '. / usually live on highly ramified Riemann surfaces over the ‘Borel plane’, or over a finite sector j arg.  0 /j < ı, or even over the positive real axis RC .

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Together, the formal model (our starting point) and the geometric models (our end goal) constitute the multiplicative models, where the product is ordinary multiplication. We go from one model to the next via algebra homomorphisms. The first of these is the Borel transform B. It acts term-wise and turns any power series e ' .z/ with coefficient growth of type Gevrey 1 into a power series b ' ./ with non-zero radius of convergence B W z  7!  1 =. / B W z 7! ı n

B We ' .z/ D

. … N/

.n/

X

an z n

7!

.n 2 N ; ı D Dirac/ X b ' ./ D an  n1 =.n1/Š

The second transform is the Laplace transform L or, for distinctiveness, L : Z ei 1 L W b ' ./ 7! ' .z/ D b ' ./ ez d  .arg   /

(2.2) (2.3) (2.4)

(2.5)

0

Here are some elementary identities for future use: B We ' 1 :e ' 2 7! b '1 b '2 B W @e ' .z/ 7! b @b ' ./ WD  b ' ./ BW

.@ WD d=dz/ .z/ D '.zC / ) b ' ./ D exp. / b./

(2.6) (2.7)

(2.8) n X ./ b ./ (2.9) B W .z/ D .' ıh/.z/ ) b./ D .b b ıb h/./ WD b./C b „n ./

nŠ This last identity (2.9) can be resorted to each time we must post-compose something by h D id C „ with „.z/ D o.1/. Minors and majors. The convolution integral (2.1) makes sense only if each factor b ' i ./ is radially integrable at 0 . When this is not the case, the germs b ' ./ – the socalled minors – have to be supplemented by companion germs, the so-called majors, which are defined only modulo the space REG of regular germs at 0 . They relate to the minors according to the formula:  1  i ' .ei / . near 0 / (2.10) z ' .e /  z 2 i Major convolution (compatible with minor convolution but of wider scope) is given by the rule: Z 1 .z ' 1 u z ' 2 /./ D z ' .1 / z ' 2 .  1 / d 1 (2.11) 2 i I. ;u/ 1 i h1 1 i i .0 <  < u  1/  C e 2 u ;  C eC 2 u with I.; u/ D 2 2 The definition makes good sense, since the small path I.; u/ keeps clear of 0 and since, modulo REG, the integral on the left-hand side of (2.11) does not depend on the choice of u. b ' ./ D 

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J. Ecalle

2.2 Convolution preserving averages. Whenever the axis arg  D  of Laplace integration carries singularities, the multivalued integrand b ' ./ must be replaced by a univalued average  b ' ./, so that the resummation scheme of Fig. 2.1 becomes: B

L



e ' .z/ ! b ' ./ ! b ' ./ ! '.z/

(2.12)



Such an average  W b ' 7!  b ' is defined via its weights . ! / : X . 1 ;:::; r / . 1 ;:::; r / b ' ./ WD  !1 ;:::; !r b ' !1 ;:::; !r ./ if !r <  < !rC1

(2.13)

i 2fC;g

where !1 ; !2 : : : are the successive singular points on arg  D  and where . 1 ;:::; r / ' ./ on the interval !r ; !rC1 Πthat corb ' !1 ;:::; !r ./ denotes the determination of b responds to the right (resp. left) circumvention of !i if i D C (resp. i D ) starting from the origin. Crucially, the average must respect convolution ' 2 / .b ' 1 / .b '2/ .b '1 b

.first local, second global/

(2.14)

Although the above requirement imposes stringent algebraic constraints on the weights  . ! / , there is still a whole zoo of such averages. Let us mention only the most useful. The trivial lateral averages. The right average C and left average  involve only one determination: 1 .!

˙ 1

;:::; r ;:::; !r

/

D 1 .resp. 0/ if 1 D    D r D ˙

.resp. otherwise/

(2.15)

These elementary ‘averages’ have simplicity going for them, but they fail to respect realness : when  D 0 and e ' .z/ is real, C b ' ./ and  b ' ./ are not, except in the trivial case when b ' ./ is regular and uniform on RC . The ‘standard’ average. Its weights, solely dependent on the i ’s, are given by the direct formula: 1 .!



;:::; r 1 ;:::; !r

/

WD

with

.p C 12 / .q C 12 /

D

.r C X X 1 ; q WD 1 p WD 1/ . 12 / . 12 /

.2 p/Š .2 q/Š pŠ qŠ .p C q/Š

4pCq

.p C q D r/

(2.16) (2.17)

i D

i DC

The ‘organic’ average. Its weights are given by the inductive formula: 1 .!



with

;:::; r 1 ;:::; !r

/

. !C /



1

1 .!

WD 

;:::; r1 1 ;:::; !r1

. ! /

WD 

1

WD

1 2

/

1 !r1  1 C r1 r 2 !r

(2.18) (2.19)

103

The Natural Growth Scale

The ‘standard’ and ‘organic’ averages both respect convolution and realness. The simpler standard average is sufficient for most intents and purposes, but in some (fairly rare) cases one must resort to the organic average (or to any one of a host of so-called well-behaved averages) in order to get a function :b ' ./ that does not grow faster than the lateral determinations ˙ :b ' ./. 2.3 Alien derivations. To capture the always important, and often remarkable, behaviour of b ' ./ near14 its singular points !, we require a system of linear operators b! carrying indices ! 2 C , behaving à la Leibniz with respect to convolution  b! b b! b b! .b '1 b ' 2 / . '1/ b '2 C b ' 1 . '2/ 

(2.20)

and yielding 0 whenever the test function b ' ./ has no singularities above !. The action of these -operators, known as alien derivations, is given15 by a formula reminiscent of (2.13): X r . 1 ;:::; r / . 1 ;:::; r / b! b ' ./ WD ' !1 ;:::; !r . C !/ .!r WD !/ (2.21)  ı !1 ;:::; !r b 2 i i D˙



with the weights ı. ! / subject to strong algebraic constraints in order to ensure (2.20). Here are the main systems of alien derivations : The ‘standard’ alien derivations. Their weights depend only on the sign sequence .1 ; : : : ; r1 /: ı

1 .!

;:::; r 1 ;:::; !r

/

WD

pŠ qŠ .p C q C 1/Š

with

p WD

1i r1 X

1I

p WD

1i r1 X

1 (2.22)

i D

i DC

The ‘organic’ alien derivations. Their weights are given by: ( ı

1 .!

;:::; r 1 ;:::; !r

/

.!pC1  !p /=.2 !r / if .1 ; :::; r / D ..C/p ; ./q ; r / .!qC1  !q /=.2 !r / if .1 ; :::; r / D ../q ; .C/p ; r / 0 otherwise

WD

Alien derivations in the multiplicative models. For use in the multiplicative models, we set : b! B ! WDB 1  b ! L1 ! WDL  ! WDe 14 or, 15

! z



!

.formal model/ .geometric models/ .formal and geometric models/

due to multivaluedness, above !. first for small on the axis arg D arg !, then in the large by analytic continuation.

(2.23) (2.24) (2.25)

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J. Ecalle

The Leibniz rule now looks even more Leibnizian : 16 ! .'1 : '2 / .! '1 /:'2 C '1 :.! '2 / ! .'1 : '2 / .! '1 /:'2 C '1 :.! '2 /

.z-plane/ .z-plane/

(2.26) (2.27)

Thanks to their exponential factor e!z , the ‘bold-face’ or ‘invariant’ operators ! have the great advantage of commuting with the ordinary z-differentiation @ WD @z b! ; b b! H) Œ! ; @ D ! ! H) Œ! ; @ D 0 Œ @ D ! 

(2.28)

and of behaving optimally simply (‘invariantly’) under post-composition by an identity tangent g: ! .f ı g/ .! f / ı g C .@f / ı g : .g/

if

g.z/  z

(2.29)

2.4 Pseudovariables and displays. Pseudovariables Z! D Z!1 ;:::;!r . The notion of pseudovariable is dual to that of alien derivation (of the bold-face or invariant sort). Pseudovarialbles carry as upper indices sequences ! WD .!1 ; :::; !r / of arbitrary length r. Multiplication for them reduces to sequence shuffling, while differentiation (ordinary or alien) and postcomposition obey the predictable rules: X   0 00 Z! (2.30) Z! : Z! D ! 2 shuffle.!0 ; !00 / @z Z !1 ;:::;!r D 0 ( !0 Z !1 ;:::;!r D

(2.31) Z !2 ;:::;!r 0

Z! ı g D Z!

if

if !0 D !1 if !0 D 6 !1

(2.32)

g.z/ D z C o.z/

(2.33)

The display. The display is best thought of as some sort of ‘alien Taylor expansion’: XX Z!1 ;:::;!r !r : : : !1 ' (2.34) Dpl ' WD ' C r

!j

It has a double character  both local (via its z-dependence) and global (via its Zdependence). It encodes, in ultra-compact and user-friendly form, a huge amount of information about the function b ' ./, describing as it does the behaviour of b ' ./ at each ! and on each of its various Riemann sheets. What is more, any relation R between resurgent functions automatically extends to their displays : f R.'1 ; : : : ; 's / 0 g H)

f R.Dpl '1 ; : : : ; Dpl 's / 0 g

(2.35)

which is fantastically helpful for establishing transcendence or independence results. 16

We drop the tilda or the polarisation angle .

105

The Natural Growth Scale

2.5 Multicritical resurgence and accelero-summation. When the full formal solution of a local analytic equation or system (say, a singular ODE) involves, alongside the familiar power series of z 1 , a mixture of several non comparable exponential blocks, for instance blocks of the form ui e ij zj with z1 WD h1 .z/  z2 WD h2 .z/      zr WD hr .z/

.e:g: zj z ˛j with 0 < ˛j "/

one is usually confronted with multi-critical resurgence. Concretely, this means that, instead of applying the simple, mono-critical resummation scheme (2.12), one must go successively through a number of distinct Borel planes j  as many as there are distinct ‘critical times’ zj . The intermediary functions b ' j .j / generically possess faster than exponential growth at 1 and each transition b ' j .j / ! b ' jC1 .jC1 / is via a so-called acceleration transform Cj;jC1 . These two complications aside, the situation in each j -plane remains much the same as in the mono-critical case : on each intermediary function b ' j .j / there act specific alien derivations, generating their own resurgence equations and contributing their own Stokes constants. The overall scheme reads: e ' 1 .z1 /

e ' .z/

'.z/

#B b ' 1 .1 /

'r .zr / L"

! b ' 2 .2 / !    ! b ' r1 .r1 / ! b ' r .r / C1 ; 2 C2 ; 3 Cr1 ; r

Fig: 2:2

2.6 Acceleration and deceleration transforms. A single pair CF ; C F of integral kernels does service for the four combinations of minor/major, ac/decelerations, but with a characteristic diagonal ‘flip’: " acceleration deceleration #

minor CF CF z1  z2 ; z1 D F .z2 / Fig:2:3 major CF CF These kernels depend on the germ F that expresses the slower ‘time’ z1 in terms of the faster one z2 . Z cCi 1 1 CF .2 ; 1 / WD ez2 2 z1 1 dz2 with z1 F .z2 / (2.36) 2 i ci 1 Z C1 F C .2 ; 1 / WD ez2 2 Cz1 1 dz2 with z1 F .z2 / and 1  u Cu

(2.37) Acceleration from 1 to 2 with z1 D F .z2 / and 1  F .x/  x: Z C1 b ' 2 .2 / D CF .2 ; 1 / b ' 1 .1 / d 1 z ' 2 .2 / D

1 2 i

(2.38)

C0 Z cCi 1 ci 1

C F .2 ; 1 / z ' 1 .1 / d 1

(2.39)

106

J. Ecalle

Deceleration from 2 to 1 with z1 D F .z2 / and 1  F .x/  x: 1 1 b ' 1 .1 / D 2 i 1 z ' 1 .1 / D

Z

02

2 b ' 2 .2 / C F .2 ; 1 / d 2

.u > 0/

(2.40)

01 Z Cv

2 z ' 2 .2 / CF .2 ; 1 / d 2

C0

(2.41)

Here again, we notice a flip of finite/infinite, path/loop integrals. Integration in (2.38) is along an infinite path, in (2.41) along a finite one. Integration in (2.39) is along an infinite loop that encircles 0 anticlockwise, in (2.40) along a finite loop from 0 to 0 that encircles 1 > 0 anticlockwise. But the basic, really useful transform is of course minor acceleration (2.38), and the crucial point to note here is that the lower kernel CF .2 ; 1 / has exactly the right faster-than-exponential rate of decrease (as 1 ! C1) to make the acceleration integral (2.38) convergent for small enough values of 2 > 0. This defines a germ b ' 2 .2 / which then must, and can, be continued in the large, over the whole of RC . 2.7 Pseudo-acceleration and -deceleration transforms. Here, the change is between two equivalent ‘times’, denoted for distinction by z1 and z1 with z1 D z1 C F .z1 / and 1  F .x/  x as above.17 The new transforms serve a totally different purpose that will be made clear in §2.8, but their integral kernels CidCF ; C idCF are closely related to the old ones: CidCF .1 ; 1 / D CF .1  1 ; 1 /

(2.42)

C idCF .1 ; 1 / D C F .1  1 ; 1 /

(2.43)

In keeping with the more elementary character of the new transforms, all integration paths/loops now become finite. Pseudodeceleration from 1 to 1 with z1 D .idCF /.z1 / : Z b ' 1 .1 / D z ' 1 .1 / D

1

C0 Z v2

1 2 i

v1

CidCF .1 ; 1 / b ' 1 .1 / d 1

(2.44)

C idCF .1 ; 1 / z ' 1 .1 / d 1

(2.45)

Pseudoacceleration from 1 to 1 with z1 D .idCF /.z1 / : 1 b ' 1 .1 / D 1 z ' 1 .1 / D 17

1 2 i

Z

02

1 b ' 1 .1 / C idCF .1 ; 1 / d 1

(2.46)

0 Z v1

0

1 z ' 1 .1 / CidCF .1 ; 1 / d 1

The case when z1 and z1 are too close, i.e. when F .x/ D o.1/, is uninteresting.

(2.47)

The Natural Growth Scale

107

2.8 Cohesiveness and the Regularity Scale. Each intermediary step b ' i .i / 7! b ' i C1 .i C1 / of the accelero-summation scheme (see Fig. 2.2) is actually three steps in one: Substep 1. If the accelerand b ' i .i / is ramified over RC , it must be averaged to b ' i .i / relative to some convolution-respecting average . Substep 2. We calculate the acceleration integral (2.38), but with :b ' i .i / in place of b ' i .i /. The integral converges for i C1 small enough and > 0. Substep 3. To turn the new germ b ' i C1 .i C1 / into a global function over RC , we must C continue it forward along R and circumvent every intervening singularity ! to the right and to the left. Obviously, the two operations in substep 3, namely continuation and circumvention of singularities, require some form of quasi-analyticity. Most of the time there is no problem, because most of the time we have analyticity – but not always. This is where cohesiveness comes in and saves the day. Cohesive functions. We define the class COHES of cohesive functions by first extending the classical Denjoy classes ˛DEN to all transfinite orders ˛ < ! ! and then going to the limit:18 n  ˛ DEN WD ff I jf .n/ .t/j < c0;f .c1;f /n log0˛C1 .n/ g (2.48) COHES WD [˛ 0 (resp. 0 < 0). Each transseries splits C e e S  .x/ with s0 2 R and with e S C .x/ resp. into three part S .x/ D S .x/ C s0 C e  e S .x/ carrying only large resp. small transmonomials. We must first define, inductively on n, the logarithm-free objects of exponential depth n. .a0 / The only log-free prime transmonomial of exp-depth 0 is x. .b0 / The only log-free transmonomials of exp-depth 0 are the x a .a 2 R /. .c0 / All log-free transseries of exp-depth 0 are well-ordered series of the form: X e a x  .0 ;  2 R ;  "/ (2.52) S .x/ D a0 x 0 C 0 . Our main generators are going to be: (gen1 ) The unit shift T. (gen2 ) The group GenT of all real shifts T W x 7! x C  . (gen3 ) The group GenP of all real power functions P W x 7! x  . (gen4 ) The exponential E WD exp. n (gen5 ) The ultra-exponentials En D E! n WD expı! (n 2 N). (gen6 ) The composition group CvgPow of convergent real power series of the form P x 7! a x .1 C an x n / with a > 0. (gen7 ) The composition group CvgTrans of gradedly-convergent transseries with a large positive leading term.29 (gen8 ) The composition group CvgUltra of gradedly-convergent ultraseries with a large positive leading term. Our extensors or enlargers will consist in demanding closure under the solving of a given type of composition equations or systems, mainly: (ext1 ) Iteration equations (see T1 in §1.1). (ext2 ) Conjugation equations (see T2 in §1.1). (ext3 ) Positive composition equations (see T3 in §1.1). (ext4 ) General composition equations (see T4 in §1.1). (ext5 ) General composition systems. We shall also pay special attention to an important subclass, the so-called twins equations or siblings systems: (ext6 ) W .f; g/ D id (ff; gg: unknown ‘twins’). (ext7 ) W1 .f1 ; ::; fr / D ::: D Wr1 .f1 ; ::; fr / D id (ffi g: unknown ‘siblings’).

29

i.e. a leading term of the form aM0 M0 .x/; aM0 > 0.

The Natural Growth Scale

115

Twins equations or siblings systems contain only unknowns and seem under-determined30 but in fact, in the most interesting cases,31 they exhibit sporadicity,32 with all the fascination that attaches to sporadic objects. Lasty, our main restrictors or qualifiers will be: (rst1 ) identity-tangency (i.e f .x/  x C o.x/). (rst2 ) shift-tangency (i.e f .x/  x C  C o.1/). (rst3 ) 0-exponentiality (i.e. lim.stat.Ln ı f ı En D id as n ! C1). (rst4 ) finite formal criticity (finitely many distinct prime transsmonomials). (rst5 ) finite analytic criticity (finitely many critical times). (rst6 ) non-polarisation (no singularities on any of the real Borel axes). (rst7 ) finite exponential depth. (rst8 ) analyticity on :::; C1Œ (as opposed to mere cohesiveness).

3 Groups of analysable germs. Complexity hierarchy 3.1 Degrees of divergence. Rising complexity. We have a neat hierarchy with seven degrees Cj of rising complexity. Each degree is defined inductively, relative to the prime transmonomials P .x/ present in a given transseries and taken in increasing order. Each degree reflects the properties of the mock power series e T .x/ of (2.59) ruled by these P .x/ and of their mock coefficients T .x/. We can legitimately drop the tildas,33 since at that stage of the inductive re-summation the T .x/ (and of course P .x/ itself) have already be re-summed. C1 Direct convergence. There is a common abscissa of convergence x0 < C1 e.x/ converge uniformly on Œx0 C ; C1Œ. such that all mock power series T e.x/ has its own abscissa x < C2 Graded convergence. Each mock power series T e T C1 of absolute convergence. These xe T may not be bounded, but there is a common e.x/ can be continued (analytically or cohesively) to the x0 < C1 such that all T whole interval x0 ; C1Œ. e.x/ ruled by P .x/ D C3 Seriable divergence. Some of the mock power series T eM.x/ may have no finite convergence abscissae, but are simultaneously Borel summable relative (i) to any time x equivalent to the ruling time M.x/ but slow enough in the class ŒM.x/, (ii) to any time x faster than M.x/, e.g. all x˛ D .P .x//˛ .˛ > 0/, (iii) with sums independent of the choice of x or x . Given this huge latitude, one cannot speak of ‘critical time classes’, nor are there any resurgence phenomena or Stokes constants attached to this very peculiar, ‘soft’ type of divergence. 30

their non-trivial solutions, when they exist at all, are determined only up to a common conjugation. when one looks for identity-tangent solutions. 32 in the sense that very few such equations or systems possess non-trivial solutions. 33 on the T .x/ and on P .x/, though not yet on e T .x/.  31 e.g.

116

J. Ecalle

C4 Non-polarising resurgent divergence. Some at least of the mock power series e T .x/ exhibit effective resurgence, mono- or even polycritical, usually of critical ˛ time(s) xe T WD P .x/ or xe T ;˛ WD .P .x// , but without singular points on the main C axis R of the corresponding Borel plane(s) e T or e T ;˛ . C5 Polarising resurgent divergence. Same as above, but with singular points on the axes RC of at least some Borel planes. When finitely (resp. infinitely) many alien derivations ! with ! > 0 act effectively on (at least) one and the same mock power series, we speak of weakly (resp. strongly) polarising resurgence.34 C6 Infinite criticity. The number of critical time classes is not bounded. C7 Infinite exponential depth. There is no bound on the height of the exponential towers present in the transseries. More precisely, the transseries carries prime e n of unbounded exponentially: lim sup expo.P e / D C1. transmonomials P Enlargement by conjugation, continuous iteration, extraction. As we shall see: (i) Composition and reciprocation (i.e. taking the composition inverse) respect each of the above seven degrees. (ii) Conjugation or continuous iteration of germs of zero-exponentiality often generates resurgence, but always of non-polarising type. (iii) ‘Extraction’ (i.e. the solving of composition equations or systems), when all the factors gi have zero-exponentiality, often generates resurgence, sometimes even of the weakly (but never strongly) polarising type. (iv) Conjugation of germs of identical but non-zero exponentiality, or more generally ‘extraction’, whenever possible in the transserial framework,35 generically introduces infinite exponential depth in the formal solutions and replaces analyticity by cohesiveness in the germ solutions. (v) We do not know at the moment of any composition equation or system that would generate seriable divergence in their solutions.36 3.2 Groups of convergent analysable germs. Convergent transseries: direct convergence. Their stability under composition and reciprocation is elementary. This applies is particular to the groups < GenT; E > and < PowSer; E > generated by the exponential and all real shifts resp. all real analytic map germs at C1. Both groups already contain transseries which, once written that infinitely many ! with ! > 0 act effectively on e '.z/ is much stronger than saying that b '. / has infinitely many singular points over RC (relative to forward analytic continuation). 34 Saying

35 That is the case iff after the substitution of E for f and E n ni for gi (ni WD expo.gi /) the composition equation has a solution n 2 Z or is trivially verified for all n. 36 but the possibility cannot be completely ruled out in the case of highly alternate composition equations or systems, since their classification in a way runs parallel to that of differential equations or systems, and these sometimes (though extremely rarely) do generate seriable divergence.

117

The Natural Growth Scale

in canonical form, are of a very general form. In particular, the first of these two groups is dense in the second, whether in the natural topology of formal or in that of convergent transseries. Convergent transseries: graded convergence. Elementary transseries of type X X X en:x :.x C n /1 D en:x .n /k x k1 .0 <  "/ (3.1) 0n

0n

0k

present us with the simplest instance of graded convergence: no uniform convergence abscissa x0 , yet no ambiguity at all as to the proper sum. To illustrate how much graded convergence differs from true divergence, let us consider two similar looking difference equations: A1 .x/  A1 .x C 1/ D a1 .x/ WD ex A2 .x/  A2 .x C 1/ D a2 .x/ WD e

˛

x 1C˛

.0 < ˛ < 1/

(3.2)

.0 < ˛ < 1/

(3.3)

with their transserial solutions expanded in canonical form: ˛

with ˇ D 1  ˛ > 0 A1 .x/ D ex x ˇ S1 .x/ X 1C˛ ˛ A2 .x/ D ex en .1C˛/ x cn .x/

(3.4) (3.5)

0n

Here, S1 .x/ is a formal, divergent and resurgent power series (in x 1 and x ˇ ) implicitely defined as the constant-free solution of the difference equation S1 .x/  R1 .x/ S1.x C 1/ D x ˇ

with

R1 .x/ WD .1 C x 1 /ˇ ex

˛ .xC1/˛

(3.6)

with R1 .x/ viewed as convergent series in CŒŒx 1 ; x ˇ . On the other hand, the coefficients cn .x/ in the expansion of A2 are convergent power series explicitely given by cn .x/ D exp. n .x//;

n .x/ D .x C n/1C˛  x 1C˛  n .1 C ˛/ x ˛

(3.7)

with n .x/ and cn .x/ viewed as series of decreasing powers of x. Their domains of convergence jxj > n, however, decrease as n increases, so that we have graded rather than direct convergence in the transseries A2 .x/ ˛

A1 .x/ 2 x ˇ ex CŒŒx 1 ; x ˇ  A2 .x/ 2 e

x 1C˛

CŒŒe

.1C˛/ x ˛

; x 1 ; x ˇ 

.ˇ WD 1  ˛ > 0/

(3.8)

.ˇ WD 1  ˛ > 0/

(3.9)

However, it would be confusing to lump A1 and A2 into the same ‘divergent’ category: A1 exhibits true resurgence, possesses genuine invariants, and can boast a non-trivial display, whereas A2 falls short on all three counts.

118

J. Ecalle m

3.3 Groups of seriable analysable germs. Let denote the multiplicative convolution Z x x dx1 m .h1 h2 /.x/ WD h1 .x1 / h2 . / (3.10) x1 x1 1 We define the ‘compensators’ x  as follows: m

m

m

x 0 ;1 ;:::;r WD x 0 x 1    x r

.i 2 R/

(3.11)

For distinct exponents i we have x 0 ;1 ;:::;r D

X 0i r

x i

Y

.j  i /1

(3.12)

j 6Di

When i occurs 1 C ni times, the formula becomes Œ1Cn0 

x 0

Œ1Cnr 

;:::;r

D .n0 Š : : : nr Š/1 .@0 /n0 : : : .@r /nr x 0 ;:::;r

(3.13)

The easy inequalities 1 j log xjr jxj rŠ ˇ z 0 ;:::;r ˇ ˇ log z ˇr ˇ z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 0 ;:::;r ˇ ˇ ˇ ˇ ˇ log x0 x0 x0 jx 0 ;:::;r j

.i > 0 ;  D inf i /

(3.14)

.x0 < jzj < 1 ; z 2 C /

(3.15)

show that a series '.z/ D

X

a z 

.each  a finite sequence/

(3.16)



can be convergent as a series of compensators, yet divergent as a power series, due to the proximity of some indices i inside the same sequences  . To deal with such series, we take our cue from the inequalities (3.14)–(3.15) and consider slowly spiralling neighbourhoods D of infinity on C : Dx0 ; 0 D fz j jzj:j log zj 0 > jx0 j:j log x0 j 0 g

(3.17)

Next, using the sup norm k:kD on these domains, we define the smaller compensation norm k:kD comp : k'kD comp WD inf

nX

o ja j:kz  kD k'kD

(3.18)

with an inf taken over all possible expansions of ' into infinite sums (3.16) of compensatrors. The compensation norm is multiplicative: D D k'1 '2 kD comp k'1 kcomp k'2 kcomp

(3.19)

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The Natural Growth Scale

That follows from the formula: 0

n D 0D

i00



i01

X

00

00

i0n

i00n ;

in C jn n ;

D r;

0D

0

z 0 ;:::;r z 0 ;:::;s D 



i0r

0

z 0 ;:::;rCs i000



i001

(3.20) (3.21)



i00r

Ds

(3.22)

00

which linearises the product z  z  of two compensators of lengths r 0 ; r 00 into 0 00 /Š compensators z  of length r 0 C r 00 . As a consequence, we can a sum of .rr 0Cr Š r 00 Š speak of the algebra of compensable power series, i.e. power series with a finite compensation norm k:kD comp for some D D Dx0 ; 0 . All well and good, except that taking the inf of all representations (3.16) is clearly not a practical re-summation strategy. Fortunately each compensable series e ' .z/ is Borel resummable, and very flexibly so: Borel summation works not only relative to all variables z WD logz  0 log logz for 0 large enough, but also relative to all variables z  z (e.g. z˛ WD z ˛ ), and in all cases yields the same sum as does the decomposition into sums of compensators. The definition of compensability, along with the re-summation procedure, extends to the mock power series, leading to the larger notion of seriability. Seriability occupies a position midway between convergence and strict resurgence. With the latter it shares divergence, but lacks precisely defined critical times, exhibits no polarisation, and generates no Stokes constants. It is of common occurence in differential geometry: to most objects ridden with Liouvillian small denominators yet having an unambiguous geometric existence,37 there tend to correspond, on the formal side, compensable or seriable expansions. 3.4 Groups of non-polarised analysable germs. Non-polarised analysable germs. This is the case when, on top of some or all of the previous complications, some of the mock power series may exhibit (mono- or polycritical) resurgence, but without any derivation ! .! 2 RC / acting directly on them.38 Displays of non-polarised germs. As a consequence, there are no singular points on RC in any of the Borel planes; no need for convolution averages; no polarisation in the re-summed germs or their displays. Frequent existence of a “geometric construction” (for the solution). Non-polarisation often goes hand in hand with the existence of a geometric construction for the solution f a given composition equation, in the form of a limit f D lim fn , with fn simply defined from the equation’s factors gi . This is definitely the case for iteration, 37

like the transit maps associated with limit cycles of ODEs in planar geometry. they may act indirectly, as initial factors in operator strings ! !r : : : !1 , with ! 2 RC ; !i 62

38 But

RC .

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conjugation, and some composition equations. These constructions, however, “do not mix” under composition, and if we want to perform regular group extensions, investigate the properties of these extensions, compare their elements pairwise, establish non-oscillation, etc, there is in the end no substitute for the transseries approach. 3.5 Groups of polarised analysable germs. Weak and strong polarisation. We say that there is weak (resp. strong) polarisation, when there is resurgence with finitely (resp. infinitely) many ! .! 2 RC / acting simultaneously on at least one mock power series. Composition equations or systems can at most generate weakly polarising resurgence. There is thus no need to resort to well-behaved convolution averages, and there is a clearly privileged sum, corresponding to the standard convolution average. The lesser display. The lesser display (if need be, polycritical) dpl, which takes into account the sole derivations ! with ! 2 RC , suffices for a comparison of all possible sums, relative to all possible choices of convolution averages, for each effective critical time. Integration of large transmonomials. The integration a.x/ 7! A.x/ D @1 a.x/ of large transmonomials a.x/, or of small transmonomials larger than some L0k .x/, is a major source of weakly polarising resurgence. Even convergent transmonomials produce resurgence,39 but of the simplest possible type, with only a single active alien derivation 1 , relative to a single critical time x0 given by ˇ   a.x/ ˇˇ

ˇ x0 D stat:limr!C1 ˇ log 0 (3.23) Lr WD logır ˇ Lr .x/ The limit here is ‘stationary’, since for r large enough the germs on the right-hand side of (3.23) become equivalent at 1. ˇ ˇ x0 D ˇ log a.x/ˇ

if

x0 D log x

if

log a.x/ C1 log x log a.x/ 0 lim 0/ 39

About the sole exceptions are e ˛ x x n or x ˛ .log x/n with n 2 N.

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The Natural Growth Scale

The critical time here is x0 D L4 .x/ and relative to that critical variable (3.26) becomes   A00 .x0 / D ex0 b0 .x0 / A0 .x0 / A.x/ ; b0 .x0 / b.x/ The formal solution is given by

  A0 .x0 / D ex0 B0 .x0 / D ex0 .1 C @x0 /1 b0 .x0 /

with .1 C @/1 expanded straightforwardly in positive powers of @, while the resummation is given by the Laplace transform of b0 .0 / D .1  0 /1 b b 0 .0 / B The definition of b b 0 here is unproblematic, since the monomial b0 .x0 / D b.E4 .x0 // is automatically subexponential in x0 , and Laplace summation too is unproblematic, since there is only one singularity on the positive real axis in the Borel plane. 3.6 Infinite exponential depth. This last complication, which takes us beyond the framework of proper transseries as defined in §2.10, creates few complications on the formal side, but tends to substitute cohesiveness for analyticity in the sums. 3.7 Accelero-summation commutes with composition. Proving this commutation establishes that large, re-summable transseries, taking in any of the seven categories Ci listed above, constitute a semi-group under composition. Then stability under reciprocation (taking the composition inverse) has to be proven. As the formal composition or reciprocation of transseries resolves itself into several steps,40 it is enough to check that accelero-summation commutes with each one of them. The checks are tedious enough (see [10] in a rather special case) but demand little more than dogged patience.

4 Conjugation=iteration of zero-exponentiality germs 4.1 The three steps of conjugation. The aim here is to show that any (large, positive) analysable germ f : f .z/ D a.z/ C A.z/

with

a.z/  A.z/I

expo.a/ D 0:

(4.1)

is analysably conjugate to the unit shift T . That will automatically take care of the mutual conjugation of any two such germs, and also settle for them the matter of continuous iteration. 40 Repeatedly

resorting to the Taylor formula; rephrasing composition and reciprocation in terms of the

operations @ and in the multiplicative plane (resp. b @ and  in the Borel plane); expelling all infinitesimals from the exponentials and logarithms; and lastly re-arranging the terms so produced in accordance with the well order of transseries.

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Conjugating f to T is a process that is best broken down to three steps. Step one: We may assume f .z/  z to be ultimately positive (if not, we replace f by f ı1 ). Regardless of whether f .z/ is  z or  .1 C const/:z or  z, there always exits large enough integers n such that the variable change z D En .z1 /;

z1 D Ln .z/

(4.2)

turns f into a strongly identity-tangent germ f1 : f1 .z1 / D Ln ı f ı En .z1 / D z1 C b.z1 / C B.z1 / I

1  b.z1 /  B.z1 /

(4.3)

Here, b denotes the leading transmonomial of f1 .z1 /  z1 (together with the real scalar in front of it) and B the remaining transseries. If the original transseries of f is convergent, the change of variable keeps it that way. If it is divergent (and resummable), it does not ‘add’ to its divergence (and keeps it resummable). Step two: A second change of variable z1 D h1;2 .z2 /;

z2 D h2;1 .z1 /

with

h2;1 WD @1

1 D b

Z

1 b

(4.4)

turns f1 into a moderately identity-tangent germ f2 , whose second transmonomial is exactly 1: f2 .z2 / D h2;1 ı f1 ı h1;2 .z2 /

  D z2 C .h02;1 b/ ı h1;2 .z2 / C o .h02;1 b/ ı h1;2 .z2 /  b ı h1;2 .z2 /  b ı h1;2 .z2 / Co D z2 C 0 h1;2 .z2 / h01;2 .z2 /

D z2 C 1 C '2 .z2 /

with

'2 .z2 / D o.1/

(4.5) (4.6) (4.7) (4.8)

In nearly all cases this step creates divergence41, but always of resurgent-resummable type $0 h2;1 D c0 D Const

with

$0 D 0 ˇ.z1 /;

0 > 0/

Dpl.h2;1 / D c0 Z$0 C h2;1

(4.9) (4.10)

Step three: We conjugate the germ f2 to the unit shift T by solving the equation: X f2 ı f2 .z2 / D 1 C f2 .z2 / with f2 .z2 / D z2 C .ın f2 /.z2 / (4.11) 1n 41 the exceptional (but obviously important) transmonomials that admit convergent indefinite integrals are z  ; z n .log z/m ; e  z z m . 2 C; n; mN/.

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The Natural Growth Scale

where ın f2 denotes the term that is n-linear in the remainder transseries '2 of (4.8). These n-linear terms are given inductively by the following system of difference equations: .ı1 f2 /.z2 /  .ı1 f2 /.z2 C 1/ D '2 .z2 / .ın f2 /.z2 /  .ın f2 /.z2 C 1/ X 1 p  .p/ .z2 C 1/ .8n > 1/ ınp f2 D '2 .z2 / pŠ 1pC1

(4.51)

The disappearance of the ‘earlier resurgence’52 was predictable in a sense, because a singularity on RC would create a polarisation, albeit of a very elementary sort (one for which all real convolution averages coincide), and that would not sit well with the existence of a privileged geometric solution. Proposition 4.2 (Iterators of general analysable germs f ). If, instead of starting from a (gradedly) convergent analysable germ f as in Proposition 4.1, we start from a general analysable germ f (but still of exponentiality 0), little changes, except that the preexisting resurgence of f (whatever the type of that resurgence) gets superimposed, in an orderly manner, to the very specific resurgence generated by the passage f 7! .f  ; f /. The neatest way to describe the resulting situation is by writing down the displays53. The earlier factorisations (4.38)-(4.40) now assume the form: Dpl f  D .Psd f  / ı .Dpl f / 



(4.52)



Dpl f D .Dpl f / ı .Psd f /

(4.53)

Dpl f ıt D .Dpl f / ı .Psd f ıt / ı .Dpl f /

(4.54)

(i) with ‘elementary factors’ .Dpl f / , .Dpl f / that carries the ‘old resurgence’ and can obtained directly by solving the conjugation equations:54 .Dpl f / ı .Dpl f / D T ı .Dpl f / .Dpl f / ı .Dpl f / D .Dpl f / ı T

(4.55) (4.56)

(ii) and with ‘non-elementary factors’ Psd f , Psd f  that carry the ‘new resurgence’, commute with the unit shift T , and of course verify: id D .Psd f / ı .Psd f  / Psd f

ıt



(4.57) 

D .Psd f / ı T ı .Psd f /

(4.58)

Observe that here, ‘elementary’ simply means ‘obtainable by purely formal manipulations on transseries’.55 But as far as the general shape is concerned, the factors (ii), being 1-periodic56, are often more ‘elementary’ than the factors (i), since there 52 i.e.

the one that appeared in step 2. See Remark 4. which the resurgence equations can be easily derived. 54 Take care to distinguish the present ‘iterators of displays’ .Dpl f / , .Dpl f / from the earlier ‘displays of iterators’ .Dpl f  /, .Dpl f /. See Proposition 4.1. 55 and thus, without recourse to analysis in the Borel plane. 56 more precisely : commuting with the unit shift T . 53 from

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is no a priori bound on the complexity of f , let alone on that of its display and its display’s iterators. A striking illustration of Proposition 4.2 shall be given in §13.5 with the ‘continued conjugation’ decomposition of a germ f . 4.4 Some examples. Example P 1: iteration of power series. For germs f given by a power series f .z/ D c z C 1n an z 1n with c > 0, the iteration pattern is well-known,57 but let us see how these results fit into the general transserial framework. In the non-identity-tangent case, we may assume c > 1. Then step 1 with n D 1 followed by step 2 with h2;1 .z1 / D z1 = log c immediately take us to the form (4.13), but with simple decreasing exponentials on the right-hand side. This is the simplest of all possible cases: the iterators are convergent. In the identity-tangent case, f .z/ D z C ap z 1p C : : : , there is no need for step 1, so that z1 D z, and steps 2 with z2 D p 1ap z1p C    C c1 z1 C  log z1 takes us to the form (4.13), with a right-hand side in the interval F1 but consisting essentially of decreasing powers.58 The only complication here is a resurgence that is governed by the general equations (4.35),(4.36),(4.37), but with invariant operators A! that depend only on the projections !. P Example 2: fractionnal iteration of monic polynomials. Let f be a real monic polynomial of degree d  2: X f .z/ WD z d C ak z k .2 d / (4.59) 0k1

P X j  Dz 1C bn0 ;n1 ;:::;np1 z  0j

0/

c ..log z/c1 / e.log log z/ 2 P2 .log log log z/

where f , despite reducing to a single transmonomial, gives rise, after normalisation by the steps 1, 2, 3, to full-fledged transseries, with some or all of the possible attendant complications.

5 Conjugation/iteration of nonzero-exponentiality germs Since in the coming five sections most germs are defined, and most constuctions make sense, only in real neighbourhoods of C1, we shall throughout call the variable x rather than z, and its conjugate variable  rather than . 5.1 Conjugation of germs with the same exponentiality. Conjugating two analysable germs of unequal exponentiality59, as we shall see in §6, is not possible in the relatively orderly framework of ordinary transseries, but requires the introduction of ultraexponentials and ultralogarithms. At the opposite end, conjugating two analysable germs f and g each of exponentiality 0 is always feasible60, via a germ h WD f ı g  itself of exponentiality 0, as we just saw in §4. That leaves only the 59 A germ f is said to be of exponentiality k if its leading transmonomial a is itself of exponentiality k, i.e. if Ln ı f ı En Ek for n large enough. 60 Provided of course they are of the same type, i.e. both ultimatily contracting or expanding.

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case when f and g have the same exponentiality k 2 Z . By considering if need be the reciprocal germs, we may assume k to be in N and it is enough to treat the case when g is the standard k-exponentialy germ. In other words, it suffices to consider pairs .f; g/ with expo.f / D k, g D Ek and then calculate the direct normaliser f Þ or its inverse Þf by solving either of the equations (5.1), (5.2): f Þ ı f D Ek ı f Þ

.expo.f Þ / D 0/

(5.1)

f ı Þf D Þf ı Ek

.expo. Þf / D 0/

(5.2)

with f .x/ D a.x/ C o.a.x// and expo.a/ D k. The solution f Þ (resp. Þf ) are defined up to pre-composition by En (resp. post-composition by Ln ) with n 2 Z 61 and the normalisers are by definition the unique pair .f Þ ; Þf / of reciprocal germs with exponentiality zero. Let us focus on the (slightly simpler) direct normaliser f Þ . We can always find n large enough to ‘normalise’ to Ek the leading transmonomial of f by a conjugation (5.3) and to make the whole transserial remainder ˛ as small as we wish. We may for example bring f to the form f1 : f1 .x1 / D .Ln ı f ı En /.x1 / D Ek .x1 / C ˛.x/ with

˛.x1 / D o.1/

(5.3)

The conjugation equation then becomes f1Þ ı .Ek C ˛/ D Ek ı f1Þ and its unique solution (both as a transseries and an analysable germ) can easily be expanded into a sum of transserial blocks k n .x1 /. Their exact expression is given at the end of the section, in Example 4, after a graded series of easier cases. This expansion P f1Þ .x1 / D x1 C n k n .x1 / converges unproblematically in the space of transseries. It also converges incredibly fast in the space of analysable germs on a suitable real neighbourhood of C1. The sum is generically non-analytic, but always cohesive, in the transfinite Denjoy class !DEN (and usually in no smaller class), irrespective of the exponentiality k. But before tackling the general situation (Example 6, infra), let us examine five simpler examples, all of them directly relevant to the numerical investigation of §15. As in the preceding section (when investigating zero-exponentiality germs), we shall e and then first assume that our analysable f has a (gradedly) convergent transseries f examine in §5.3 what changes for a general analysable f . 5.2 Graded examples. Example 1. General f of exponentiality 1. f .x/ D ex C ˛.x/ with ˛.x/ D o.1/ or O.1/ The direct normaliser f Þ admits a fast converging expansion X n .x/ f Þ .x/ D x C 1n 61

In the framework of transseries; in that of ultraseries, n may range through R.

(5.4)

(5.5)

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The Natural Growth Scale

with summands n given by the induction

1 .x/ D log 1 C ex ˛.x/ id C  C  C : : : 

1 2 n1 n .x/ D log ı f .x/ .8n  2/ id C 1 C 2 C : : : n2

(5.6) (5.7)

They are uniformly bounded by jn .x/j < Const=@x En .x/ on a real neigbourhood of C1 and their derivatives too admit similar bounds. The sum (5.4) converges to a cohesive germ f Þ in the Denjoy class !DEN. Example 2. Special f of exponentiality 1. f .x/ WD c .ex  1/ .c > 0/ I f Þ D id C

X

n I

f D id C

Þ

0n

X

n

(5.8)

0n

Here, f and its reciprocal f ı1 being equally simple, the direct and reciprocal normalisers have equally explicit expansions: 0 .x/ D log.c/

 log.c/  c  1 .x/ D log 1 C ı f .x/ id C c

  n1 n .x/ D log 1 C ı f .x/ id C 0 C : : : n2

.8n  2/

(5.9)

0 .x/ D  log.c/

 c  log.c/  ı exp.x/ 1 .x/ D log 1 C id

  n1 n .x/ D log 1 C ı exp.x/ .8n  2/ (5.10) c C id C 0 C : : : n2 Example 3. Special f of exponentiality 1. f .x/ WD c x ex .c > 0/ I f Þ D id C

X

n

(5.11)

0n

0 .x/ D log.c x/

 0  1 .x/ D log 1 C ı f .x/ id

  n1 n .x/ D log 1 C ı f .x/ id C 0 C : : : n2

.8n  2/ (5.12)

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Example 4. Special f of exponentiality 1. f .x/ WD c sinh.x/ D

X c x c x .c > 0/ I f Þ D id C n e  e 2 2 0n

0 .x/ D log.c=2/  log.c=2/ x 

e  e2x 1 .x/ D log 1 C .c=2/

  n1 ı f .x/ n .x/ D log 1 C id C 0 C : : : n2

(5.13)

.8n  2/ (5.14)

Example 5. General f of exponentiality 2. x

f .x/ D ee C ˛.x/ with ˛.x/ D o.1/ or O.1/ Here, the direct normaliser admits an expansion X 2 n .x/ f Þ .x/ D x C

(5.15)

(5.16)

1n

with summands 2n whose leading terms are small of exponentiality 2n for ˛ D O.1/, and of exponentiality 2n C n0 for ˛ small of exponentiality n0 : 2 .x/ D log

log.f .x/ 

ex log.id C  C  : : : 

2 4 2 .n1/ / 2 n .x/ D log ı f .x/ .8n  2/ log.id C 2 C 4 : : : 2 .n2/ /

(5.17) (5.18)

Using the induction, these identities may also be re-written in a form better suited for majorising the (exceedingly small) terms 2n :  ˛.x/ 

2 .x/ D log 1 C ex log 1 C ex e   2 n2

log 1 C idC2C::: 2 n4 2 n .x/ D log 1 C ı f .x/ log.id C 2 C : : : 2 n4 /

(5.19) .8n  3/

(5.20)

Example 6. General f of exponentiality k  1. f .x/ D Ek .x/ C ˛.x/ with ˛.x/ D o.1/ or O.1/ .k  1/ Here, the direct normaliser admits an expansion X k n .x/ f Þ .x/ D x C 1n

(5.21)

(5.22)

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The Natural Growth Scale

with summands k n whose leading terms are small of exponentiality k n for ˛ D O.1/, and of exponentiality k n C n0 for ˛ small of exponentiality n0 : L .f .x/ 

k1 k .x/ D log (5.23) Ek1 .x/ L .id C  C  : : : 

k1 k 2k .n1/ k / ı f .x/ .8n  2/ (5.24) k n .x/ D log Lk1 .id C k C 2 k : : : .n2/ k / with alternative, easier-to-majorise expressions analogous to (5.19), (5.20). 5.3 Resurgence and displays. The mapping f 7! .f Þ ; Þf /, as just seen, creates no resurgence in case of a resurgent-free f ; and when f is resurgent, it creates no new resurgence. As usual, this is best seen at the level of the displays: the displays of the normalisers coincide with the normalisers of the displays, and as such, are directly obtainable from the following composition identities: .Dpl f Þ / D .Dpl f /Þ H) .Dpl f Þ / ı .Dpl f / D Ek ı .Dpl f Þ /

(5.25)

.Dpl Þf / D

(5.26)

.Dpl f / H) .Dpl f / ı .Dpl Þf / D .Dpl Þf / ı Ek

Þ

Thus, Dpl f Þ may be calculated simply by replacing f Þ and k n by their respective displays in (5.22), (5.23), (5.24), and then formally expanding everything in series of pseudovariables. We may note in passing that there is no contradiction between the fact that f Þ ; Þf are generically non-analytic (merely cohesive) and the presence of resurgence, for the resurgence in question always attaches to specific sub-transseries of f Þ ; Þf which, when separately re-summed, are analytic.

6 Universal asymptotics of ultra-slow germs62 6.1 The bialgebra of iso-differentiations. An iso-differential operator or iso-differentiation of iso-degree n is a non-linear operator of the form: Df WD

Xr Dn X n1 C:::n 1rn

WD

with

H D log.1=f 0 /

(6.1)

1ni

Xr Dn X n1 C:::n 1rn

an1 ;:::;nr H .n1 / : : : H .nr /

1ni

bn1 ;:::;nr

f .1Cnr / f .1Cn1 / : : : f0 f0

(6.2)

These operators are uniquely adapted to the description of “universal asymptotics" since, as we shall see in a moment, they always produce the same asymptotic series when made to act on ultra-slow germs. 62 This algebra was first introduced by us in 1991 (see [E5]), under a different label (“post-homogeneous operators”) but already in connection with ultra-slow germs.

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Due to their double homogeneousness (– the iso part of their name alludes to that –) they are essentially invariant under pre- and post-composition by simulitudes S : D.S ı f / Df I

D.f ı S / ˛ n

.S.z/ D ˛z C ˇ/

(6.3)

They also generate an interesting bialgebra, since they possess (i) a commutative product , distinct from the non-commutative operator composition and additive with respect to the iso-degree: .D1  D2 / f WD .D1 f /:.D2 f / ideg.D1  D2 / D ideg.D1 / C ideg.D2 / (ii) a non-commutative coproduct D 7!  .D/: X D ;D a D 1 2 D1 ˝ D2 D D ˝ 1 C 1 ˝ D C : : :  .D/ WD

(6.4) (6.5)

(6.6)

degDDdegD1 CdegD2

that reflects the action of iso-differentiations on composition products: X D1 ;D2 D.f2 ı f1 / WD aD .D1 f1 / .D2f2 /ıf1 : .f10 /n2 .n2 WD idegD2 / (6.7) idegDDidegD1 CidegD2

e (iii) an involution D 7! D: e / ı g : .g 0 /n Dg .Df

.n D idegD ; f ı g D id/

(6.8)

that reflects the action of iso-differentiations on functional inverses. All three operations verify the predictable rules, namely:

C

e1  D e2 D1  D2 D D

and

 .D1  D2 / D  .D1/   .D2/

(6.9)

with .Di ˝ Dj /  .Di 0 ˝ Dj 0 / .Di  Di 0 / ˝ .Dj  Dj 0 /

(6.10)

The resulting bialgebra ISO differs advantageously from the so-called Faa di Bruno bialgebra (-multiplicatively generated by all powers @n ) in that the latter lacks a “degree" with nice stability properties under both product and co-product.63 6.2 The first two main bases Dnfg and Dsfg of ISO. The operators D fn1 g W f 7! f .1Cn1 / =f 0 clearly constitute a -multiplicative basis of ISO, but their simplicity is deceptive. The operators Dnfn1g W f 7! .@/n1 log.1=f 0 / lead to far simpler formulae for all operations: co-product, involution etc. They constitute the so-called natural generators of ISO, to which there answers the additive basis:

Y Dnfn1 ;:::;nr g WD Dnfn1g      Dnfnr g W f 7! .@/ni: log.1=f 0 / (6.11) i 63

It differs even more from the co-commutative Leibniz bialgebra that simply reflects the Leibniz rule.

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The Natural Growth Scale

Here and throughout the sequel, the brackets fng signal that the sequence n inside is non ordered (defined only up to order). Ordered sequences n will be within sharp brackets < n > or remain unbracketed. We also require the symmetric generators Dsfn1g , so-called because they react to involution in the simplest way possible: e fn1g D Dsfn1g I Ds

e fn1 ;:::;nr g D .1/r Dsfn1 ;:::;nr g Ds

(6.12)

fnfn1g / would also produce such a symmetric Although the half-sums 1=2 .Dnfn1g  D basis, the following definition (6.13) of Dsf g in terms of Dnf g is to be preferred, not least because it admits an almost identical inverse (6.14), expressing Dnf g in terms of Dsf g : n0 f1g Dn  Dsfn0g 2 n0 f1g Ds  Dnfn0g D r Dnfn0g C 2

Dsf1Cn0g D r Dsfn0g  Dnf1Cn0g

with

Dsf1g D Dnf1g

(6.13)

with

Dnf1g D Dsf1g

(6.14)

Here, r and r denote operators acting as derivations on ISO relative to the natural product : X r:Dnfn1 ;:::;nrg WD Dnfn1 ;:::;1Cnj ;:::;nrg .r WD @/ (6.15) j

fn1 ;:::;nr g

r :Ds

WD

X

Dsfn1 ;:::;1Cnj ;:::;nrg

(6.16)

j

The equivalence of the identities (6.13), (6.14), as well as the “symmetry" relations (6.12), follow from the formula: fnf1Cn0g D r D fnfn0g  n0 Dnf1g  D fnfn0g D

with

fnf1g D Dnf1g D

(6.17)

which is itself a direct consequence of (6.8). The corresponding analytical expressions read fnfn0g D Dsfn0g Dnfn0g

X

n1 n 2 :::nr X

1r

n0 Dn1 C:::Cnr

X

n1 n 2 :::nr X

1r

n0 Dn1 C:::Cnr

X

n1 n 2 :::nr X

1r

n0 Dn1 C:::Cnr

.1/r Hnn10;:::;nr Dnfn1 ;:::;nrg

(6.18)

.2/1r Hnn10;:::;nr Dnfn1 ;:::;nrg

(6.19)

.C2/1r Hnn10;:::;nr Dsfn1 ;:::;nrg

(6.20)

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with the same positive, integer-valued structure constants Hnn10;:::;nr in all three formulae. We may remark in passing that if we set X

t nC1 A.t/ WD Ct C ˛n .n C 1/Š 1n B.t/ WD t C

X 1n

ˇn

t nC1 .n C 1/Š

r1 X

with ˛n0 WD

Hnn10;:::;nr

(6.21)

.1/r Hnn10;:::;nr

(6.22)

n1 C:::nr Dn0 r1 X

with ˇn0 WD

n1 C:::nr Dn0

the integers ˛n ; ˇn possess tree-theoretical interpretations64 and the generating series A and B verify first order ODEs: 1  A0 .t/ D log.1  A.t//I

1  B 0 .t/ D log.1  B.t//

(6.23)

As for the co-product  , the identities (6.6), (6.7) lead to the induction65 n o  .Dnf1Cn1g / D r ˝ id C id ˝ r C Dnf1g ˝ Ideg : .Dnfn1g / (6.24) n o 1 1  .Dsf1Cn1g / D r ˝ id C id ˝ r C Dsf1g ˝ Ideg  Ideg ˝ Dsf1g : .Dsfn1g / 2 2 which in turn yields the analytical expression X 1 1  .Dnfn0g / D K n ;n2 Dnfn g ˝ Dfn2g jn jCn2 Dn0 1

fn0g

 .Ds

/ D

X

Kn

1

;n2

Kn

1

;n2

jn1 jjn2 jDn0

D

X



(6.25)

1 2 e fn2g ˝ Ds e fn1g Dsfn g ˝ Dsfn g  Ds

Dsfn g ˝ Dsfn g  .1/r.n 1

2

1

:n2 /

(6.26)

Dsfn g ˝ Dsfn 2

g

1



jn1 jjn2 jDn0

For the symmetric basis, the right-hand side is (unsurprisingly) alternately symmetric or antisymmetric. For the natural basis, it is linear in the second argument,66 although semi-linearity, by itself, does not suffice to characterise the natural basis.67

64 Thus

the integer ˛n1 is the number of increasing trees with n nodes and cyclically ordered branches. An increasing tree is a rooted tree whose n nodes carry distinct labels ranging over f1; :::; ng, with the labels increasing along any branch starting from   the root. 65 with the notation Op ˝ Op : D ˝ D 1 2 WD .Op1 D1 / ˝ .Op2 D2 / for any two linear operators on 1 2 ISO, and with Ideg denoting the scalar multiplication of any D in ISO by its iso-degree ideg.D/. 66 it involves the single-indexed Dnfn2g rather than the multi-indexed Dsfn2 g of (6.26). 67 The pseudo-natural operators D fn1 g W f 7! f 1Cn1 =f 0 , mentioned and then dismissed at the beginning of this section, also possess right semi-linearity with respect to .

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The Natural Growth Scale

6.3 Universal asymptotics. The algebras Isolog  ]Isolog. Let us consider the algebras68 ]Isologk resp. ]Isolog spanned by the formal series X 1 ;:::;nr i .x/ D .L0p1 .x//n1 : : : .L0pr .x//nr if r k (6.27) `ehn k 1p1 0/ The critical time here is x0 D L.L.x//: it belong to the growth type of ŒŁ C1 . Relative to this critical variable, (10.28) becomes   A0 .x0 / A.x/ ; b0 .x0 / b.x/ (10.30) A00 .x0 / D ex0 b0 .x0 / The formal solution is given by   A0 .x0 / D ex0 B0 .x0 / D ex0 .1 C @x0 /1 b0 .x0 /

(10.31)

with .1 C @/1 expanded straightforwardly in positive powers of @, and the sum is given by the Laplace transform b0 .0 / D .1  0 /1 b b 0 .0 / B

(10.32)

The definition of b b 0 here is unproblematic, since the monomial b0 .x0 / D b.E.E.x0/// is automatically subexponential in x0 , and Laplace summation too is unproblematic, since there is only one singularity on the positive real axis in the Borel plane.

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10.5 Further remarks. Resurgent ultraseries and their displays. The above example (10.31)–(10.32) shows that the display DplA is going to depend on the initial choice of ultraexponential tower, since the real residue b b 0 .1/ in (10.32) clearly depends on it. This is a general feature: the displays of resurgence-carrying ultraseries depend on the choice of ultraexponential tower underlying the construction.95 However, any two exponential towers are connected by a periodic Witt tower (see §8.6) and, based on these, one can produce conversion formulae for the corresponding displays. These formulae show in particular that the independence relations implied by the displays do not, unlike the displays themselves, depend on the choice of ultraexponential tower. Sweeping closure properties. The variable and/or the unknown germ could even be allowed to sit inside the iteration orders of our functional equations – and we still would have closure! In fact, it is hard to think of meaningful problems in analysis that would take us beyond the range of ultraseries, with ! ! as natural upper limit for the iteration orders.

11 Composition equations: resurgence and displays This and the next two sections take up the subject of general composition equations and also, occasionally, composition systems. Though these problems make sense in the general transserial setting, we shall restrict ourselves mostly to germs expressible as power series (often identity-tangent ones), not only to avoid unnecessary – and on the whole notational rather than substantial – complications, but also because this more familiar setting already presents us with the typical difficulties inherent in the subject and with the main methods required for overcoming them. Moreover, since our data and unknowns, though still real germs and defined as usual on Œ:::; C1Œ, will extend to sectorial neighbourhoods of C1 in C , we shall revert to calling z the variable (in multplicative plane) and  the conjugate variable (in the convolutive or Borel plane). 11.1 Composition equations: alternance. For the most general composition equations, i.e. for equations of type T4 (see at the beginning of §1.1), there exist various notions of k-alternance, which roughly measure the number of free parameters present in the general (oscillation-free) solution. But all useful definitions agree in assigning 0-alternance to the “positive" composition equations (i.e. those of type T3 ), 1-alternance to the conjugation equations (type T2 ), and k-alternance to those very special equations involving k imbricated commutators: f:::fff; f1 g; f2 g; : : : ; fk g D f0 95 to

with

ff; gg WD f ı g ı f 1 ı g  1

the extent that one and the same ultraseries may be convergent or divergent depending on that choice:

think again of A in (10.28), which will be convergent or divergent according as the residue b b 0 .1/ vanishes or not, which again depends on the choice of ultraexponential tower.

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The Natural Growth Scale

For a thorough discussion in the case of “twin" equations, see [16]. In any case, the present section is devoted to 0-alternance equations, whose definition is entirely unproblematic. 11.2 Composition equations: resurgence and displays. Let us examine the general 0-alternance composition equation with data gi real-analytic at C1: W .f / D id W .f / WD f

ımr

ı gr : : : f

ım1

ı g1

.mi 2 Z ;

X

(11.1) mi 6D 0/

The factors gi are given, and the unknown f is sought, of the form:   1 / (11.2) gi .z/ D z C i C i .z/ D z C i C i z 1 C : : : i .z/ 2 O.z   1 1 '.z/ 2 O.z / (11.3) f .z/ D z C  C '.z/ D z C  C  z C : : : P P P P with a real shift  WD  i = mi and a real residue  WD  i = mi . Crucial to the discussion are these two exponential polynomials: 

mi X

SW ./ D

sgn.mi /e

i;m i

1i r

D

˛i " X

si e ˛ i



mi X

TW ./ D

1i r

.si 2 Z; ˛i 2 R/

(11.4)

1i m

sgn.mi /e

i;m i

i;mi D

˛i " X

ti e˛i .ti 2 R; ˛i 2 R/

(11.5)

1i m

with SW .0/ D s1 C    C sm D m1 C    C mr 6D 0

.m

X

jmi j/ (11.6)

i;mi D i;mi D

.1 C ::: C i / C .m1 C ::: C mi 1 C mi /  .1 C ::: C i / C .m1 C ::: C mi 1 C mi / 

(11.7) (11.8)

Here, each mi ranges through Œ0; mi  1 or Œmi ; 1 depending on the sign of mi , and m is the number of distinct frequencies i;mi . The second exponential polynomial TW ./ is also second in importance. It merely e.z; u/ of determines the ramification factors z n in the parameter saturated solution f W .f / D id. It vanishes when all residues i vanish, in which case there is no ramification.

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The first exponential polynomial SW ./ is the one that really matters, because its roots j determine (i) the nature of the exponentials in the saturated solution of W .f / D id (ii) the location of the singularities in the Borel plane (iii) the set of active alien derivations.96 The roots j of SW ./, with j running through an enumerable set J , are all 6D 0 (due to (11.6)) and located within a vertical strip  (11.10) with n running through the set J0N of all J -indexed,Pfinitely supported, integer-valued sequences of the form n D fnj j j 2 J ; nj 2 N; nj < 1g and with generically divergent, but always resurgent power series X RW .j /

en .z/ 2 z CŒŒz 1  < n;  >D nj j ; j WD j 0 (11.11) f SW .j / whose ramification factor z n D z is always 1 when the residues j vanish.98 The resurgence support  contains the additive semi-group  generated by the j , but is larger than  : it also contains all elements of the form   j , with  2  . As usual, all resurgence properties are accounted for by the Bridge Equation, which here asumes the form: e.z; u/ D A! f e.z; u/ ! f

.8 !P 2 /

with ! D e! z ! X Aj! uj @uj and A! D un or A! D un Aj! @uj

(11.12)

.as always/ .8 !P D < n; >2  /

.8 !P D < n;  > j 2    /

(11.13) (11.14)

96 That is to say, the set of all  liable to act (with a non-vanishing result) either on f or on some of its ! successive alien derivatives. 97 The normalisation condition is e f nj .z/ D z j C o.z j /. It bears on the pilot components e f nj .z/ j z preceded by the factors uj e . 98 The scalars  in (11.11) are well defined. Indeed, S 0 . / 6D 0 since we assumed all zeros of S W to j j W be simple.

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The Natural Growth Scale

The component-by-component interpretation of the Bridge Equation yields X j  00 0 en0 .z/ D un en00 .z/ A! uj @uj un f un ! f

(11.15)

with n0 D n C n00 to ensure the simultaneous elimination of the exponential terms and u-factors. Eventually, 11:15 reduces to the identities

X en0 .z/ D en0 n .z/ with !P D < n;  > .n0j  nj / Aj! f (11.16) ! f with only a finite number of terms on the right-hand side. In the special case n0 D 0, we have the identities e.z/ D Aj f e j .z/ j f j n

with

P j D j ; nj D fnji jnji D ıij ; i 2 J g

(11.17)

e.z; u/ and f e.z/ with a single term on the right-hand side.99 Lastly the displays of f are given by: X X e.z; u/ e.z; u/ D f e.z; u/ C (11.18) Dpl f Z!1 ;:::;!r A!1 : : : A!r f !i 2

r

e.z/ D f e.z/ C Dpl f

X X r

!i 2

  e.z; u/ Z!1 ;:::;!r A!1 : : : A!r f uD0

(11.19)

Sketch of the proof. First, a few words about the interpretation of the Bridge equation. Although (11.16) and (11.17) show that only alien derivations ! of a very en , yet for any ! 2 n   a special sort can act (effectively) on P any given f derivation chains !r : : : !1 with !i D ! can always be found that will act e or f en . As a consequence, in the Borel plane the functions f bn ./ or effectively on f b./ generally possess singularities over all points of n   resp.  . Moref over, barring the exceptional cases when some of the special (and pairwise commuting) operators Aj D Ajj @uj (with P j D j ) have vanishing coefficients Ajj , the identity100 Y . /nj

Y

j e.z/ en .z/ D f .Ajj /nj f nj Š j

.Pj D j /

(11.20)

j

e, via en from the sole knowledge of f makes it possible to recover all components f some analysis in the Borel plane. Of course, if one knows the composition equation e is the solution, it is far more economical to get these f en by W .f / D id of which f e formally calculating its saturated solution f .z;u/. But if one does not know W .f /, the en from f e, which of course identity (11.20) shows how to retrieve all components f e would be impossible if f were convergent. In other words, composition equations 99 nj0

100

is the sequence fnj I nj0 D 1; nj D 0 if j 6D j0 g. which results from a repeated application of the Bridge equation.

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with strictly resurgent solutions exhibit a far greater ‘inner cohesion’: knowing even a small part of the saturated solution, one can retrieve everything, including (modulo some hard work) the original equation W .f / D id itself. e.z; u/ offers no As for proving Proposition 11.1, calculating the formal integral f e and f en : difficulty, since the coefficients of f X e.z/ D z C  C (11.21) f ak z 1k X en .z/ D z f an;k z k (11.22) are given by inductions of the form101 SW .0/ ak D earlier terms SW .!/ an;k D earlier terms 0 .j / anj ;k D earlier terms .j  k/ SW

.!P D< n;  >/

(11.23) (11.24)

.j D< nj;  >/

(11.25)

As for the analysis part of Proposition 11.1, the shortest way is to solve the perturbed composition equation W .f / D id derived from W .f / D id by viewing both its data and unknown as perturbations of simple shifts: gi ! gi; with f ! f

with

gi; .z/ D z C i C  i .z/ .gi;1 D gi / (11.26) P f .z/ D z C  C 1k  k 'k .z/ .f1 D f / (11.27)

Expanding W .f / in powers of , we get as coefficient of  k the identity (11.28), which translates to (11.29) in the Borel plane: ' k .z/ D polyn. in earlier terms @ p e ' k 0 .z C ˛q / .k 0 < k/ SW .@/ e ' k ./ D convol. polyn. in earlier terms ./ e SW ./ b p

˛q

b ' k 0 ./

(11.28) (11.29)

Repeated division by the exponential polynomial SW ./ and repeated convolutions make clear where the singular points of each b ' k ./ are going to be. Moreover, the right-hand side of (11.29), though more complicated than in the case of pure iteration equations (type T1 , see §1.1), are essentially similar, and surprisingly easy to majorize, especially above RC . By duplicating the argument used for iteration equab ./ is tions,102 one sees that each  and in particular for  D 1, the function f 103 endlessly continuable, with only isolated singularities and (at most) exponential growth along any non-vertical104 axis arg  D  . And this is all the analysis we need in order to establish Proposition 11.1: the algebraic machinery of resurgence takes care of the rest, and leads straightaway to the Bridge equation (11.12)–(11.13). 101 The

“earlier terms" in (11.24) cover all coefficients an0 ;k 0 such that n0  n; k 0  k and jn0 j C k 0
(11.30) and with subexponential ramification factors flanked by ramified powers z : P

en .z/ 2 z e . f e j; .z/ 2 f n

z j .0/ e

0< are not zeros of SW , the same induction holds as in (11.25), except that now J replaces J . e j; , which due to (11.37) reduce to f e j;0 : For the pilot components f n n

P X k e j;0 .z/ D z j .0/ e . 0< is of the form j (due to the to all components f resonances, this may happen even if n is not of type nj; ). . /

.< n; j .0/ > k/ Constn SW j .j / an;k D earlier terms

(11.42)

Lastly, when some of the factors .j .0/  k/ or .< n; j .0/ > k/ in (11.41) 1

or (11.42) vanish, the essential part107 of fnj; or fn , instead of living in CŒŒz   , 1 now lives in CŒŒz    ˝ CŒlog z. clearly does not determine n, and n determines only !, P but not !. i.e. the series part, as opposed to the subexponential factor that precedes it.

106 ! 107

The Natural Growth Scale

175

11.3 Some remarks. Remark 1: Display and saturated solution. There is a vague kinship between the saturated solution f .z; u/ and the display Dpl f : both verify the composition equation108 W .f / D id and both involve a mixture of power series and exponential terms.109. But the display is a far richer and more complex object. For one thing, the saturated solution f .z; u/ has its power series indexed by elements n of J0N or JKN 0, whereas the power series in the display Dpl f are indexed by the incomparably more numerous sequences of !i in . Secondly, whereas f .z; u/ can be derived from the composition equation by purely formal manipulations on power series, Dpl f carries j j;k scalars like A! or A!;n , which, being generically transcendental Stokes constants, are beyond the reach of formal deduction: their calculation necessarily involve some (and often a good deal of) analysis, be it analytic continuation in the Borel plane110 or the recourse to closed (but highly multiple) expansions involving two ingredients: universal monics on the one hand, and the Taylor coefficients of the data gi in the composition equation. Remark 2: A priori constraints on the holomorphic invariants. If, like for the 0-alternance composition equation (11.1) when all the residues i vanish, the components fn .z/ of the saturated solution f .z;u/ are ordinary power series of z 1 , the action of the derivations ! , and by way of consequence the values of the invariant operators A! , will not depend on ! as an element of C , but only on the projection !P on C . This entails a drastic simplification of the display Dpl f .z;u/, whose pseudovariables Z!1 ;:::;!r may themselves be indexed by projections !P i . Even when the fn .z/ are not themselves power series, they are often simply related to power series hn .z/, via an elementary monomial factor fn .z/ D z hn .z/

with

hn .z/ 2 CŒŒz 1 

(11.43)

This in turn implies A !  A!

.8! 2 C ; 8  D e2i k 2 C ; k 2 Z/

(11.44)

so that, here again, it suffices to know the operators A! for ! ranging through a single sheet of C . If, instead, the hn .z/ are ramified power series111 in CŒŒz 1=p , what is required is the knowledge of operators A! for ! ranging through p consecutive sheets of C . More complex situations may arise, but it is exceedingly rare for all values of A! (with !P fixed) to be truly independent, unless of course one starts from fully ramified data gi , e.g. gi .z/ 2 Rfz 1 ; z 1 log zg. 108 As noted in the introduction, the pseudovariables behave like constants under ordinary differentiation or composition, and multiply according to the shuffle product: see §2.4. 109 in the display, the exponential terms enter via the alien derivations ! D e ! z ! . 110 via the (wholly constructive) definition of the alien derivations: see §2.3. 111 That would be the case if in the composition equation (11.1) we were to consider factors g of the form i gi .z/ D z C i z 1p C : : :

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Remark 3: Composition equations with resurgent data. If we now consider composition equations W .f; g1 ; : : : ; gr / D id whose data gi are themselves resurgent (with one or several critical times), the earlier argument112 shows that resurgence will survive, with the specific resurgence generated by the composition equation simply getting ‘grafted’ onto the pre-existing resurgence carried by the data. But how exactly will the two combine? This is where the displays come in handy, since we can calculate Dpl f , which exhaustively describes both resurgences, old and new, in their exact combination, by formally solving the ‘displayed’ composition equation: W .Dpl f; Dpl g1 ; : : : ; Dpl gr / D id

(11.45)

with Dpl f as unknown and the Dpl gi as data. Remark 4: Twin-related composition equations. When all shifts i in (11.3), and so too all frequencies ˛i in (11.4), are commensurate (this is always the case for twinrelated composition equations – see §13), the sum SW ./ is a polynomial of degree d in e˛ for some maximal ˛ . The roots j of SW ./ are therefore of the form 1 C

2 i 2 i m1 ; : : : ;  d C md ˛ ˛

.mj 2 Z/

(11.46)

If the  j are non-resonant,113 the j are not resonant either; but if the  j are, then the relations (11.46) massively amplify that resonance. Iteration and conjugation equations are a striking case in point. So let us have a closer look at them. 11.4 Iteration and conjugation equations: what is so special about them. For the purpose of comparison, let us write the resurgence formulae for the solutions of iteration and composition equations, first in the standard form, then based on parametersaturated solutions. Let f; f1 ; f2 be real-analytic germs of the form z 7! z C 1 C O.z 1 /, with their invariant operators A! ; A1;! ; A2;! , and let h2;1 WD  f2 ı f1 be the conjugator of f2 to f1 , normalised by the condition h2;1 .z/ D z C O.z 1 /. For simplicity, we drop the tildas everywhere. Iteration related resurgence: the standard form. f  ı f D T ı f I

f ı f D f ı T I

Resurgence support:  WD 2 i Z . Complete system of invariants: fA! j !P 2 g

112 See 113

the proof of Proposition 11.1, towards the end. in the sense of (11.9).

f ıt D f ı T ıt ı f 

(11.47)

177

The Natural Growth Scale

Resurgence equations: ! f  .z/ D A! e! f

 .z/

(11.48)  d  @ WD dz

! f .z/ D CA! e! z @ f .z/ 

 ! t  e! f .z/ ! f ıt .z/ 1 e D CA ! @f ıt .z/ @f  .z/





0 if t 2 Z

(11.49) (11.50)

Conjugation related resurgence: the standard form. h2;1 D  f2 ı f1 ! h2;1 @ h2;1

!r :::!1



!0

h2;1

@ h2;1

D

Dr iY i D1

h2;1 ı f1 D f2 ı h2;1

(11.51)



D .A2;!  A1;! /

e! f1 @f1

(11.52)

 

e! f1 .!i !i / A1;!i / .A2;!0 A1;!0 / (11.53) @f1

 D !0 C ::: C !r . with !i WD !0 C ::: C !i 1 and !  WD !rC1

Iteration related resurgence: the parameter-saturated form. The above formulae give the complete resurgence picture with all the Stokes constants, and cannot be bettered for simplicity. However, to get a real grasp of the difference with generic composition equations, we must re-write these results in the general, necessarily clumsier form, based on the parameter-saturated solutions. If we introduce formal periodic functions Pu ; Pu ; Puıt of the form: X uj e.2i /jz (11.54) Pu .z/ D z  j 2Z 

Pu .z/ D z C

X

Pu ı Pu D id

(11.55)

wj .tI u/ e.2i /jz D .Pu ı T ı t ı Pu /.z/

(11.56)

vj .u/ e.2i /jz

with

j 2Z

Puıt .z/ D z C

X

j 2Z

there is no difficulty in expressing the coefficients vj .u/ and wjt .u/ as formal power series of u, and we find that the saturated solutions attached to f  ; f; f ıt admit factorisations of the form f  .z; u/ D .Pu ı f  /.z/  f .z; u/ D .f ı Pu /.z/ f ıt .z; u/ D .f ı Puıt ı f  /.z/

(11.57) (11.58) (11.59)

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J. Ecalle

and analytical expressions of the form X  f  .z; u/ D f  .z/  uj e2ijf .z/ 

f .z; u/ D f .z/ C

X

nj vj1 1

nj r Š

n

:::

vjrjr nj 1 Š

n

ıt

ıt

f .z; u/ D f .z/ C

(11.60)

X wj j1 1

nj r Š

 f .z/ e2i z @jnj z

(11.61)

n

:::

wjrjr nj 1 Š

e2i z Hnt .f ; f /.z/ (11.62)

In view of the factorisations (11.57), (11.58), (11.59), we find that all three saturated solutions verify the same Bridge Equation ! f  .z; u/ D A! f  .z; u/ ! f .z; u/ D A! f .z; u/

.!P 2 2 i Z / .!P 2 2 i Z /

(11.63) (11.64)

! f ıt .z; u/ D A! f ıt .z; u/

.!P 2 2 i Z /

(11.65)

but with invariant operators A! of the form X A! D 2 iA! .j C k/ uj Ck @uj

if

!P D .2 i /k .k 2 Z /

(11.66)

k2Z

These A! are much simpler than the A! predicted by the general theory (see (11.38)). In the present instance, the general A! would be of the form: X   n n A! D uj1jr : : : uj1jr Aj!;n @uj !P D .2 i / k/ (11.67) j Dk

Comparison with generic composition equations. To grasp the scope of the simplification, let us start from r analytic germs: X g1 .z/ D z C 1 C 1 a1;nC1 z n (11.68) gi .z/ D z C i

X

2n

ai;nC1 z n

.2 i < r/

(11.69)

2n

and consider these similar-looking composition equations: id D W .f / D f ır ı g1 id D W .f / D f ı gr ı    ı f ı g1 ı

(11.70) (11.71)

Both equations admit a unique, generically divergent, and always resurgent solution f , with the same resurgence support  WD .2 i / .Z  r Z /, the same exponential polynomials SW ./ D SW ./ D e =r

1  e ; 1  e =r

RW ./ D RW ./ 0

(11.72)

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The Natural Growth Scale

In both cases the saturated solutions f .z; u/ have unramified114 components fn .z/ 2 CŒŒz 1  and verify the Bridge Equation. But whereas for the iteration equation (11.70) the corresponding invariant operators are of the elementary form (11.66), in the case of the mixed equation (11.71) they are (as soon as r  3) of the general form (11.67), without any universally valid a priori relations115 between the various scalars Aj!;n . 11.5 Stokes constants and coefficient asymptotics. Let e ' .z/ be a resurgent power series and b ' ./ its Borel transform. Let prox the finite set of its ‘closest singular points’ ! in the Borel plane, i.e. those lying on the boundary of the convergence disk of './, O and let ! e ' .z/ be the corresonding alien derivatives. ' .z/ D A! e ' ! .z/ D A! e

! .! z/ ! e e ' .z/ D './ O D

X X

an z n I n1

an

 I .n  1/Š

e

! .z/ D A!

O ! ./ D A!

.! 2 prox ; z  1/ X X

b!;m z m

(11.73) (11.74)

m1

b!;m

 .m  1/Š

(11.75)

In all instances of ‘equational resurgence’, in particular in all cases of resurgence ' .z/ as well as the coeffiresulting for composition equations, the coefficients an of e cients b!;m of the alien derivatives are easily accessible (by formal calculations) – the former exactly, the latter up to multiplication by the invariants (Stokes constants) A! in front of them. This leads, for the calculation of the dominant Stokes constants A! (those with ‘closest’ indices !), to a method which relies solely on the asymptotics of the an ’s and the knowledge of a few first coefficients b!;m , and is not just simpler, but numerically more efficient than analytic continution in the Borel plane. Simple contour integration in the Borel plane shows that Z 1C X an n b !

! .t  1/ t n dt C O.j ! .1 C /jn / D A! .n  1/Š 1 prox !2

X ! n Z an z

! .1  t/ t n dt C O.j ! .1 C /jn / D A! .n  1/Š 2 i  prox

(11.76)

!2

(11.77) The second variant, which relies on the majors116 z

! ./ and on a contour integration  that avoids the origin, applies even when the minor b

! ./ fails to be integrable because RW . / D RW . /  0. than the trivial relations pointed out in Remark 3 above (in this case: dependence on !P alone). The shortest way to prove this is to expand each Aj!;n as an entire function of  WD .1 ; : : : ; r / and to push the Taylor expansion in  far enough to disprove the possibility of any given a priori constraints between the scalars Aj!;n . 116 See (2.10). 114

115 Other

180

J. Ecalle

there, due to positive powers of z in e

! .z/. The same recourse to majors makes it possible to extend the identity Z C1 .n  m  1/Š .t  1/m1 n if n > m > 1 (11.78) D t dt .n  1/Š .m  1/Š 1 to all real pairs m; n with n > m. Assuming that sole condition, the contibution of b!;m to an is thus ! n .n  m  1/Š. Therefore, for any fixed m0 > 0, as n goes to C1, we have X X ! n A! .n  m  1/Š b!;m C Rem.n; m0 / (11.79) an D prox

!2 1

m