Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations [1 ed.] 3540437827, 9783540437826

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Lecture Notes in Mathematics Editors: J.–M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1787

3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo

Stefaan Caenepeel Gigel Militaru Shenglin Zhu

Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations

13

Authors Stefaan Caenepeel Faculty of Applied Sciences Vrije Universiteit Brussel, VUB Pleinlaan 2 1050 Brussels Belgium e-mail: [email protected] http://homepages.vub.ac.be/˜scaenepe/welcome.html Gigel Militaru Faculty of Mathematics University of Bucharest Strada Academiei 14 70109 Bucharest 1 Romania e-mail: [email protected]

Shenglin ZHU Institute of Mathematics Fudan University Shanghai 200433 China e-mail: [email protected]

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Caenepeel, Stefaan: Frobenius and separable functors for generalized module categories and nonlinear equations / Stefaan Caenepeel ; Gigel Militaru ; Shenglin Zhu. Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; Vol. 1787) ISBN 3-540-43782-7

Mathematics Subject Classification (2000): primary 16W30, secondary 16D90, 16W50, 16B50 ISSN 0075-8434 ISBN 3-540-43782-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10878510

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Dedicated to Gilda, Lieve and Xiu

Preface

One of the key tools in classical representation theory is the fact that a representation of a group can also be viewed as an action of the group algebra on a vector space. This has been (one of) the motivations to introduce algebras, and modules over algebras. During the passed century, it has become clear that several different notions of module can be introduced, with a variety of applications in different mathematical disciplines. For example, actions by group algebras can also be used to develop Galois descent theory, with its applications in number theory. Graded modules originated from projective algebraic geometry. In fact a group grading can be considered as a coaction by the group algebra, i.e. the dual of an action. One may then consider various types of modules over bialgebras and Hopf algebras: Hopf modules (in integral theory), relative Hopf modules (in Hopf-Galois theory), dimodules (when studying the Brauer group). Perhaps the most important ones are the Yetter-Drinfeld modules, that have been studied in connection with the theory of quantum groups, the quantum Yang-Baxter equation, braided monoidal categories, and knot theory. Frobenius fuctors generalize the classical concept of Frobenius algebra that appeared first 100 years ago in the work of Frobenius on representation theory. The study of Frobenius algebras has seen a revival during the passed five years, serving as an important tool in problems arising from different fields: Jones theory of subfactors of von Neumann algebras ([98], [100]), topological quantum field theory ([3], [8]), geometry of manifolds and quantum cohomology ([79], [129] and the references indicated there), the quantum Yang-Baxter equation ([15], [42]), and Yetter-Drinfeld modules ([49], [88]). Separable functors are a generalization of the theory of separable field extensions, and of separable algebras. Separability plays a crucial role in several topics in algebra, number theory and algebraic geometry, for example in classical Galois theory, ramification theory, Azumaya algebras and the Brauer group theory, Hochschild cohomology and ´etale cohomology. A more recent application can be found in the Jones theory of subfactors of von Neumann algebras, already mentioned above with respect to Frobenius algebras. In this monograph, we present - from a purely algebraic point of view - a unification schedule for actions and coactions and their properties, where we are mainly interested in generalizations of Frobenius and separability prop-

VIII

Preface

erties. The unification theory takes place at four different levels. First, we have a unification on the level of categories of modules: DoiKoppinen modules were introduced first, and all modules mentioned above can be viewed as special cases. Entwined modules arose from noncommutative geometry; they are at the same time more general and easier to deal with, and provide new fields of applications. Secondly, there is a unification at the level of functors between module categories: one can introduce morphisms of entwining structures, and then associate such a morphism a pair of adjoint functors. Many “classical” pairs of adjoint functors (the induction functor, forgetful functors, restriction of (co)scalars, functors forgetting a grading, and their adjoints) are in fact special cases of this construction. A third unification takes place at the level of the properties of these pairs of adjoint functors. Here the inspiration comes from two at first sight completely different algebraic notions, having their roots in representation theory: separable algebras and Frobenius. We give a categorical approach, leading to the introduction of separable functors and Frobenius functors. Not only this explains the at first sight mysterious fact that both separable and Frobenius algebras can be characterized using Casimir elements, it also enables us to prove Frobenius and separability type properties in a unified framework, with several new versions of Maschke’s Theorem as a consequence. The fourth unification is based on the theory of Yetter-Drinfeld modules, their relation with the quantum Yang-Baxter equation, and the FRT Theorem. The pentagon equation has appeared in the theory of duality for von Neumann algebras, in connection with C ∗ -algebras. Here we explain how they are related to Hopf modules. In a similar way, another nonlinear equation which we called the Long equation is related to the category of Long dimodules, that finds its origin in generalizations of the Brauer-Wall group. Finally, the FS equation can be used to characterize Frobenius algebras, as well as separable algebras, providing yet another explanation of the relationship between the two notions. For all these equations, we have a version of the FRT Theorem. In Chapter 1, some preliminary results are given. We have included a Section about coalgebras and bialgebras, and one about adjoint functors. Section 1.2 deals with a classical treatment of Frobenius and separable algebras over fields, and we explain how they are connected to classical representation theory. Chapter 2 provides a discussion of entwining structures and their representations, entwined modules, and we discuss how they generalize other types of modules and how they are related to the smash (co)product and the factorization problem of an algebra through two subalgebras. We also give the general pair of adjoint functors mentioned earlier. First properties of the category of entwined modules are discussed, for example we discuss when the category of entwined modules is a monoidal category. We use entwining structures mainly as a tool to unify all kinds of modules, but we want to point

Preface

IX

out that they were originally introduced with a completely different motivation, coming from noncommutative geometry: one can generalize the notion of principal bundles to a very general setting in which the role of coordinate functions on the base is played by a general noncommutative algebra A, and the fibre of the principal bundle by a coalgebra C, where A and C are related by a map ψ : A ⊗ C → C ⊗ A, called the entwining map, that has to satisfy certain compatibility conditions (see [32] and [33]). Entwined modules, as representations of an entwining structure, were introduced by Brzezi´ nski [23], and he proved that Doi-Koppinen Hopf modules and, a fortiori, graded modules, Hopf modules and Yetter-Drinfeld modules are special cases. Entwined modules can also be applied to introduce coalgebra Galois theory, we come back to this in Section 4.8, where we also explain the link to descent theory. The starting points of Chapter 3 are Maschke’s Theorem from Representation Theory (a group algebra is semisimple if and only if the order of the group does not divide the characteristic of the base field), and the classical result that a finite group algebra is Frobenius. Larson and Sweedler have given Hopf algebraic generalizations of these two results, using integrals. Both the Maschke and Frobenius Theorem can be restated in categorical terms. Let us first look at Maschke’s Theorem. If we replace the base field k by a commutative ring, then we obtain the following result: if the order of the group G is invertible in k, then every exact sequence of kG-modules that splits as a sequence of k-modules is split as a sequence of kG-modules. If k is field, this implies immediately that kG is semisimple; in fact it turns out that all variations of Maschke’s Theorem that exist in the literature admit such a formulation. In fact we have more: the kG-splitting maps are constructed deforming the k-splitting maps in a functorial way. A proper definition of functors that have this functorial Maschke property was given by N˘ ast˘asescu, Van den Bergh, and Van Oystaeyen [145]. They called these functors separable functors because for a given ring extension R → S, the restriction of scalars functor is separable if and only if S/R is separable in the classical sense. A Theorem of Rafael [158] gives necessary and sufficient conditions for a functor with an adjoint to be separable: the unit (or counit) of the adjunction has to be split (or cosplit). We will see that the separable functor philosophy can be applied successfully to any adjoint pair of functors between categories of entwined modules. We will focus mainly on the functors forgetting the action and the coaction, as this is more transparent and leads to several interesting results. A similar functorial approach can be used for the Frobenius property. It is well-known that a k-algebra S is Frobenius if and only if the restriction of scalars functors is at the same time a left and right adjoint of the induction functor. This has lead to the introduction of Frobenius functors: this is a functor which has a left adjoint that is also a right adjoint. An adjoint pair of Frobenius functors is called a Frobenius pair.

X

Preface

Let η : 1 → GF be the unit of an adjunction; as we have seen, to conclude that F is separable, we need a natural transformation ν : GF → 1. Our strategy will be to describe all natural transformations GF → 1; we will see that they form a k-algebra, and that the natural transformations that split the unit are idempotents (separability idempotents) in this algebra. A look at the definition of adjoint pairs of functors tells us that we have to investigate natural transformations GF → 1 and 1 → F G; the difference is that the normalizing properties for the separability property and the Frobenius property are not the same. But still we can handle both problems in a unified framework, and this is what we will do in Chapter 3. In Chapter 4, we will apply the results from Chapter 3 in some important subcases. We have devoted Sections to relative Hopf modules and Hopf-Galois theory, graded modules, Yetter-Drinfeld modules and the Drinfeld double, and Long dimodules. For example, we prove that, for a finitely generated projective Hopf algebra H, the Drinfeld double D(H) is a Frobenius extension of H if and only if H is unimodular. Part I tells us that Hopf modules, Yetter-Drinfeld modules and Long dimodules over a Hopf algebra H can be regarded as special cases as DoiKoppinen Hopf modules and entwined modules, and that a unified theory can be developed. In Part II, we look at these three types of modules from a different point of view: we will see how they are connected to three different nonlinear equations. The celebrated FRT Theorem shows us the close relationship between Yetter-Drinfeld modules and the quantum Yang-Baxter equation (QYBE) (see e.g. [115], [108], [128]). We will discuss how the two other types of modules, Hopf modules and Long dimodules, are related to other nonlinear equations. It comes as a surprise that the nonlinear equation related to the category of Hopf modules H MH is the pentagon (or fusion) equation, which is even older, and somehow more basic then the quantum Yang-Baxter equation. Using Hopf modules, we will present two different approaches to solving this equation: a first approach is to prove an FRT type Theorem for the pentagon equation; a second, completely different, approach was developed by Baaj and Skandalis for unitary operators on Hilbert spaces ([10]) and, more recently, by Davydov ([65]) for arbitrary vector spaces. We will conclude Chapter 6 with a few open problems that may have important consequences: from a philosophical point of view the theory presented herein views a finite dimensional Hopf algebra H simply as an invertible matrix R ∈ Mn2 (k) ∼ = Mn (k) ⊗ Mn (k) that is a solution for the pentagon equation R12 R13 R23 = R23 R12 . Furthermore, in this case dim(H)|n. This point of view could be crucial in reducing the problem of classifying finite dimensional Hopf algebras (currently in full development and using different and complex techniques) to the elementary theory of matrices from linear algebra. At this point a new Jordan theory (we called it restricted Jordan theory) has to be developed. In Chapter 8, we will focus on the Frobenius-separability equation, all solu-

Preface

XI

tions of which are also solutions of the braid equation. An FRT type theorem will enable us to clarify the structure of two fundamental classes of algebras, namely separable algebras and Frobenius algebras. The fact that separable algebras and Frobenius algebras are related to the same nonlinear equation is related to the fact that separability and Frobenius properties studied in Chapters 3 and 4 are based on the same techniques. As we already indicated, the quantum Yang-Baxter equation has been intensively studied in the literature. For completeness sake, and to illustrate the similarity with our other nonlinear equations, we decided that to devote a special Chapter to it. This will also allow us to present some recent results, see Section 5.5. The three authors started their common research on Doi-Koppinen Hopf modules in 1995, with a three month visit by the second and third author to Brussels. The research was continued afterwards within the framework of the bilateral projects “Hopf algebras and (co)Galois theory” and “Hopf algebras in Algebra, Topology, Geometry and Physics” of the Flemish and Romanian governments, and “New computational, geometric and algebraic methods applied to quantum groups and differential operators” of the Flemish and Chinese governments. We benefitted greatly from direct and indirect contributions from - in alphabetical order - X-TO-Status: 00000003 Margaret Beattie, Tomasz Brzezi´ nski, Sorin Dˇ ascˇalescu, Jose (Pepe) G´omez Torrecillas, Bogdan Ichim, Bogdan Ion, Lars Kadison, Claudia Menini, Constantin Nˇ astˇasescu, S¸erban Raianu, Peter Schauenburg, Mona Stanciulescu, Dragos S ¸ tefan, Lucien Van hamme, Fred Van Oystaeyen, Yinhuo Zhang, and Yonghua Xu. Chapters 2 and 3 are based on an seminar given by the first author in Brussels during the spring of 1999. The first author wishes to thank Sebastian Burciu, Corina Calinescu and Erwin De Groot for their useful comments. Finally we wish to thank Paul Taylor for his kind permission to use his “diagrams” software. A few words about notation: in Part I, we work over a commutative ring k; unadorned Hom, ⊗, M etc. are assumed to be taken over k. In Part II, we are always assuming that we work over a field k. For k-modules M and N , IM will be the identity map on M , and τ : N ⊗ M → M ⊗ N will be the switch map mapping m ⊗ n to n ⊗ m. Also it is possible to read part II without reading part I first: one needs the generalities of Chapter 1, and the definitions in the first Sections of Chapter 2.

Brussels, Bucharest, Shanghai, February 2002

Stefaan Caenepeel Gigel Militaru Shenglin Zhu

Table of Contents

Part I Entwined modules and Doi-Koppinen Hopf modules 1

Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Coalgebras, bialgebras, and Hopf algebras . . . . . . . . . . . . . . . . . 3 1.2 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 Separable algebras and Frobenius algebras . . . . . . . . . . . . . . . . . 28

2

Doi-Koppinen Hopf modules and entwined modules . . . . . . 2.1 Doi-Koppinen structures and entwining structures . . . . . . . . . . 2.2 Doi-Koppinen modules and entwined modules . . . . . . . . . . . . . . 2.3 Entwined modules and the smash product . . . . . . . . . . . . . . . . . 2.4 Entwined modules and the smash coproduct . . . . . . . . . . . . . . . 2.5 Adjoint functors for entwined modules . . . . . . . . . . . . . . . . . . . . 2.6 Two-sided entwined modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Entwined modules and comodules over a coring . . . . . . . . . . . . 2.8 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 48 50 59 64 68 71 78

3

Frobenius and separable functors for entwined modules . . . 3.1 Separable functors and Frobenius functors . . . . . . . . . . . . . . . . . 3.2 Restriction of scalars and the smash product . . . . . . . . . . . . . . . 3.3 The functor forgetting the C-coaction . . . . . . . . . . . . . . . . . . . . . 3.4 The functor forgetting the A-action . . . . . . . . . . . . . . . . . . . . . . . 3.5 The general induction functor . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 99 124 137 146

4

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Relative Hopf modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hopf-Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Doi’s [H, C]-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Yetter-Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Long dimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Modules graded by G-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Two-sided entwined modules revisited . . . . . . . . . . . . . . . . . . . . . 4.8 Corings and descent theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 168 179 181 193 195 198 204

XIV

Table of Contents

Part II Nonlinear equations 5

6

Yetter-Drinfeld modules and the quantum Yang-Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The quantum Yang-Baxter equation and the braid equation . . 5.3 Hopf algebras versus the QYBE . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The FRT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The set-theoretic braid equation . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 218 225 235 238

Hopf modules and the pentagon equation . . . . . . . . . . . . . . . . . 6.1 The Hopf equation and the pentagon equation . . . . . . . . . . . . . 6.2 The FRT Theorem for the Hopf equation . . . . . . . . . . . . . . . . . . 6.3 New examples of noncommutative noncocommutative bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The pentagon equation versus the structure and the classification of finite dimensional Hopf algebras . . . . . . . . . . . .

245 245 253

7

Long dimodules and the Long equation . . . . . . . . . . . . . . . . . . . 7.1 The Long equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The FRT Theorem for the Long equation . . . . . . . . . . . . . . . . . . 7.3 Long coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301 301 304 311

8

The Frobenius-Separability equation . . . . . . . . . . . . . . . . . . . . . . 8.1 Frobenius algebras and separable algebras . . . . . . . . . . . . . . . . . 8.2 The Frobenius-separability equation . . . . . . . . . . . . . . . . . . . . . . 8.3 The structure of Frobenius algebras and separable algebras . . 8.4 The category of FS-objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317 320 332 339

267 277

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

1 Generalities

1.1 Coalgebras, bialgebras, and Hopf algebras In this Section, we give a brief introduction to Hopf algebras. A more detailed discussion can be found in the literature, see for example [1], [63], [140] or [172]. Throughout, k will be a commutative ring. In some specific cases, we will assume that k is a field. k M = M will denote the category of (left) k-modules (we omit the index k if no confusion is possible). ⊗ and Hom will be shorter notation for ⊗k and Homk . Let M and N be k-modules. IM : M → M will be the identity map, and τM,N : M ⊗ N → N ⊗ M the switch map. Indices will be omitted if no confusion is possible. M ∗ = Hom(M, k) is the dual of the k-module M . For m ∈ M and m∗ ∈ M ∗ , we will often use the duality notation m∗ , m = m∗ (m) Let M be a finitely generated and projective k-module. Then there exists a (finite) dual basis {mi , m∗i | i = 1, · · · , n} for M . This means that m=

n


m∗i , mmi and m∗ =

i=1

n


m∗ , mi m∗i

i=1

for all m ∈ M and m∗ ∈ M ∗ . Algebras and coalgebras Recall that a k-algebra (with unit) is a k-module together with a multiplication map m = mA : A⊗A → A and a unit element 1A ∈ A satisfying the conditions m ◦ (m ⊗ I) = m ◦ (I ⊗ m) m(a ⊗ 1A ) = m(1A ⊗ a) = a for all a ∈ A. The map η = ηA : k → A mapping x ∈ k to x1A is called the unit map of A and satisfies the condition

S. Caenepeel, G. Militaru, and S. Zhu: LNM 1787, pp. 3–37, 2002. c Springer-Verlag Berlin Heidelberg 2002


4

1 Generalities

m ◦ (η ⊗ I) = m ◦ (I ⊗ η) = I The opposite Aop of an algebra A, is equal to A as a k-module, with multiplication mAop = mA ◦ τ . A is commutative if A = Aop , or m ◦ τ = m. k-alg will be the category of k-algebras, and multiplicative maps. Coalgebras are defined in a similar way: a k-coalgebra C is a k-module together with k-linear maps ∆ = ∆C : C → C ⊗ C and ε = εC : C → k satisfying (∆ ⊗ I) ◦ ∆ = (I ⊗ ∆) ◦ ∆

(1.1)

(ε ⊗ I) ◦ ∆ = (I ⊗ ε) ◦ ∆ = I

(1.2)

∆ is called the comultiplication or the diagonal map, and ε is called the counit or augmentation map. (1.2) tells us that the comultiplication is coassociative. We will use the Sweedler-Heyneman notation for the comultiplication: for c ∈ C, we write
c(1) ⊗ c(2) = c(1) ⊗ c(2) ∆(c) = (c)



The summation symbol will usually be omitted. The coassociativity can then be reformulated as follows: c(1)(1) ⊗ c(1)(2) ⊗ c(2) = c(1) ⊗ c(2)(1) ⊗ c(2)(2) and therefore we write ∆2 (c) = (∆ ⊗ I)(∆(c)) = (I ⊗ ∆)(∆(c)) = c(1) ⊗ c(2) ⊗ c(3) and, in a similar way, ∆3 (c) = c(1) ⊗ c(2) ⊗ c(3) ⊗ c(4) The counit property (1.2) can be restated as ε(c(1) )c(2) = ε(c(2) )c(1) = c The co-opposite C cop of a coalgebra C is equal to C as a k-module, with comultiplication ∆C cop = τ ◦ ∆C . C is called cocommutative if C = C cop , or τ ◦ ∆ = τ , or c(1) ⊗ c(2) = c(2) ⊗ c(1) for all c ∈ C. A k-linear map f : C → D between two coalgebras C and D is called a morphism of k-coalgebras if ∆D ◦ f = (f ⊗ f )∆C and εD ◦ f = εC

1.1 Coalgebras, bialgebras, and Hopf algebras

5

or f (c)(1) ⊗ f (c)(2) = f (c(1) ) ⊗ f (c(2) ) and εD (f (c)) = εC (c) for all c ∈ C. We also say that f is comultiplicative. The category of kcoalgebras and comultiplicative map is denoted by k-coalg. The tensor product of two coalgebras C and D is again a coalgebra. The comultiplication and counit are given by the formulas ∆C⊗D = (IC ⊗ τC,D ⊗ ID ) ◦ (∆C ⊗ ∆D ) and εC⊗D = εC ⊗ εD Example 1. Let X be an arbitrary set, and C = kX the free k-module with basis X. On C we define a comultiplication and counit as follows: ∆C (x) = x ⊗ x and εC (x) = 1 for all x ∈ X. kX is called the grouplike coalgebra. The convolution product Let C be a coalgebra, and A an algebra. Then we can define a multiplication on Hom(C, A) in the following way: for f, g : C → A, we let f ∗ g = mA ◦ (f ⊗ g) ◦ ∆C , that is, (f ∗ g)(c) = f (c(1) )g(c(2) ) This multiplication is called the convolution. ηA ◦ εC is a unit for the convolution. In particular, if A = k, we find that C ∗ is a k-algebra, with unit ε, and comultiplication given by c∗ ∗ d∗ , c = c∗ , c(1) d∗ , c(2)  In fact, the multiplication on C ∗ is the dual of the comultiplication on C. If A is an algebra, which is finitely generated and projective as a k-module, then A∗ is a coalgebra. The comultiplication is given by ∗

mA ⊗ A)∗ ∼ A∗ −→(A = A∗ ⊗ A∗

This means that ∆(a∗ ) = a∗(1) ⊗ a∗(2) if and only if a∗ , ab = a∗(1) , aa∗(2) , b for all a ∈ A and b ∈ B. The comultiplication can be described in terms of a dual basis {ai , a∗i | i = 1, · · · , n} of A: ∆(a∗ ) =

n


i,j=1

a∗ , ai aj a∗i ⊗ a∗j

(1.3)

6

1 Generalities

for all a∗ ∈ A∗ . From (1.3), it also follows that n


ai aj ⊗ a∗i ⊗ a∗j =

n


ai ⊗ ∆(a∗i )

(1.4)

i=1

i,j=1

For later use, we rewrite this formula in terms of coalgebras: put C = A∗ , and let {ci , c∗i | i = 1, · · · , n} be a finite dual basis for C. Then

ci ⊗ cj ⊗ c∗i ∗ c∗j (1.5) ∆(ci ) ⊗ c∗i = i,j

i

Bialgebras and Hopf algebras Proposition 1. For a k-module H that is at once a k-algebra and a kcoalgebra, the following assertions are equivalent: 1. mH and ηH are comultiplicative; 2. ∆H and εH are multiplicative; 3. for all h, g ∈ H, we have ∆(gh) = g(1) h(1) ⊗ g(2) h(2) ε(gh) = ε(g)ε(h)

(1.6) (1.7)

∆(1) = 1 ⊗ 1 ε(1) = 1

(1.8) (1.9)

In this situation, we call H a bialgebra. A map between bialgebras that is multiplicative and comultiplicative is called a morphism of bialgebras. Proof. This follows from the following observations: mH is comultiplicative ⇐⇒ (1.6) and (1.8) hold; ηH is comultiplicative ⇐⇒ (1.7) and (1.9) hold ∆H is multiplicative ⇐⇒ (1.6) and (1.7) hold; εH is multiplicative ⇐⇒ (1.8) and (1.9) hold Definition 1. A bialgebra H is called a Hopf algebra if the identity IH has an inverse S in the convolution algebra Hom(H, H). Thus we need a map S : H → H satisfying S(h(1) )h(2) = h(1) S(h(2) ) = η(ε(h))

(1.10)

The map S is called the antipode of H. Let f : H → K be a morphism of bialgebras between two Hopf algebras H and K. It is well-known that f also preserves the antipode, that is, SK ◦ f = f ◦ SH and f is called a morphism of Hopf algebras.

1.1 Coalgebras, bialgebras, and Hopf algebras

7

Example 2. Let G be a semigroup. Then kG is a coalgebra (see Example 1), and a k-algebra. It is easy to see that kG is a bialgebra. If G is a group, then kG is a Hopf algebra. The antipode is given by S(g) = g −1 , for all g ∈ G. If H is bialgebra, then H op , H cop and H opcop are also bialgebras. If H antipode S, then S is also an antipode for H opcop . An antipode S for also an antipode for H cop , and is called a twisted antipode. S has to the property S(h(2) )h(1) = h(2) S(h(1) ) = η(ε(h))

has an H op is satisfy (1.11)

for all h ∈ H. Proposition 2. Let H be a Hopf algebra. Then S is a bialgebra morphism from H to H opcop . If S is bijective, then S −1 is a twisted antipode. If H is commutative or cocommutative, then S ◦ S = IH , and consequently S = S. Proof. Consider the maps ν, ρ : H ⊗ H → H given by ν(h ⊗ k) = S(k)S(h) and ρ(h ⊗ k) = S(hk) It is easy to prove that both ν and ρ are convolution inverses of the multiplication map m, and ν = ρ, and S(hk) = S(k)S(h) for all h, k ∈ H. Furthermore 1 = η(ε(1)) = (I ∗ S)(1) = I(1)S(1) = S(1) and we find that S : H → H op is multiplicative. In a similar way, we prove that S : H → H cop is comultiplicative: the maps ψ, ϕ : H → H ⊗ H given by ψ(h) = ∆(S(h)) and ϕ(h) = S(h(2) ) ⊗ S(h(1) ) are both convolution inverses of ∆H , and therefore ψ = ϕ and ∆(S(h)) = S(h(2) ) ⊗ S(h(1) ) for all h ∈ H. Finally ε(h) = ε((η ◦ ε)(h)) = ε(S(h(1) )h(2) ) = ε(S(h(1) ))ε(h(2) ) = ε(S(h)) Assume that S is bijective. Then S −1 (hk) = S −1 (k)S −1 (h), and S −1 (1) = 1. Applying S −1 to (1.10), we find (1.11), and S −1 is a twisted antipode. Finally, if H is commutative or cocommutative, then S is also a twisted antipode, and we have for all h ∈ H that (S ∗ (S ◦ S))(h) = S(h(1) S(S(h(2) ))   = S S(h(2) )h(1) = S((η ◦ ε)(h)) = (η ◦ ε)(h) proving that S ◦ S is a convolution inverse for S, and S ◦ S = I.

8

1 Generalities

Modules Let A be a k-algebra. A left A-module M is a k-module, together with a map l ψ = ψM : A ⊗ M → M, ψ(a ⊗ m) = am such that a(bm) = (ab)m and 1m = m for all a, b ∈ A and m ∈ M . We say that ψ is a left A-action on M , or that A acts on M from the left. Let M and N be two left A-modules. A k-linear map f : M → N is called left A-linear if f (am) = af (m), for all a ∈ A and m ∈ M . The category of left A-modules and A-linear maps is denoted by A M. In a similar way, we can introduce right A-modules, and the category of right A-modules MA . Let B be another k-algebra. A k-module M that is at once a left A-module and a right B-module such that a(mb) = (am)b for all a ∈ A, b ∈ B and m ∈ M is called an (A, B)-bimodule. A MB will be the category of (A, B)-bimodules. Observe that we have isomorphisms of categories ∼ ∼ A MB = A⊗B op M = MAop ⊗B Take M ∈ MA and N ∈ A M. The tensor product M ⊗A N is by definition l r the coequalizer of the maps IM ⊗ ψN and ψM ⊗ IN , that is, we have an exact sequence ✲ M ⊗ N −→M ⊗A M −→0 M ⊗A⊗N ✲ If H is a bialgebra, then the tensor product of two (left) H-modules M and N is again an H-module. The action on M ⊗ N is given by h(m ⊗ n) = h(1) m ⊗ h(2) n We also write M H = {m ∈ M | hm = ε(h)m, for all h ∈ H} Module algebras and module coalgebras Assume that H is a bialgebra. Let A be a left H-module, and a k-algebra. We call A a left H-module algebra if the unit and multiplication are left H-linear, or h(ab) = (h(1) a)(h(2) b) and h1A = ε(h)1A

(1.12)

for all h ∈ H, and a, b ∈ A. In a similar way, we introduce right H-module algebras. If A is a left H-module algebra, then Aop is a right H opcop -module algebra. A k-coalgebra that is also a left H-module is called a left H-module coalgebra if the counit and the comultiplication are left H-linear. This is equivalent to ∆C (hc) = h(1) c(1) ⊗ h(2) c(2) and εC (hc) = εH (h)εC (c)

(1.13)

1.1 Coalgebras, bialgebras, and Hopf algebras

9

for all h ∈ H and c ∈ C. We can also introduce right module coalgebras, and if C is a left H-module coalgebra, then C cop is a right H opcop -module coalgebra. If C is a right H-module coalgebra, then C ∗ is a left H-module algebra. The left H-action on C ∗ is given by the formula h · c∗ , c = c∗ , ch

(1.14)

In a similar way, if C is a left H-module coalgebra, then C ∗ is a right Hmodule algebra, with c∗ · h, c = c∗ , hc (1.15) Example 3. Let G be a group, and X a right G-set. This means that we have a map X × G → X : (x, g) →  xg such that (xg)h = x(gh), for all g, h ∈ G. Then the coalgebra kX is a right kG-module coalgebra. Comodules Let C be a coalgebra. A right C-comodule M is a k-module together with a map ρ = ρrM : M → M ⊗ C such that (ρ ⊗ IC ) ◦ ρ = (IM ⊗ ∆C ) ◦ ρ and (IC ⊗ εC ) ◦ ρ = IM

(1.16)

We will say that C acts from the right on M . We will use the SweedlerHeyneman notation ρ(m) = m[0] ⊗ m[1] and (ρ ⊗ IC )(ρ(m)) = (IM ⊗ ∆C )(ρ(m)) = m[0] ⊗ m[1] ⊗ m[2] The second identity in (1.16) can be rewritten as ε(m[1] )m[0] = m for all m ∈ M . A map f : M → N between two right comodules is called a morphism of C-comodules, or a right C-colinear map if ρrN ◦ f = (f ⊗ IC ) ◦ ρrM or f (m)[0] ⊗ f (m)[1] = f (m[0] ) ⊗ m[1] for all m ∈ M . MC will be the category of right C-comodules and right C-colinear maps.

10

1 Generalities

Example 4. Let C = kX, with X an arbitrary set. Let M be a k-module graded by X, that is  Mx M= x∈X

where every Mx is a k-module. Then M is a kX-comodule, the coaction is given by ρr (m) = mx ⊗ x

if m = mx with mx ∈ Mx . Conversely, every kX-comodule M is graded by X, one defines the grading by Mx = {m ∈ M | ρ(m) = m ⊗ x} Thus we have an equivalence between MkX and the category of X-graded modules. We have a functor F : MC → C ∗ M defined as follows: for a right C-comodule M , we let F (M ) = M , with left C ∗ -action given by c∗ · m = c∗ , m[1] m[0] for all c∗ ∈ C ∗ and m ∈ M ; if f : M → N is right C-colinear, then it is easy to prove that f is also left C ∗ -linear, and we let F (f ) = f . Proposition 3. The functor F : MC → C ∗ M is faithful. If C is projective as a k-module, then F is fully faithful. If C is finitely generated and projective, then F is an isomorphism of categories. Proof. Take two right C-comodules M and N . Obviously HomC (M, N ) → C ∗ Hom(F (M ), F (N )) is injective, so F is faithful. Assume that C is k-projective, and let {ci , c∗i | i ∈ I} be a dual basis. Let M and N be C-comodules, and assume that f : M → N is left C ∗ -linear. We claim that f is also right C-colinear. Indeed, for all m ∈ M , we have
f (m[0] ) ⊗ c∗i , m[1] ci f (m[0] ) ⊗ m[1] = =




i∈I

f (c∗i · m) ⊗ ci

i∈I

=




c∗i · f (m) ⊗ ci

i∈I

=




c∗i , f (m)[1] f (m)[0] ⊗ ci

i∈I

= f (m)[0] ⊗ f (m)[1]

1.1 Coalgebras, bialgebras, and Hopf algebras

11

Assume moreover that C is finitely generated, and let {ci , c∗i | i = 1, · · · , n} be a dual basis for C. We define a functor G : C ∗ M → MC as follows: G(M ) = M as a k-module, with right C-coaction ρ(m) =

n


c∗i · m ⊗ ci

i=1

We will show that ρ defines a coaction, and leave all other verifications to the reader. We obviously have (IM ⊗ ε)(ρ(m)) =

n


ε(ci )c∗i · m = ε · m = m

i=1

Next we want to prove that (ρ ⊗ IC ) ◦ ρ = (IM ⊗ ∆C ) ◦ ρ

(1.17)

For all c∗ , d∗ ∈ C ∗ , we have   (IM ⊗ c∗ ⊗ d∗ ) ◦ (ρ ⊗ IC ) ◦ ρ (m)  
∗ ∗ = (IM ⊗ c∗ ⊗ d∗ ) (cj ∗ ci ) · m ⊗ cj ⊗ ci i,j


c∗ , cj d∗ , ci (c∗j ∗ c∗i ) · m = i,j ∗

= (c ∗ d∗ ) · m = c∗ ∗ d∗ , ci c∗i · m   = (IM ⊗ c∗ ⊗ d∗ ) c∗i · m ⊗ δ(ci )   = (IM ⊗ c∗ ⊗ d∗ ) ((IM ⊗ ∆C ) ◦ ρr )(m)

and (1.17) follows after we apply Lemma 1

Lemma 1. Let M, Nbe k-modules, and  assume that N is finitely generated and projective. Take j mj ⊗ pj and k m′k ⊗ p′k in M ⊗ N . If

n∗ , pj mj = n∗ , p′k m′k j

k

for all n∗ ∈ N ∗ , then
j

mj ⊗ p j =




m′k ⊗ p′k

k

Proof. Let {ni , n∗i | i = 1, · · · , n} be a dual basis for N . Then



mj ⊗ p j = mj ⊗ n∗i , pj ni = m′k ⊗ n∗i , p′k ni = m′k ⊗ p′k j

i,j

i,k

k

12

1 Generalities

Let H be a bialgebra. If M and N are right H-comodules, then M ⊗ N is again a right H-comodule. The H-coaction is given by ρrM ⊗N (m ⊗ n) = m[0] ⊗ n[0] ⊗ m[1] n[1] We call M coH = {m ∈ M | ρ(m) = m ⊗ 1} the submodule of coinvariants of M . We can also introduce left C-comodules. For a left C-comodule M , the Sweedler-Heyneman notation takes the following form: ρlM (m) = m[−1] ⊗ m[0] ∈ C ⊗ M The category of left C-comodules and left C-colinear maps is denoted by C M. We have an isomorphism of categories C

M∼ = MC

cop

If M is at once a left C-comodule and a right D-comodule in such a way that (ρl ⊗ ID ) ◦ ρr = (IC ⊗ ρr ) ◦ ρl then we say that M is a (C, D)-bicomodule. We then write, following the Sweedler-Heyneman philosophy: (m[0] )[−1] ⊗ (m[0] )[0] ⊗ m[1] = m[−1] ⊗ (m[0] )[0] ⊗ (m[0] )[1] = m[−1] ⊗ m[0] ⊗ m[1] = ρlr (m) Observe that C itself is a (C, C)-bicomodule. C MD is the category of (C, D)bicomodules and left C-colinear right C-colinear maps. We have isomorphisms cop cop C MD ∼ = MC ⊗D = C⊗D M ∼ Proposition 4. Let C be a coalgebra, and M a finitely generated projective k-module. Right C-coaction on M are in bijective correspondence with left C-coactions on M ∗ . Proof. Let {mi , m∗i | i = 1, · · · , n} be a dual basis for M , and let ρr : M → M ⊗ C be a right C-coaction. We define ρl = α(ρr ) : M ∗ → C ⊗ M ∗ by ρl (m∗ ) =

n
i=1

This is a coaction on M ∗ since

mi[1] ⊗ m∗ , mi[0] m∗i

(1.18)

1.1 Coalgebras, bialgebras, and Hopf algebras

(IC ⊗ ρl )(ρl (m∗ )) =

n


13

mi[1] ⊗ mj[1] ⊗ m∗ , mi[0] m∗i , mj[0] m∗j

i,j=1

=

n


mj[1] ⊗ mj[2] ⊗ m∗ , mj[0] m∗j

j=1

= (∆C ⊗ IM ∗ )(ρl (m∗ )) n n

ε, mi[1] m∗ , mi[0] m∗i ε(m∗[−1] )m∗[0] = i=1

i=1

=

n


m∗ , mi m∗i = m∗

i=1

Conversely, given ρl : M ∗ → C ⊗ M ∗ , we define ρr = α (ρl ) : M → M ⊗ C by n
m∗i[0] , mmi ⊗ m∗i[−1] ρr (m) = i=1

An easy computation shows that α and α  are each others inverses.

The category of comodules over a coalgebra over a field k is a Grothendieck category. Over a commutative ring, we have the following generalization of this result, due to Wisbauer [187]. Proposition 5. Let C be a coalgebra over a commutative ring k. The following assertions are equivalent: 1. C is flat as a k-module; 2. MC is a Grothendieck category and the forgetful functor MC → M is exact; 3. MC is an abelian category and the forgetful functor MC → M is exact. Proof. 1. ⇒ 2. It is clear that MC is additive. Let f : M → N be a map in MC . To prove that Ker (f ) is a C-comodule, we need to show, for any m ∈ Ker (f ): ρ(m) ∈ Ker (f ) ⊗ C = Ker (f ⊗ IC ) (using the fact that C is k-flat). This is obvious, since (f ⊗ IC )ρ(m) = f (m[0] ) ⊗ m[1] = ρ(f (m)) = 0 On Coker (f ), we put a C-comodule structure as follows: ρ(n) = n[0] ⊗ n[1] for all n ∈ N . This is well-defined: if n = f (m), then n[0] ⊗ n[1] = f (m)[0] ⊗ f (m)[1] = f (m[0] ) ⊗ m[1] = 0

14

1 Generalities

It is clear that every monic in MC is the kernel of its cokernel, and that every epic is the cokernel of its cokernel, so MC is an abelian category. Let us next see that MC is an AB3-category. If {Mλ | λ ∈ Λ} is a family in MC , then M = ⊕λ Mλ is again a comodule: we have maps Mλ

✲ Mλ ⊗ C

ρλ

✲ M ⊗C

iλ ⊗IC

and therefore a unique map ρ : M → M ⊗ C making M into a comodule, and iλ into a right C-colinear map. The fact that MC is an AB5-category follows easily since M is AB5, and the functor forgetting the C-coaction is exact. Let us finally show that MC has a family of generators. First observe that every right C-comodule of the form M ⊗ C, with C-coaction induced by C, is generated by C. Indeed, for any k-module M , we can find an epimorphism k (λ) → M in M, and therefore an epimorphism k (λ) ⊗ C = C (λ) → M ⊗ C in MC . Now we claim that the C-subcomodules of C form a family of generators of MC . It suffices to show that for every right C-comodule M and m ∈ M , there exists a C-subcomodule D of C and a C-colinear map f : D → M such that m ∈ Im (f ). ρ : M → M ⊗ C is a monomorphism in MC , so M is isomorphic to ρ(M ) = {n[0] ⊗ n[1] | n ∈ M }. C generates M ⊗ C, so there exists a Ccolinear map f : C → M ⊗ C and c ∈ C such that f (m) = ρ(m). Now let D = {d ∈ C | f (d) ∈ ρ(M )} Indeed, for d ∈ D, we can find n ∈ N such that f (d) = n[0] ⊗ n[1] , and we see that (ρ ⊗ IC )(f (d)) = n[0] ⊗ n[1] ⊗ n[2] ∈ ρ(M ) ⊗ C Now look at the diagram with exact rows that defines D: 1

1

✲ D

✲ C

f

f

❄ ✲ ρ(M )

❄ ✲ M ⊗C

C is flat, so we have a commutative diagram with exact rows 1

✲ D⊗C f ⊗ IC

1

✲ C ⊗C f ⊗ IC

❄ ❄ ✲ ρ(M ) ⊗ C ✲ M ⊗ C ⊗ C

1.1 Coalgebras, bialgebras, and Hopf algebras

15

and D ⊗ C = {x ∈ C ⊗ C | (f ⊗ IC )(x) ∈ ρ(M ) ⊗ C It follows that ρ(d) ∈ D ⊗ C, and D is a right C-comodule. We now have f : D → ρ(M ) ∼ = M in MC , and f (c) = m[0] ⊗ m[1] ∼ = m. 2. ⇒ 3. is trivial. 3. ⇒ 1. The forgetful functor F : MC → M is a left adjoint of • ⊗ C : M → MC . The unit and counit of the adjunction are given by ρ : M → M ⊗ C ; ρ(m) = m[0] ⊗ m[1] εN : N ⊗ C → N

: εN (n ⊗ c) = ε(c)n

for all M ∈ MC and N ∈ M. It is well-known that a functor between abelian categories that is a right adjoint of a covariant functor is left exact (see e.g. [11, I.7.1]), and it follows that • ⊗ C : M → MC is exact. Now the forgetful functor MC → M is also left exact, by assumption, so the composition • ⊗ C : M → M is left exact, and C is flat, as needed. Remark 1. The assumption that the forgetful functor is exact, in the second and third condition of the Proposition, means the following: for a C-colinear map f : MC → MC , the (co)kernel of f in MC has to be equal as a k-module to the kernel of f viewed as a map between k-modules. J. G´ omez Torrecillas kindly pointed out to us that this condition is missing in Wisbauer’s paper [187]. For an example of a coalgebra C such that MC is abelian, while C is not flat, and the functor forgetting the coaction is not exact, we refer to [80]. The cotensor product Take M ∈ MC and N ∈ C M. The cotensor product M
C N = M ⊗C N is defined as the equalizer 0−→M
C N −→M ⊗ N

✲ ✲ M ⊗C ⊗N

Example 5. Let C = kX, and M and N X-graded modules. Then  Mx ⊗ N x M
C N = x∈X

For a fixed right C-comodule M , we have a functor M
C • :

C

M→M

If M is flat as a k-module, then M ⊗ • is an exact functor, and it follows easily that M
C • is left exact, but not necessarily right exact. Definition 2. A right C-comodule M is called right C-coflat if it is flat as a k-module, and if M
C • is an exact functor. A similar definition applies to left C-comodules.

16

1 Generalities

Now take M ∈ MC , N ∈ C M, and P ∈ M. We then have a natural map f : (M
C N ) ⊗ P → M
C (N ⊗ P )   given by f (( i mi ⊗ ni ) ⊗ p) = i mi ⊗ (ni ⊗ p).

Lemma 2. With notation as above, the natural map

f : (M
C N ) ⊗ P → M
C (N ⊗ P ) is an isomorphism in each of the following cases: 1. P is k-flat (e.g. if k is a field); 2. M is right C-coflat. Proof. 1. M
C N is defined by the exact sequence 0−→M
C N −→M ⊗ N ✲ ✲ M ⊗C ⊗N Using the fact that P is k-flat, we obtain a commutative diagram with exact rows ✲ 0 −→ (M
C N ) ⊗ P −→ M ⊗ N ⊗ P ✲ M ⊗C ⊗N ⊗P    ∼ ∼ f = =  ✲ 0 −→ M
C (N ⊗ P ) −→ M ⊗ N ⊗ P ✲ M ⊗C ⊗N ⊗P and the result follows from the Five Lemma (see e.g. [123, Sec. VIII.4]). 2. Recall the definition of the tensor product: N ⊗ P = N × P/I, where I is the ideal generated by elements of the form (n, p + q) − (n, p) − (n, q) ; (n + m, p) − (n, p) − (m, p) ; (nx, p) − (n, xp) and we have an exact sequence of left C-comodules 0−→I−→N × P −→N ⊗ P −→0 and, using the right C-coflatness of M , we find a commutative diagram with exact rows 0

0

−→ M
C I  = 

−→

J

−→ M
C (N × P ) −→ M
C (N ⊗ P )  ∼ f =  −→ (M
C N ) × P

−→ (M
C N ) ⊗ P

−→ 0

−→ 0

and the result follows again from Five Lemma.

Assume that A is a k-algebra, C a k-coalgebra, P ∈ A M, M ∈ MC and N ∈ C MA . By this we mean that N is a left C-comodule and a right Amodule such that the right A-action is left C-colinear, i.e. ρl (na) = n[−1] ⊗ n[0] a for all n ∈ N and a ∈ A.

1.1 Coalgebras, bialgebras, and Hopf algebras

17

Lemma 3. With notation as above, the natural map f : (M
C N ) ⊗A P → M
C (N ⊗A P ) is an isomorphism in each of the following situations: 1. P is left A-flat; 2. M is right C-coflat. Proof. 1) The proof is identical to the proof of the first part of Lemma 2 2) The right A-action on M
C N is given by

mi ⊗ ni )a = mi ⊗ n i a ∈ M
C N ( i

for every



i

i

mi ⊗ ni ∈ M
C N . Now (M
C N ) ⊗A P is the equalizer of (M
C N ) ⊗ A ⊗ P −→ −→ (M
C N ) ⊗ P

which is by Lemma 2 isomorphic to the equalizer of M
C (N ⊗ A ⊗ P ) −→ −→ M
C (N ⊗ P ) and this equalizer is isomorphic to M
C (N ⊗A P ) because M is right Ccoflat. In some situations, the cotensor product can be computed explicitely. Proposition 6. Let M and N be right C-comodules, and assume that M is finitely generated and projective as a k-module. Then we have a natural isomorphism HomC (M, N ) ∼ = N
C M ∗ Proof. We use notation as in Proposition 4. We know from (1.18) that M ∗ is a left C-comodule. From (1.18), we deduce that m∗[0] , mm∗[−1] = m∗ , m[0] m[1]

(1.19)

M is finitely generated projective, so we have an isomorphism α : Hom(M, N ) → N ⊗ M ∗ given by α(f ) =

n


f (mi ) ⊗ m∗i and α−1 (n ⊗ m∗ )(m) = m∗ , mn

i=1

We will show that α restricts to the required isomorphism. Assume first that f is right C-colinear. Using (1.18) we find that

18

1 Generalities




f (mi ) ⊗ m∗i[−1] ⊗ m∗i[0] =

i

=







f (mi ) ⊗ mj[1] ⊗ m∗i , mj[0] m∗j

i,j

f (mj[0] ) ⊗ mj[1] ⊗ m∗j

j

=




f (mj )[0] ⊗ f (mj )[1] ⊗ m∗j

j

 and it followsthat α(f ) ∈ N
C M ∗ . Now take k nk ⊗ m∗k ∈ N
C M ∗ , and let f = α−1 ( k nk ⊗ n∗k ). f is then right C-colinear, since for all m ∈ M , we have
f (m[0] ) ⊗ m[1] = n∗k , m[0] nk ⊗ m[1] (1.19)

=




k

n∗k[0] , mnk ⊗ n∗k[−1]

k

=




n∗k , mnk[0] ⊗ nk[1]

k

= ρ(f (m)) Coflatness versus injectivity Let C be a coalgebra over a field. We will show that a C-comodule is an injective object in the category of C-comodules if and only if it is C-coflat. Our proof is based on the approach presented in [63]. First we need some Lemmas. Lemma 4. Let C be a coalgebra over a field k, and M a right C-comodule. For every m ∈ M , there exists a finite dimensional subcomodule M ′ of M containing m. Consequently there exists an index set J and a set {Mj | j ∈ J} consisting of finite dimensional right C-comodules, and an epimorphism φ : ⊕j∈J Mj → M in MC . Proof. Let {ci | i ∈ I} be a basis for C as a k-vector space, and write
mi ⊗ ci ρ(m) = i∈I

where only a finite number of the mi are nonzero - for a change, we do not use the Sweedler notation. Let M ′ be the k-subspace of M generated by the mk . M ′ is finite dimensional, and
ε(ci )mi ∈ M ′ m= i∈I

We can write ∆(ci ) =




j,l∈I

ajl i cl ⊗ cm

1.1 Coalgebras, bialgebras, and Hopf algebras

19

where only a finite number of the ajl i ∈ k are different from 0. We now compute that

mi ⊗ ∆(ci ) ρ(mi ) ⊗ ci = i∈I

i∈I

=




ajl i mi ⊗ cl ⊗ cm

i,j,l∈I

=




aji l ml ⊗ cl ⊗ ci

i,j,l∈I

Since the ci form a basis of C, we have
ji ρ(mi ) = al ml ⊗ cl ∈ M ′ ⊗ C j,l∈I

for all i ∈ I, and this proves that M ′ is a subcomodule of M . Consider two right C-comodules M and Q. We say that Q is M -injective if for every subcomodule M ′ ⊂ M , the canonical map HomC (M, Q) → HomC (M ′ , Q) is surjective. Clearly Q is an injective comodule (i.e. an injective object of MC ) if and only if Q is M -injective for every M ∈ MC . Lemma 5. If {Mi | i ∈ I} is a collection of C-comodules, and Q ∈ MC is Mi -injective for all i ∈ I, then Q is also ⊕i∈I Mi -injective. Proof. Write M = ⊕i∈I Mi . Let M ′ be a subcomodule of M , and f : M ′ → Q C-colinear. Consider P = {(L, g) | M ′ ⊂ L ⊂ M in MC , g : L → Q in MC , g|M ′ = f } P is nonempty since (M ′ , f ) ∈ P, and P is ordered: (L, g) ≤ (L′ , g ′ ) if L ⊂ L′ ′ = g. It is easy to show that this ordering is inductive, so P has a and g|L maximal element, by Zorn’s Lemma. We call this element (L0 , g0 ), and we claim that Mi ⊂ L0 , for all i ∈ I. Assume Mi is not contained in L0 , and consider h = g0|Mi ∩L0 : Mi ∩ L0 → Q Since Q is Mi -injective, we have a C-colinear map h : Mi → Q such that h|Mi ∩L0 = h Now define g : Mi + L0 → Q as follows g(x + y) = h(x) + g0 (y)

20

1 Generalities

for x ∈ Mi and y ∈ L0 . g is well-defined, since h and g0 coincide on Mi + L0 . Now g|L0 = g0 and Mi + L0 strictly contains L0 , so (L0 , g0 ) < (Mi + L0 , g) in P which is a contradiction. We conclude that Mi ⊂ L0 , so M = ⊕i∈I Mi ⊂ L0 , and g0 : M = L0 → Q extends f . Theorem 1. Let C be a coalgebra over a field k. For a right C-comodule Q, the following assertions are equivalent. 1. Q is injective as a C-comodule; 2. Q is M -injective, for every finite dimensional C-comodule M ; 3. Q is right C-coflat. Proof. 1. ⇒ 3. Assume that Q is injective. The coaction ρQ is monomorphic, so we have a C-colinear map νQ : Q ⊗ C → Q splitting  ρQ . Let f : X → Y be a surjective morphism of left C-comodules, and take i qi ⊗ yi ∈ Q
C Y . As f is surjective, we find xi  ∈ X such that f (xi ) = yi , and our problem is that we don’t know whether i qi ⊗ xi ∈ Q
C X. We have

qi[0] ⊗ qi[1] ⊗ f (xi ) = qi ⊗ xi[−1] ⊗ f (xi[0] ) i

i

so
i

qi ⊗ yi =




νM (qi[0] ⊗ qi[1] ) ⊗ f (xi )

i

= (IM ⊗ f )(νM (qi ⊗ xi[−1] ) ⊗ xi[0] )

Using the fact that νQ is C-colinear, we find (ρQ ⊗ IX )(νQ (qi ⊗ xi[−1] ) ⊗ xi[0] ) = νQ (qi ⊗ xi[−2] ) ⊗ xi[−1] ⊗ xi[0] = (IQ ⊗ ρX )(νQ (qi ⊗ xi[−1] ) ⊗ xi[0] ) so νQ (qi ⊗ xi[−1] ) ⊗ xi[0] ∈ M
C X, and this shows that IQ
C f : Q
C X → Q
C Y is surjective. 3. ⇒ 2. Let M ∈ MC be finite dimensional, and take a subcomodule M ′ ⊂ M . Then M ∗ and M ′∗ are left C-comodules, and Proposition 6 implies that Q
C M ∗ ∼ = HomC (M, Q) and Q
C M ′∗ ∼ = HomC (M ′ , Q) Now M ∗ → M ′∗ is surjective, so Q
C M ∗ → Q
C M ′∗ is also surjective since Q is C-coflat, and we find that HomC (M, Q) → HomC (M ′ , Q) is surjective, as needed. 2. ⇒ 1. Take an arbitrary N ∈ MC . From Lemma 4, we know that there

1.1 Coalgebras, bialgebras, and Hopf algebras

21

exists a collection {Mi | i ∈ I} of finite dimensional C-comodules and a Ccolinear surjection φ : ⊕i∈I Mi → N . Let P = Ker φ. Now take a subcomodule N ′ ⊂ N , and let M ′ = φ−1 (N ′ ). Then P ⊂ M ′ , so we have the following commutative diagram with exact rows in MC : 0

0

✲ P ✻

✲ M ✻

=

⊂

✲ P

✲ M′

φ ✲ N ✻

✲ 0

⊂ φ ✲ ′ N

✲ 0

Applying HomC (•, Q) to this diagram, we find 0

✲ HomC (N, Q)

✲ HomC (M, Q) ✲ HomC (P, Q) ✻ =

0

❄ ❄ ✲ HomC (N ′ , Q) ✲ HomC (M ′ , Q) ✲ HomC (P, Q)

HomC (M, Q) → HomC (M ′ , Q) is surjective, by Lemma 5. An easy diagram argument shows that HomC (N, Q) → HomC (N ′ , Q) is surjective, as needed. Comodule algebras and comodule coalgebras Let H be a bialgebra. A right H-comodule A that is also a k-algebra is called a right H-comodule algebra , if the unit and multiplication are right H-colinear, that is ρr (ab) = a[0] b[0] ⊗ a[1] b[1] and ρr (1A ) = 1A ⊗ 1H

(1.20)

for all a, b ∈ A. Left H-comodule algebras are introduced in a similar way, and if A is a right H-comodule algebra, then Aop is a left H opcop -comodule algebra. A k-coalgebra C that is also a right H-comodule is called a right H-comodule coalgebra if the comultiplication and the counit are right H-colinear, or c[0](1) ⊗ c[0](2) ⊗ c[1] = c(1)[0] ⊗ c(2)[0] ⊗ c(1)[1] c(2)[1]

(1.21)

εC (c[0] )c[1] = εC (c)1H

(1.22)

and for all c ∈ C. Example 6. Let G be a (semi)group, and take H = kG. Then a kG-comodule algebra is nothing else then a G-graded k-algebra (see [146] for an extensive study of graded rings). A kG-comodule coalgebra is a G-graded coalgebra (see [144]).

22

1 Generalities

Proposition 7. Let C be a coalgebra which is finitely generated and projective as a k-module. There is a bijective correspondence between right Hcomodule coalgebra structures on C and left H-comodule algebra structures on C ∗ . Proof. Let {ci , c∗i | i = 1, · · · , n} be a finite dual basis of C, and assume that C is a right H-comodule coalgebra. We know from Proposition 4 that C ∗ is a left H-comodule, with left H-coaction given by ρl (c∗ ) =

n


ci[1] ⊗ c∗ , ci[0] c∗i

i=1

This makes C ∗ into a left H-comodule algebra since c∗[−1] d∗[−1] ⊗ c∗[0] d∗[0] =

n


ci[1] cj[1] ⊗ c∗ , ci[0] d∗ , cj[0] c∗i ∗ c∗j

i,j=1

(1.5)

=

n


ci(1)[1] ci(2)[1] ⊗ c∗ , ci(1)[0] d∗ , ci(2)[0] c∗i

i=1

(1.21)

=

n


ci[1] ⊗ c∗ ∗ d∗ , ci[0] c∗i

i=1 l ∗

= ρ (c ∗ d∗ ) ρl (εC ) =

n


ci[1] ⊗ εC , ci[0] c∗i

i=1

(1.22)

=

n


1H ⊗ εC , ci c∗i = 1H ⊗ εC

i=1

The further details of the proof are left to the reader.

1.2 Adjoint functors We give a brief discussion of properties of pairs of adjoint functors; of course these results are well-known, but we have organized them in such a way that they can be applied easily to Frobenius and separable functors in Chapter 3. We will occasionally use the Godement product of two natural transformations. Let us introduce the Godement product briefly, refering the reader to [21] for more detail. Let C, D and E be categories, and consider functors F, G : C → D and H, K : D → E and natural transformations

1.2 Adjoint functors

23

α : F → G and β : H → K The Godement product β ∗ α : HF → KG is defined by (β ∗ α)C = βG(C) ◦ H(αC ) = K(αC ) ◦ βF (C) : HF (C) → KG(C) If F = G, and α = 1F , then we find (β ∗ 1F )C = βF (C) If H = K, and β = 1H , then we find (1H ∗ α)C = H(αC ) Now consider, in addition, functors L : C → D and M : D → E and natural transformations γ : G → L and δ : K → M then we have the following formula: (δ ∗ γ) ◦ (β ∗ α) = (δ ◦ β) ∗ (γ ◦ α) Pairs of adjoint functors Let A, B, C and D be categories, and consider functors F : A → C, G : B → C, H : A → D, and K : B → D We have functors HomC (F, G), HomD (H, K) : Aop × B → Sets and we can consider natural transformations θ : HomC (F, G) → HomD (H, K) The naturality of θ can be expressed as follows: given a : A′ → A in A, b : B → B ′ in B, and f : F (A) → G(B) in C, we have θA′ ,B ′ (G(b) ◦ f ◦ F (a)) = K(b) ◦ θA,B (f ) ◦ H(a)

(1.23)

Proposition 8. For two functors F : C → D and G : D → C, we have the following isomorphisms of classes of natural transformations: Nat(1C , GF ) ∼ = Nat(HomD (F, •), HomC (•, G)) Nat(F G, 1D ) ∼ = Nat(HomC (•, G), HomD (F, •))

(1.24) (1.25)

24

1 Generalities

Proof. (Sketch) Consider a natural transformation η : 1C → GF . The corresponding natural transformation θ : HomD (F, •) → HomC (•, G) is defined by θC,D (f ) = G(f ) ◦ ηC (1.26) for all f : F (C) → D in D. Conversely, given θ, the corresponding η is given by ηC = θC,F (C) (IF (C) ) for all C ∈ C. Lemma 6. Let F and G be as in Proposition 8, and consider natural transformations θ : HomD (F, •) → HomC (•, G) and ψ : HomC (•, G) → HomD (F, •). let η : 1C → GF and ε : F G → 1D be natural transformations from Proposition 8. 1. ψ ◦ θ is the identity natural transformation if and only if (ε ∗ F ) ◦ (F ∗ η) = 1F

(1.27)

2. θ ◦ ψ is the identity natural transformation if and only if (G ∗ ε) ◦ (η ∗ G) = 1G

(1.28)

Proof. 1. Take f : F (C) → D in D. We easily compute that ψC,D (θC,D (f )) = εD ◦ F G(f ) ◦ F (ηC ) Now take D = F (C) and f = IF (C) . Then ψC,F (C) (θC,F (C) (IF (C) )) = εF (C) ◦ F (ηC ) and, under the assumption that ψ ◦ θ is the identity natural transformation, we find (1.27). Conversely, assume that (1.27) holds. ε is natural, so we have the following commutative diagram for any f : F (C) → D in D: F GF (C)

F G(f✲)

εF (C) ❄ F (C)

F G(D) εD

F G(f ) ✲ ❄ D

and we find that f = f ◦ εF (C) ◦ F (ηC ) = εD ◦ F G(f ) ◦ F (ηC ) = ψC,D (θC,D (f )) The proof of 2. is similar.

1.2 Adjoint functors

25

Recall that (F, G) is an adjoint pair of functors if HomD (F, •) and HomC (•, G) are naturally isomorphic, or, equivalently, if there exists natural transformations η : 1C → GF and ε : F G → 1D satisfying (1.27-1.28). In this case, F is called a left adjoint of G, and G is called an adjoint of F . η is called the unit of the adjunction, while ε is called the counit. It is well-known that the left or right adjoint of a functor is unique up to natural isomorphism; we include a proof for completeness sake. Proposition 9. (Kan) [101]. If G and G′ are both adjoints of a functor F : C → D, then G and G′ are naturally isomorphic. Proof. We have two adjunctions (F, G) and (F, G′ ). Let (η, ε) and (η ′ , ε′ ) be the unit and counit of both adjunctions, and consider the natural transformations γ = (G′ ∗ ε) ◦ (η ′ ∗ G) : G → G′ γ ′ = (G ∗ ε′ ) ◦ (η ∗ G′ ) : G′ → G η is natural, so for any D ∈ D, we have a commutative diagram G′ (εD ) ✲ ′ G (D)

G′ F G(D) ηG′ F G(D)

ηG′ (D)

❄ ❄ GF G′ (εD✲) GF G′ F G(D) GF G′ (D) or (η ∗ G′ ) ◦ (G′ ∗ ε) = (GF G′ ∗ ε) ◦ (η ∗ G′ F G) Now η is natural, and we have a commutative diagram G(D)

′ ηG(D)

✲ G′ F G(D) ηG′ F G(D)

ηG(D)

❄ ❄ GF (η ′ G(D) ) ✲ GF G′ F G(D) GF G(D) or (η ∗ G′ F G) ◦ (η ′ ∗ G) = (GF ∗ η ′ ∗ G) ◦ (η ∗ G) The naturality of ε′ gives a commutative diagram F G′ F G(D)

F G′ (εD✲)

ε′F G(D) ❄ F G(D)

F G′ (D) ε′D

εD

❄ ✲ D

26

1 Generalities

or ε′ ◦ (F G′ ∗ ε) = ε ◦ (ε′ ∗ F G) and it follows that (G ∗ ε′ ) ◦ (GF G′ ∗ ε) = (G ∗ ε) ◦ (G ∗ ε′ ∗ F G) Combining all these formulas, we find γ ′ ◦ γ = (G ∗ ε′ ) ◦ (η ∗ G′ ) ◦ (G′ ∗ ε) ◦ (η ′ ∗ G) = (G ∗ ε′ ) ◦ (GF G′ ∗ ε) ◦ (η ∗ G′ F G) ◦ (η ′ ∗ G) = (G ∗ ε) ◦ (G ∗ ε′ ∗ F G) ◦ (GF ∗ η ′ ∗ G) ◦ (η ∗ G)   = (G ∗ ε) ◦ G ∗ ((ε′ ∗ F ) ◦ (F ∗ η ′ )) ∗ G ◦ (η ∗ G)  = (G ∗ ε) ◦ G ∗ 1F ∗ G) ◦ (η ∗ G) = (G ∗ ε) ◦ (η ∗ G) = 1G In a similar way, we obtain that γ ◦ γ ′ = 1G′ , and it follows that G and G′ are naturally isomorphic. Recall the following properties of adjoint pairs: Theorem 2. Let (F, G) be an adjoint pair of functors. F preserves colimits, and, in particular, coproducts, initial objects and cokernels. G preserves limits, and, in particular, products, final objects and kernels. If C and D are abelian categories, then F is right exact, and G is left exact. If F is exact, then G preserves injective objects. If G is exact, then F preserves projective objects. Here is another well-known property of adjoint functors that will be useful in the sequel. Proposition 10. Let (F, G) be an adjoint pair functors, then we have isomorphisms Nat(F, F ) ∼ = Nat(G, G) ∼ = Nat(1C , GF ) ∼ = Nat(F G, 1D ) Proof. We will show that Nat(G, G) ∼ = Nat(1C , GF ), the proof of the other assertions is left to the reader. For a natural transformation θ : 1C → GF , we define α = X(θ) : G → G by αD = G(εD ) ◦ θG(D)

(1.29)

Conversely, for α : G → G, θ = X −1 (α) : 1C → GF is defined by θC = αF (C) ◦ ηC

(1.30)

1.2 Adjoint functors

27

We are done if we can show that X and X −1 are each others inverses. First take α : G → G, and θ = X −1 (α). The diagram G(D)

ηG(D) ✲ GF G(D) G(εD✲) G(D)

❅ ❅ αF G(D) αD ❅ θG(D) ❅ ❘ ❅ ❄ ❄ G(ε ) D ✲ G(D) GF G(D) commutes: the triangle is commutative because of (1.30), and the square commutes because α is natural. From (1.27), it follows that the composition of the two maps in the top row is IG(N ) , and then we see from the diagram that α = X(θ). Conversely, take θ : 1C → GF , and let α = X(θ). Then θ = X −1 (α) because the following diagram commutes: C

ηC

✲ GF (C)

❅ ❅ θGF (C) ❅αF (C) ❅ ❅ ❘ ❄ GF (η ) ❄ G(ε F (C) ) C ✲ GF (C) ✲ GF GF (C) GF (C) θC

A result of the same type is the following: Proposition 11. Let (F, G) be an adjoint pair of functors. Then we have isomorphisms Nat(GF, 1C ) ∼ = Nat(HomD (F, F ), HomC (•, •))

(1.31)

Nat(1D , F G) ∼ = Nat(HomC (G, G), HomD (•, •))

(1.32)

Proof. We outline the proof of the first statement. Given a natural transformation ν : GF → 1C , we define θ = α(ν) : HomD (F, F ) → HomC (•, •) as follows: take g : F (C) → F (C ′ ) in D, and put θC,C ′ (g) = νC ′ ◦ G(g) ◦ ηC Straightforward arguments show that θ is natural. Conversely, given θ : HomD (F, F ) → HomC (•, •) we define α−1 (θ) = ν : GF → 1C by

28

1 Generalities

νC = θGF (C),C (εF (C) ) : GF (C) → C We leave it as an exercise to show that ν is natural, as needed, and that α and α−1 are inverses. The proof of the second statement is similar. Let us just mention that, given ζ : 1D → F G, we define β(ζ) = ψ : HomC (G, G) → HomD (•, •) as follows: given f : G(D) → G(D′ ) in C, we put ψD,D′ (f ) = εD′ ◦ F (f ) ◦ ζD

1.3 Separable algebras and Frobenius algebras In this Section, we give the classical Definition, and elementary properties of separable and Frobenius algebras. We will refer to it in Chapter 3, where we will introduce separable and Frobenius functors, and show that they are generalizations of the classical concepts. The Section on separable algebras is based on [109], and the one on Frobenius algebras on [113]. Separable algebras Let k be a commutative ring, A a k-algebra and M an A-bimodule. Recall that M can be viewed as a left Ae -module, where Ae = A ⊗ Aop is the enveloping algebra of A. A derivation of A in M is a k-linear map D : A → M such that D(ab) = D(a)b + aD(b)

(1.33)

for all a, b ∈ A. Derk (A, M ) will be the k-module consisting of all derivation of A into M . For any m ∈ M , we have a derivation Dm , given by Dm : A → M,

Dm (a) = am − ma

called the inner derivation asociated to m. It is clear that Dm = 0 if and only if m ∈ M A = {m ∈ M | am = ma, ∀a ∈ A}, so we have an exact sequence 0 → M A → M → Derk (A, M )

(1.34)

We also note that MA ∼ = HomAe (A, M ),

M∼ = HomAe (Ae , M )

(1.35)

The multiplication mA on A induces an epimorphism A ⊗ Aop → A of left Ae -modules, still denoted by mA , and we have another exact sequence 0 → I(A) = Ker (mA ) → A ⊗ Aop → A → 0

(1.36)

1.3 Separable algebras and Frobenius algebras

29

We have a derivation δ : A → I(A),

δ(a) = a ⊗ 1 − 1 ⊗ a

for all a ∈ A. It is clear that δ(a) ∈ I(A) and Aδ(A) = I(A) = δ(A)A Indeed, take x =



i




x=

ai ⊗ bi ∈ I(A), then ai (1 ⊗ bi − bi ⊗ 1) = −




ai δ(bi ) ∈ Aδ(A)

i

i

Lemma 7. Let M be an A-bimodule over a k-algebra A. Then we have an isomorphism of k-modules HomAe (I(A), M ) ∼ = Derk (A, M )

(1.37)

Proof. We define φ : HomAe (I(A), M ) → Derk (A, M ), φ−1 is given by

φ(f ) = f ◦ δ



φ−1 (D)( ai D(bi ) ai ⊗ bi ) = − i

i

We show that φ−1 (D) is left Ae -linear, and leave the other details to the reader.

aai ⊗ bi b) ai ⊗ bi )) = φ−1 (D)( φ−1 (D)((a ⊗ b)( i

=−




aai D(bi b) = −


i

i

i

aai D(bi )b −


ai ⊗ bi ) = (a ⊗ b)φ−1 (D)(




aai bi D(b)

i

i

Applying the functor HomAe (•, M ) to the exact sequence (1.36) and taking (1.35) and (1.37) into account, we find a long exact sequence 0 → M A → M → Derk (A, M ) → Ext1Ae (A, M ) → 0

(1.38)

extending (1.34). Indeed, Ext1Ae (Ae , M ) = 0, since Ae is projective as a left Ae -module. H 1 (A, M ) = Ext1Ae (A, M ) is another notation, and H 1 (A, M ) is called the first Hochschild cohomology group of A with coefficients in M . For more information on Hochschild cohomology, we refer to [55, Ch. IX]. Thus (1.38) tells us that H 1 (A, M ) ∼ = Derk (A, M )/InnDerk (A, M ) A is called a separable k-algebra if it satisfies the equivalent conditions of the following theorem:

30

1 Generalities

Theorem 3. For a k-algebra A the following statements are equivalent: 1. A is projective as a left Ae -module; 2. the exact sequence splits as a sequence of left Ae -modules;  (1.36) 1 2 3. there exists e = e ⊗ e ∈ A ⊗ A such that
(1.39) ae = ea and e1 e2 = 1

for all a ∈ A. 4. H 1 (A, M ) = 0, for any A-bimodules M . 5. the derivation δ : A → I(A), δ(a) = a ⊗ 1 − 1 ⊗ a is inner; 6. every derivation D : A → M is inner, for any A-bimodule M .

An element e ∈ Ae satisfying  1 2 ae = ea for all a ∈ A is called a Casimir element. If, in addition, e e = 1, then e is an idempotent, and it is called a separability idempotent. Proof. 1. ⇔ 2. is obvious. 2. ⇒ 3. If ψ : A → Ae is a left Ae -module map and section of mA then, e = ψ(1) satisfies (1.39). e e 3. ⇒ 2. Define  ψ1 : 2 A → A , ψ(a) = ae = ea. ψ is left A -module map and mA ψ(a) = ae e = a. 1. ⇔ 4. is obvious. 4. ⇔ 6. follows from the exact sequence (1.38). 6. ⇒ 5. is trivial. 5. ⇒ 6. Let D : A → M be a derivation. From the above Lemma we know that there is a f ∈ HomAe (I(A), M ) such that D = f ◦ δ. δ is inner, so we can write δ = Dx , with x ∈ I(A). Now, D(a) = f (δ(a)) = f (ax − xa) = af (x) − f (x)a = Df (x) (a) i.e. D is inner. Let us now prove some immediate properties of separable algebras. Proposition 12. Any projective separable algebra A over a commutative ring k is finitely generated. Proof. We take a dual basis {si , s∗i | i ∈ I} for A. This means that, for all s ∈ A, the set I(s) = {i ∈ I | s∗i , s =  0} is finite, and s=


s∗i , ssi i∈I

For all i ∈ I, we define φi : A ⊗ A

op

→ A by

φi (s ⊗ t) = s∗i , ts

1.3 Separable algebras and Frobenius algebras

31

such that φi (s′ s ⊗ t) = s∗i , ts′ s = s′ φi (s ⊗ t) and φi is left A-linear. We now claim that {zi = 1 ⊗ si , φi | i ∈ I} is a dual basis of A ⊗ Aop as a left A-module. Take z = s ⊗ t ∈ A ⊗ Aop . If φi (z) = s∗i , ts = 0, then s∗i , t =  0, so i ∈ I(t), and we conclude that I(z) = {i ∈ I | φi (z) = 0} ⊂ I(t) is finite. Moreover s⊗t =




s ⊗ s∗i , tsi =

i∈I

i∈I

=





s∗i , ts ⊗ si

φi (s ⊗ t) ⊗ si =




φi (s ⊗ t)(1 ⊗ si )

i∈I

i∈I

A is separable, so we have a separability idempotent e = e1 ⊗ e2 ∈ A ⊗ Aop . Our next claim is that I(et) ⊂ I(e) for all t ∈ A. Indeed, we compute φi (et) = φi (e1 ⊗ e2 t) = φi (te1 ⊗ e2 ) = tφi (e) so i ∈ I(et), or φi (et) = 0, implies φi (e) = 0 and i ∈ I(e). For all t ∈ A, we finally compute 
 t = 1t = m(e)t = m(et) = m φi (et)zi =m




i∈I(e)

1

2

φi (e ⊗ e t)zi = m

i∈I(e)

= Write e =

r




s∗i , e2 te1 m(zi ) i∈I(e)

j=1 ej



=







s∗i , e2 te1 zi

i∈I(e)

s∗i , e2 te1 si



i∈I(e)

⊗ e′j . We have shown that {ej si , s∗i , e′j • | i ∈ I(e), j = 1, · · · , r}

is a finite dual basis for A. Proposition 13. A separable algebra A over a field k is semisimple.  1 Proof. Let e = e ⊗ e2 ∈ A ⊗ A be a separability idempotent and N an A-submodule of a right A-module M . As k is a field, the inclusion i : N → M splits in the category of k-vector spaces. Let f : M → N be a k-linear map such that f (n) = n, for all n ∈ N . Then

32

1 Generalities

f˜ : M → N,

f˜(m) :=




f (me1 )e2

is a right A-module map that splits the inclusion i. Thus N is an A-direct factor of M , and it follows that M is completely reducible. This shows that A is semisimple. Examples 1. 1. Let k be a field of characteristic p, and a ∈ k \ k p . l = k[X]/(X p − a) is then a purely inseparable field extension of k, and l is not a separable k-algebra in the above sense. Indeed, d : l→l dX is a derivation that is not inner. More generally, one can prove that a finite field extension l/k is separable in the classical sense if and only if l is separable as a k-algebra, see [66, Proposition III.3.4]. 2. Let k be a field. It can be show that a separable k-algebra is of the form A = Mn1 (D1 ) × · · · × Mnr (Dr )

(1.40)

where Di is a division algebra with center a finite separable field extension li of k. See [66, Theorem III.3.1] for details. 3. Any nmatrix ring Mn (k) is separable as a k-algebra: for any i = 1, · · · , n, ei = j=1 eji ⊗eij is a separablity idempotent. More generally, any Azumaya algebra A is separable as a k-algebra. Frobenius algebras In this Section we will recall the classical definition of a Frobenius algebra, thus showing how it came up in representation theory. We will work over a a field k. For a k-algebra A, the k-dual A∗ = Homk (A, k) is an A-bimodule via the actions r∗ · r, r′  = r∗ , rr′ ,

r · r∗ , r′  = r∗ , r′ r

(1.41)

for all r, r′ ∈ A and r∗ ∈ A∗ . Definition 3. A finite dimensional k-algebra A is called a Frobenius algebra if A ∼ = A∗ as right A-modules. Remarks 1. 1. A finite dimensional k-algebra A is Frobenius if and only if there exists a k-linear map λ : A → k such that for any ψ ∈ A∗ there exists a unique element r = rψ ∈ A such that ψ(x) = λ(rx) for all x ∈ A. In particular, the matrix algebra Mn (k) is Frobenius: take λ = Tr, the trace map. 2. The concept of Frobenius algebra is left-right symmetric: that is A ∼ = A∗ ∗ A in M. in MA if and only if A ∼ = A It suffices to observe that there exists a one to one correspondence between the following data:

1.3 Separable algebras and Frobenius algebras

33

– the set of all isomorphisms of right A-modules f : A → A∗ ; – the set of all bilinear, nondegenerate and associative maps B : A × A → k; – the set of all isomorphisms of left A-modules g : A → A∗ , given by the formulas f (x)(y) = B(x, y) = g(y)(x) (1.42) for all x, y ∈ A. Let us now explain how the original problem of Frobenius arises naturally in representation theory, as explained in the book of Lam [113]. We fix a basis {e1 , · · · , en } of a finite dimensional algebra A. Then for any r ∈ A we can (r) (r) find scalars aij and bij such that ei r =

n


(r)

aij ej ,

rei =

n


(r)

bji ej

(1.43)

j=1

j=1

for all i = 1, · · · , n. Hence we have constructed k-linear maps α, β : A → Mn (k),

(r)

α(r) = (aij ),

(r)

β(r) = (bij )

(1.44)

for all r ∈ A. It is straightforward to prove that α and β are algebra maps, i.e. they are representations of the k-algebra A. The problem of Frobenius: When are the above representations α and β equivalent? We recall that two representations α, β : A → Mn (k) are equivalent if there exists an invertible matrix U ∈ Mn (k) such that β(r) = U α(r)U −1 , for all r ∈ A. Before giving the answer to the problem we present one more construction: let (clij )i,j,l=1,n be the structure constants of the algebra A, that is ei ej =

n


ckij ek

k=1

for all i, j = 1, · · · , n. For a = (a1 , · · · , an ) ∈ k n , let Pa ∈ Mn (k) be the matrix given by n
ak ckij (Pa )i,j = k=1

The matrix Pa is called the paratrophic matrix. In the next Theorem, the equivalence 2. ⇔ 3. was the original theorem of Frobenius, while the equivalence 1. ⇔ 2. translate the problem from representation theory into the language of modules. Theorem 4. For an n-dimensional algebra A, the following statements are equivalent:

34

1 Generalities

1. A is Frobenius; 2. the representations α and β : A → Mn (k) constructed in (1.44) are equivalent; 3. there exists a ∈ k n such that the paratrophic matrix Pa is invertible; 4. there exists a bilinear, nondegenerate and associative map B : A×A → k, i.e. B(xy, z) = B(x, yz), for all x, y, z ∈ A; 5. there exists a hyperplane of A that does not contain a nonzero right ideal of A; 6. thereexists a pair (ε, e), called a Frobenius pair , where ε ∈ A∗ and e= e1 ⊗ e2 ∈ A ⊗ A such that

ae = ea, and ε(e1 )e2 = e1 ε(e2 ) = 1. (1.45)

Before proving the Theorem, let us recall some well-known facts. First of all, let V be a n-dimensional vector space with basis B = {v1 , · · · , vn }. Let canV : Endk (V )op → Mn (k),

canV (f ) = MB (f )

be the canonical isomorphism of algebras; here, for f ∈ Endk (V ), MB (f ) = (aij ), is the matrix asociated to f with respect to the basis B written as follows n
f (vi ) = aij vj j=1

for all i = 1, · · · , n. Secondly, a k-vector space M has a structure of right A-module if and only if there exists an algebra map ϕM : A → Endk (M )op ϕM is called the representation associated to M . The correspondence between the action ” · ” and the representation is given by ϕM (r)(m) = m · r. In particular, if dimk (M ) = n, M has a structure of right A-module if and only if there exists an algebra map ϕ˜M (= canM ◦ ϕM ) : A → Mn (k). Finally, let M and N be two right A-modules and ϕM : A → Endk (M ), ϕN : A → Endk (N ) the associated representations. Then M ∼ = N (as right A-modules) if and only if there exists an isomorphism of k-vector spaces θ : M → N such that ϕM (r) = θ−1 ◦ ϕN (r) ◦ θ for all r ∈ A. Indeed, a k-linear map θ : M → N is a right A-module map if and only if θ(m · r) = θ(m) · r for all m ∈ M , r ∈ A. This is equivalent to

1.3 Separable algebras and Frobenius algebras

35

θ(ϕM (r)(m)) = ϕN (r)(θ(m)) or θ ◦ ϕM (r) = ϕN (r) ◦ θ for all r ∈ A. Proof. (of Theorem 4). 1. ⇔ 2. This follows from the remarks made above if we can prove that α = ϕ˜A , β = ϕ˜A∗ . Let us prove first that α = ϕ˜A , where A ∈ MA , via right multiplication. The representation associated to this structure is ϕA : A → Endk (A),

ϕA (r)(r′ ) = r′ r

hence, ϕA (r)(ei ) = ei r =

n


(r)

aij ej

j=1

i.e. α = ϕ˜A . Let us show next that β = ϕ˜A∗ . Let {e∗i } be the dual basis of {ei } and ϕA∗ : A → Endk (A∗ ), ϕA∗ (r)(r∗ ) = r∗ (r). Now β(r) = ϕ˜A∗ (r) if and only if e∗i

·r =

n


(r)

bij e∗j

j=1

or

n
(r) e∗i , rek  =  bij e∗j , ek  j=1

(r)

for all k. Both sides are equal to bik . 1. ⇔ 3. Any right A-module map f : A → A∗ has the form f (r) = λ · r, for some λ ∈ A∗ . Thus, there exists a1 , · · · , an ∈ k such that f (r) = (a1 e∗1 + · · · + an e∗n ) · r for any r ∈ A. Using the dual basis formula we have e∗k · ei , ej  = e∗k , ei ej  = ckij Hence e∗k · ei =

n

k ∗ j=1 cij ej ,

f (ei ) =

and it follows that

n


k=1

ak e∗k · ei =

n n

( ckij ak )e∗j j=1 k=1

36

1 Generalities

for all i = 1, · · · , n. This means that the matrix associated to f in the pair of basis {ei , e∗i } is just the paratrophic matrix Pa , where a = (a1 , · · · , an ) ∈ k n . 1. ⇔ 4. follows from (1.42). 4. ⇒ 5. H = {a ∈ A | B(1, a) = 0} is a k-subspace of A of codimension 1. Assume that J is a right ideal of A and J ⊂ H, and take x ∈ J. using the fact that xA ⊂ J ⊂ H, and that B is associative, we obtain 0 = B(1, xA) = B(x, A) As B is nondegenerate we obtain that x = 0. 5. ⇒ 1. Let H be a such a hyperplane. As k is a field, we can pick a k-linear map λ : A → k such that Ker (λ) = H. Then f = fλ : A → A∗ ,

f (x), y = λ(xy)

for all x, y ∈ A, is an injective right A-linear map. Indeed, for x, y, z ∈ A we have f (xy), z = λ(xyz) = f (x), yz = f (x) · y, z On the other hand, from f (x) = 0 it follows that λ(xA) = 0, hence xA ⊂ Ker(λ) = H. We obtain, that xA = 0, i.e. x = 0. Thus, f is an injective right A-module map, that is an isomorphism as A and A∗ have the same dimension. 1. ⇒ 6. Let (ei , e∗i ) be a dual basis of A and → A∗ an isomorphism  f : A −1 (e∗i )) is a Frobenius of right A-modules. Then (ε = f (1), e = i ei ⊗ f pair. This is an elementary computation left to the reader at this point; in Theorem 28, we give proof in a more general situation.  1the same 6. ⇒ 1. If (ε, e = e ⊗ e2 ) is a Frobenius pair, then f : A → A∗ ,

f (x), y = ε(xy)

is an isomorphism of right A-modules with inverse
f −1 : A∗ → A, f −1 (a∗ ) = a∗ , e1 e2

for all a∗ ∈ A∗ .

Examples 2. 1. Theorem 4 gives an elementary way to check whether an algebra A is Frobenius. Let A = k[X, Y ]/(X 2 , Y 2 ). Then A has a basis e1 = 1, e2 = x, e3 = y and e4 = xy. Through a trivial computation we find that the paratrophic matrix is   a1 a2 a3 a4    a2 0 a4 0    Pa =   a a 0 0  4   3 a4

0

0

0

1.3 Separable algebras and Frobenius algebras

37

Thus, if a4 is non-zero, then Pa is invertible, so A is a Frobenius algebra. 2. A similar computation shows that the k-algebra A = k[X, Y ]/(X 2 , XY 2 , Y 3 ) is not Frobenius. 3. Using the criterium 5) given by Theorem 4 we can see that any finite dimensional division k-algebra D is a Frobenius algebra. It can be proved that Mn (D) is also a Frobenius k-algebra. Using (1.40) and the fact that a product of Frobenius algebras is Frobenius algebra, we obtain that any separable algebra over a field is Frobenius.

2 Doi-Koppinen Hopf modules and entwined modules

In this Chapter, we introduce entwining structures and entwined modules. We show how various kinds of modules that appear in ring theory are special cases of entwined modules. We also show that there is a close analogy, based on duality arguments, with the factorization problem for algebras, and the smash product of algebras. Entwined modules themselves can be viewed as special cases of comodules over corings. Pairs of adjoint functors between categories of entwined modules are investigated, and it is discussed how one can make the category of entwined modules into a monoidal category.

2.1 Doi-Koppinen structures and entwining structures Entwining structures Throughout this Section, k is a commutative ring. A (right-right) entwining structure on k consists of a triple (A, C, ψ), where A is a k-algebra, C a k-coalgebra, and ψ : C ⊗ A → A ⊗ C a k-linear map satisfying the relations (ab)ψ ⊗ cψ = aψ bΨ ⊗ cψΨ (1A )ψ ⊗ cψ = 1A ⊗ c

(2.1) (2.2)

ψ aψ ⊗ ∆C (cψ ) = aψΨ ⊗ cΨ (1) ⊗ c(2)

(2.3)

ψ

εC (c )aψ = εC (c)a

(2.4)

Here we used the sigma notation ψ(c ⊗ a) = aψ ⊗ cψ = aΨ ⊗ cΨ A morphism (α, γ) : (A, C, ψ) → (A′ , C ′ , ψ ′ ) consists of an algebra map α : A → A′ and a coalgebra map γ : C → C ′ such that (α ⊗ γ) ◦ ψ = ψ ′ ◦ (γ ⊗ α) or, equivalently, α(aψ ) ⊗ γ(cψ ) = α(a)ψ′ ⊗ γ(c)ψ

(2.5) ′

(2.6)

E•• (k) will denote the category of entwining structures. The category E•• (k) is monoidal. E∗ •• (k) is the full subcategory of E•• (k) consisting of entwining

S. Caenepeel, G. Militaru, and S. Zhu: LNM 1787, pp. 39–87, 2002. c Springer-Verlag Berlin Heidelberg 2002


40

2 Doi-Koppinen Hopf modules and entwined modules

structures (A, C, ψ) with ψ invertible. Left-right, right-left, and left-left versions can also be introduced. For example, • E• (k) is the category with objects (A, C, ψ), where now ψ : A ⊗ C → A ⊗ C, ψ(a ⊗ c) = aψ ⊗ cψ is a map satisfying (2.2,2.3,2.4) and (ab)ψ ⊗ cψ = aψ bΨ ⊗ cΨ ψ

(2.7)

In • E• (k), we need maps ψ : C ⊗ A → C ⊗ A, satisfying (2.1,2.2,2.4) and aψ ⊗ ∆C (cψ ) = aψΨ ⊗ (c(1) )ψ ⊗ (c(2) )Ψ • • E(k),

In (2.8).

(2.8)

we will need maps ψ : A ⊗ C → C ⊗ A satisfying (2.2,2.4,2.7) and

Proposition 14. The categories E•• (k), • E• (k), • E• (k), and •• E(k) are isomorphic. Proof. It is easy to see that the isomorphism between E•• (k) and • E• (k) is given by sending (A, C, ψ) to (Aop , C, ψ ◦ τ ). The other isomorphisms are left to the reader. Obviously the isomorphisms in Proposition 15 restrict to the subcategories consisting of structures with invertible ψ. For these subcategories, there exists alternative isomorphisms. Proposition 15. The categories E∗ •• (k) and •• E∗ (k) are isomorphic via the functor S given by S(A, C, ψ) = (A, C, ψ −1 ) (2.9) Proof. Asssume that ψ : C ⊗ A → A ⊗ C satisfies (2.1-2.4). We have to show that ϕ = ψ −1 satisfies (2.2,2.4, 2.7) and (2.8). (2.2) and (2.4) are obvious. (2.1) is equivalent to commutativity of the diagram C ⊗A⊗A

ψ ⊗ I✲ A

A⊗C ⊗A

IA ⊗✲ ψ

A⊗A⊗C mA ⊗ IC

I C ⊗ mA ❄ C ⊗A

❄ ✲ A⊗C

ψ

This is equivalent to commutativity of the following diagram A⊗A⊗C

IA ⊗✲ ϕ

A⊗C ⊗A

mA ⊗ IC ❄ A⊗C

ϕ ⊗ I✲ A

C ⊗A⊗A IC ⊗ mA

ϕ

❄ ✲ C ⊗A

2.1 Doi-Koppinen structures and entwining structures

41

which is equivalent to cϕφ ⊗ aφ bϕ = cϕ ⊗ (ab)ϕ and this tells us that ϕ satisfies (2.7). In a similar way (2.3) implies that ϕ satisfies (2.8). Doi-Koppinen structures Let H be a bialgebra, A a right H-comodule algebra, and C a right H-module coalgebra. We call (H, A, C) a right-right Doi-Koppinen structure or DK structure over k. A morphism between two DK structures consists of a triple ϕ = (
, α, γ) : (H, A, C) → (H ′ , A′ , C ′ ), where
: H → H ′ , α : A → A′ , and γ : C → C ′ are respectively a bialgebra map, an algebra map, and a coalgebra map such that ρA (α(a)) = α(a[0] ) ⊗
(a[1] ) γ(ch) = γ(c)
(h)

(2.10) (2.11)

for all a ∈ A, c ∈ C, and h ∈ H. The category of right-right Doi-Hopf structures over k is denoted by DK•• (k). DK•• (k) is a monoidal category, if we define (H, A, C) ⊗ (H ′ , A′ , C ′ ) = (H ⊗ H ′ , A ⊗ A′ , C ⊗ C ′ ) with the obvious structure maps. The unit element is (k, k, k). We will also consider the full subcategories H•• (k), HA•• (k), and HC•• (k) of DK•• (k), consisting of objects respectively of the form (H, H, H), (H, A, H), (H, H, C) The subcategory of DK•• (k) consisting of objects (H, A, C) and morphisms (
, α, γ) where H has a twisted antipode S, and where
preserves the twisted antipode, is denoted by DKs•• (k). In a similar way, we introduce the categories • DK• (k), • DK• (k), and •• DK(k), and their various subcategories. For example, • DK• (k) has objects (H, A, C), where A is a right H-comodule algebra, and C is a left H-module coalgebra. • In the definition of • DK∗ (k) and • DK∗ • (k), we require that the bialgebra H in each object is a Hopf algebra (i.e., it has an antipode). In the left-left case, we want a twisted antipode. Proposition 16. The categories DK•• (k), • DK• (k), • DK• (k), and •• DK(k) are isomorphic. Similar statements hold for the respective subcategories introduced above. Proof. Let (H, A, C) ∈ DK•• (k). Then the opposite algebra Aop with the original right H-coaction is a right H op -comodule algebra. The coalgebra C with left H op -action defined by hop · c = ch

42

2 Doi-Koppinen Hopf modules and entwined modules

is a left H op -module coalgebra. The functor DK•• (k) → • DK• (k), mapping (H, A, C) to (H op , Aop , C) is easily seen to be an isomorphism of categories. Observe also that H op has an antipode in case H has a twisted antipode, so we also find an isomorphism between the categories DKs•• (k) and • DKs• (k). The other statements follow in a similar way, let us mention that the objects corresponding to (H, A, C) ∈ DK•• (k) are (H cop , A, C cop ) ∈ • DK• and (H opcop , Aop , C cop ) ∈ •• DK(k). Proposition 17. We have faithful functors F : DK•• (k) → E•• (k) and

•

F : DK∗ • (k) → E∗ •• (k)

Proof. We define F by F (H, A, C) = (A, C, ψ) ; F (
, α, γ) = (α, γ) with ψ : C ⊗ A → A ⊗ C given by ψ(c ⊗ a) = a[0] ⊗ ca[1] We leave it to the reader to check that ψ satisfies (2.1-2.4), and that (α, γ) satisfies (2.5). If (H, A, C) ∈ DKs•• (k), then H has a twisted antipode S, and the inverse of ψ is given by the formula ψ −1 (a ⊗ c) = cS(a[1] ) ⊗ a[0] Alternative Doi-Koppinen structures These structures were recently introduced by Schauenburg [163]. A left-right alternative Doi-Koppinen structure consists of a triple (H, A, C), where H is a bialgebra, A is left H-module algebra, and C is a right H-comodule coalgebra. We write • aDK• (k) for the category left-right alternative Doi-Koppinen structures. The morphisms are defined in the obvious way, and analogous definitions can be given in the right-left, left-left and right-right situations. The alternative version of Proposition 17 is the following: Proposition 18. We have a faithful functor Fa :

• • aDK (k)

→ • E• (k)

Proof. We put Fa (H, A, C) = (A, C, ψ) and Fa (
, α, γ) = (α, γ) with ψ : A ⊗ C → A ⊗ C, ψ(a ⊗ c) = c[1] a ⊗ c[0] A straightforward computation shows that (A, C, ψ) is a left-right entwining structure.

2.1 Doi-Koppinen structures and entwining structures

43

Doi-Koppinen structures versus entwining structures An obvious question is the following: is an entwining structure (A, C, ψ) defined by a Doi-Koppinen structure, i.e. can we find a bialgebra H, an H-coaction on A, and an H-action on C such that (A, C, ψ) = F (H, A, C) We will see that a sufficient condition is that A is finitely generated and projective as a k-module. If C is finitely generated and projective, then every entwining structure comes from an alternative Doi-Koppinen structure. A recent counterexample due to Schauenburg shows that there exist entwining structures that do not arise from Doi-Koppinen structures. We start with a construction due to Sweedler [172, p.155], reformulated by Tambara [178]. Let A be finitely generated projective, with dual basis {ai , a∗i | i = 1, · · · , n}, and write H = H(A) = T (A∗ ⊗ A)/I the tensor algebra of A∗ ⊗ A divided by the ideal I generated by elements of the form a∗ , 1A  − a∗ ⊗ 1A ∗

a ⊗ ab −

(a∗(1)

(2.12)

⊗ a) ⊗

(a∗(2)

⊗ b)

(2.13)

where a∗ ∈ A∗ and a, b ∈ A. We write [a∗ ⊗ a] for the class represented by a∗ ⊗ a. Proposition 19. Let A be a finitely generated projective k-algebra. Then H = H(A) is a bialgebra, with comultiplication and counit given by ∆H [a∗ ⊗ a] =

n


[a∗ ⊗ ai ] ⊗ [a∗i ⊗ a] and εH [a∗ ⊗ a] = a∗ , a

i=1

Proof. A straightforward calculation; we will show that ∆H is well-defined, i.e. ∆H = 0 on I. First, ∆H (a∗ ⊗ 1A ) =

n


[a∗ ⊗ ai ] ⊗ [a∗i ⊗ 1A ]

i=1

=

n


[a∗ ⊗ ai ] ⊗ a∗i , 1A  = a∗ , 1A 1 ⊗ 1

i=1

Next ∆H ([(a∗(1) ⊗ a) ⊗ (a∗(2) ⊗ b)]) =

n 
i=1

n  
[a∗(2) ⊗ aj ] ⊗ [a∗j ⊗ b] [a∗(1) ⊗ ai ] ⊗ [a∗i ⊗ a] j=1

44

2 Doi-Koppinen Hopf modules and entwined modules

= = (1.4)

=

n


[a∗(1) ⊗ ai ][a∗(2) ⊗ aj ] ⊗ [a∗i ⊗ a][a∗j ⊗ b]

i,j=1 n


[a∗ ⊗ ai aj ] ⊗ [a∗i ⊗ a][a∗j ⊗ b]

i,j=1 n


[a∗ ⊗ ai ] ⊗ [a∗i(1) ⊗ a][a∗i(1) ⊗ b]

i=1

=

n


[a∗ ⊗ ai ] ⊗ [a∗i ⊗ ab]

i=1

= ∆H [a∗ ⊗ ab] Remark 2. As above, let A be finitely generated and projective, and consider the functor F : k-Alg → k-Alg ; F (B) = A ⊗ B Tambara [178] observes that F has a right adjoint G, and, as a k-algebra, H(A) = G(A). For any k-algebra B, we write G(B) = a(A, B) = T (A∗ ⊗ B)/I with I defined as above (1A is replaced by 1B , and a, b ∈ B). The unit and counit of the adjunction are given by ηB : B → a(A, A ⊗ B) ; ηB (b) =

n


[a∗i ⊗ (ai ⊗ b)]

i=1

εB : A ⊗ a(A, B) → B ; εB (a ⊗ [a∗ ⊗ b]) = a∗ , ab The comultiplication and counit on a(A, A) = H(A) can be defined using the adjunction properties. Proposition 20. Let A be a finitely generated projective k-algebra, and H = H(A). Then A is a right H-comodule algebra, and A∗ is a left H-comodule coalgebra. The structure maps are ρr (a) = ρl (a∗ ) =

n


i=1 n


ai ⊗ [a∗i ⊗ a]

(2.14)

[a∗ ⊗ ai ] ⊗ a∗i

(2.15)

i=1

Proof. A is a right H-comodule since (ρr ⊗ IH )(ρr (a)) =

n


i,j=1

aj ⊗ [a∗j ⊗ ai ] ⊗ [a∗i ⊗ a] = (IA ⊗ ∆H )(ρr (a))

2.1 Doi-Koppinen structures and entwining structures

45

A is a right H-comodule algebra since r

ρ (ab) = (2.13) (1.4)

= =

n


ai ⊗ [a∗i ⊗ ab]

i=1 n


ai ⊗ [a∗i(1) ⊗ a][a∗i(2) ⊗ b]

i=1 n


ai aj ⊗ [a∗i ⊗ a][a∗j ⊗ b]

i,j=1 r

= ρ (a)ρr (b) n
ρr (1A ) = ai ⊗ [a∗i ⊗ 1A ] i=1

(2.12)

=

n


ai ⊗ a∗i , 1A  = 1A ⊗ 1H

i=1

From Proposition 7, it follows that A∗ is a left H-comodule coalgebra, with ρl (a∗ ) = = =

n


ai[1] ⊗ a∗ , ai[0] a∗i

i=1 n


[a∗j ⊗ ai ] ⊗ a∗ , aj a∗i

i,j=1 n


[a∗ ⊗ ai ] ⊗ a∗i

i=1

Theorem 5. Let A be a finitely generated projective algebra, and C a coalgebra. There is a bijective correspondence between left H(A)-module coalgebra structures on C, and left-right entwining structures of the form (A, C, ψ). Consequently every entwining structure (A, C, ψ) with A finitely generated and projective can be derived from a Doi-Koppinen structure. Proof. First consider an entwining structure (A, C, ψ) ∈ • E• (k). As before, we write H = H(A). On C, we define the following left H-action: [a∗ ⊗ a] · c = a∗ , aψ cψ This action is well-defined since [a∗ ⊗ 1A ] · c = a∗ , (1A )ψ cψ = a∗ , 1A c and [a∗ ⊗ ab] · c = a∗ , (ab)ψ cψ = a∗ , aψ bΨ cΨ ψ = a∗(1) , aψ a∗(2) , bΨ cΨ ψ   = [a∗(1) , a] · [a∗(2) , b] · c

(2.16)

46

2 Doi-Koppinen Hopf modules and entwined modules

The comultiplication and counit of C are left H-linear since n


[a∗ ⊗ ai ] · c(1) ⊗ [a∗i ⊗ a] · c(2) =

n


∗ Ψ a∗ , aiψ cψ (1) ⊗ ai , aΨ c(2)

i=1

i=1

∗

= a

, aΨ ψ cψ (1)

⊗

cΨ (2)

= a∗ , aψ ∆(cψ ) = ∆([a∗ ⊗ a] · c)

and εC ([a∗ ⊗ a] · c) = εC (a∗ , aψ cψ ) = a∗ , aεC (c) = εH ([a∗ ⊗ a])εC (c) Conversely, let C be a left H-module coalgebra. We know from Proposition 20 that A is a right H-comodule algebra, so we have (H, A, C) ∈ • DK• (k) and (A, C, ψ) = F (H, A, C) ∈ • E• (k). Recall that ψ : A ⊗ C → A ⊗ C is given by ψ(a ⊗ c) = a[0] ⊗ a[1] c Let us check that we have a bijective correspondence, as needed. Let (A, C, ψ) be an entwining structure, (H, A, C) the corresponding Doi-Koppinen structure, and write F (H, A, C) = (A, C, ψ ′ ). Then ψ ′ (a ⊗ c) = a[0] ⊗ a[1] c =

n


ai ⊗ [a∗i ⊗ a] · c

i=1

=

n


ai ⊗ a∗i , aψ cψ = aψ ⊗ cψ = ψ(a ⊗ c)

i=1

If C is a left H(A)-module coalgebra, then we have a left-right Doi-Koppinen structure (H(A), A, C), and an entwining structure (A, C, ψ) = F (H(A), A, C). This entwining structure defines a left H(A)-module coalgebra structure on C, we denote this action temporarily by ⇀. This action equals the original one, since [a∗ ⊗ a]⇀c = a∗ , aψ cψ = a∗ , a[0] a[1] · c n
a∗ , ai [a∗i ⊗ a] · c = [a∗ ⊗ a] · c = i=1

Adapting our arguments, we see that every entwining structure (A, C, ψ), with C finitely generated and projective, comes from an alternative DoiKoppinen structure. This time, the involved bialgebra is H ′ = H(C ∗ )cop . Observe that H(C ∗ )cop = T (C ∗ ⊗ C)/I, with I generated by elements of the form εC , c − εC ⊗ c and c∗ ∗ d∗ ⊗ c − (c∗ ⊗ c(1) ) ⊗ (d∗ ⊗ c(2) ) and

2.1 Doi-Koppinen structures and entwining structures

∆H ′ ([c∗ ⊗ c]) =

n


47

[c∗ ⊗ ci ] ⊗ [c∗i ⊗ c] and εH ′ ([c∗ ⊗ c]) = c∗ , c

i=1

From Proposition 20, it follows that C is a right H-comodule algebra, with ρr (c) =

n


ci ⊗ [c∗i ⊗ c]

i=1

Given a left-right entwining structure (A, C, ψ), we define a left H ′ -action on A as follows: [c∗ ⊗ c] · a = c∗ , cψ aψ and this makes A into a left H ′ -module algebra. (H ′ , A, C) is a left-right alternative Doi-Koppinen structure. Further verifications are left to the reader. We summarize our results as follows: Theorem 6. Let C be a finitely generated projective coalgebra, and H ′ = H(C ∗ )cop . Then C is a right H ′ -comodule coalgebra. For a given k-algebra A, there is a bijective correspondence between left-right entwining structures (A, C, ψ) and left H ′ -module algebra structures on A. Consequently every entwining structure comes from an alternative Doi-Koppinen structure. We will now show that not every entwining structure arises from a DoiKoppinen structure. Let k be a field. For a left-right entwining structure (A, C, ψ), c ∈ C and c∗ ∈ C ∗ , we consider the transformation Tc,c∗ : A → A ; Tc,c∗ (a) = c∗ , cψ aψ If (A, C, ψ) = F (H, A, C) arises from a Doi-Koppinen structure, then Tc,c∗ (a) = c∗ , a[1] ca[0] and then every H-subcomodule of A is Tc,c∗ -invariant. As every a ∈ A is contained in a finite dimensional H-subcomodule of A (cf. [172, Theorem 2.1.3b]), the Tc,c∗ -invariant subspace of A generated by a is finite dimensional. Example 7. (Schauenburg [163]) Let C = k ⊕ kt, with t primitive, and let A be the free algebra with generators Xi , where i ranges over the integers. We define ψ : A ⊗ C → A ⊗ C by ψ(a ⊗ 1) = a ⊗ 1 for all a ∈ k, and ψ(Xi1 Xi2 · · · Xin ⊗ t) = Xi1 +1 Xi2 +1 · · · Xin +1 ⊗ t A straightforward computation shows that ψ is entwining. Now take c∗ ∈ C ∗ such that c∗ , t = 1. Then Tt,c∗ (Xi ) = Xi+1 and the Tt,c∗ -invariant subspace of A generated by X0 is infinite dimensional, and (A, C, ψ) cannot be derived from a Doi-Koppinen structure.

48

2 Doi-Koppinen Hopf modules and entwined modules

2.2 Doi-Koppinen modules and entwined modules Entwined modules Let (A, C, ψ) ∈ E•• (k). An (A, C, ψ)-entwined module is a k-module with a right A-action and a right C-coaction such that ρr (ma) = m[0] aψ ⊗ mψ [1]

(2.17)

The category of (A, C, ψ)-entwined modules and A-linear C-colinear maps is denoted by M(ψ)C A . We also have left-left, left-right and right-left versions: For (A, C, ϕ) ∈ •• E(k), C A M(ϕ) consists of left A-modules and left Ccomodules such that ρl (am) = mϕ (2.18) [−1] ⊗ aϕ m[0] For (A, C, ψ) ∈ • E• (k), comodules such that For (A, C, ϕ) ∈ • E• (k), comodules such that

C A M (ψ)

consists of left A-modules and right C-

ρr (am) = aψ m[0] ⊗ mψ [1] C

(2.19)

MA (ϕ) consists of right A-modules and left C-

ρl (ma) = mϕ [−1] ⊗ m[0] aϕ

(2.20)

E•• (k),

Examples 3. 1. For (A, C, ψ) ∈ A ⊗ C and C ⊗ A are right-right entwined modules. The structure is given by the formulas (c ⊗ a)b = c ⊗ ab (a ⊗ c)b = abψ ⊗ cψ

ρr (c ⊗ a) = (c(1) ⊗ aψ ) ⊗ cψ (2) ρr (a ⊗ c) = a ⊗ c(1) ⊗ c(2)

ψ : C ⊗ A → A ⊗ C is a morphism in E•• (k). See also Examples 13 and 14 2. Let H be a bialgebra with twisted antipode S, and consider the map ψ : H ⊗ H → H ⊗ H with ψ(h ⊗ k) = h(2) ⊗ h(3) kS(h(1) ) (H, H, ψ) ∈ • E• (k), and the objects of H M(ψ)H are k-modules with a left H-action and right H-coaction such that ρr (hm) = h(2) m[0] ⊗ h(3) m[1] S(h(1) ) These modules are known under the name Yetter-Drinfeld modules . YetterDrinfeld modules will be investigated in Section 4.4 and Chapter 5. 3. For any bialgebra H, (H, H, IH⊗H ) ∈ • E• (k). H M(ψ)H consists of kmodules with a left H-action and right H-coaction such that ρr (hm) = hm[0] ⊗ m[1] i.e. the right H-coaction is left H-linear. In the situation where H is commutative and cocommutative, this type of modules has been considered first by Long in [118]; in the sequel, we will refer to them as Long dimodules. If H is commutative and cocommutative, then Long dimodules and Yetter-Drinfeld modules coincide. We will come back more extensively to Long dimodules in Section 4.5 and Chapter 7.

2.2 Doi-Koppinen modules and entwined modules

49

Doi-Koppinen Hopf modules Let (H, A, C) ∈ DK•• (k), and (A, C, ψ) = F (H, A, C) the corresponding object in E•• (k). M(H)C A will be another notation for M(ψ)C . (2.17) takes the form A ρr (ma) = m[0] a[0] ⊗ m[1] a[1]

(2.21)

The objects of M(H)C A are called unified Hopf modules, Doi-Hopf modules or Doi-Koppinen Hopf modules. If H has a twisted antipode, then ψ is bijective, C −1 and (A, C, ψ −1 ) ∈ •• E(k). C ). This A M(H) will be a new notation for A M(ψ category consists of left A-modules and left C-comodules such that ρl (am) = m[−1] S(a[1] ) ⊗ a[0] m[0]

(2.22)

Similar constructions apply to left-right, right-left and left-left Doi-Koppinen structures. Examples 4. 1. (k, A, k) ∈ DK•• (k), and the corresponding entwining structure is (A, k, IA ) ∈ E•• (k). The category of entwined modules is nothing else then the category of right A-modules. 2. (k, k, C) ∈ DK•• (k) corresponds to (k, C, IC ) ∈ E•• (k). An entwined module is now simply a right C-comodule. 3. Let H be a bialgebra. (H, H, H) ∈ DK•• (k) corresponds to (H, C, ψ) ∈ E•• (k), with ψ(h ⊗ k) = k(1) ⊗ hk(2) An entwined module is now a Hopf module in the sense of Sweedler [172]. 4. If H is a bialgebra, and A is a right H-comodule algebra, then (H, A, H) ∈ DK•• (k) is a right-right Doi-Koppinen structure. The corresponding DoiKoppinen modules are the well-known (A, H)-relative Hopf modules (see e.g. [67]). 5. In a similar way, if C is a right H-module coalgebra, then (H, H, C) ∈ DK•• (k) is a right-right Doi-Koppinen structure, and the corresponding DoiKoppinen modules are the [H, C])-Hopf modules studied in [67]. 6. Let G be a group, and A a G-graded k-algebra. Then (kG, A, kG) is a Doi-Koppinen structure, and the corresponding category of Doi-Koppinen modules is the category of G-graded right A-modules. 7. Now let X be a right G-set. Then kX is a right kG-module coalgebra, and (kG, A, kX) is a Doi-Koppinen structure. M(kG)kX A consists of right Amodules graded by the G-set X (see [143] and [147] for a study of modules graded by G-sets). 8. Yetter-Drinfeld modules and Long dimodules are special cases of Doi-Hopf modules. This will be explained in Section 4.4 and 4.5. Proposition 21. For (A, C, ψ) ∈ E•• (k), the categories M(ψ)C A , Aop M(ψ ◦ cop cop τ )C , C M(τ ◦ ψ)A and C M(τ ◦ ψ ◦ τ ) are isomorphic. In particular, for Aop op C C cop cop op M(H ) , M(H )A , (H, A, C) ∈ DK•• (k), the categories M(H)C A A cop opcop M(H ) are isomorphic. and C Aop

50

2 Doi-Koppinen Hopf modules and entwined modules

Proof. Everything is straightforward. For example, if M ∈ M(ψ)C A , then the corresponding object in Aop M(ψ ◦ τ )C is equal to M as a right C-comodule, but with left Aop -action given by aop · m = ma All the other isomorphisms are defined in a similar way, and we leave further details to the reader.

2.3 Entwined modules and the smash product Let A and B be k-algebras, and consider a map R : B ⊗ A → A ⊗ B. We will use the following notation (summation understood): R(b ⊗ a) = aR ⊗ bR = ar ⊗ br

(2.23)

We put A#R B = A ⊗ B as a k-module, but with a new multiplication: mA#R B = (mA ⊗ mB ) ◦ (IA ⊗ R ⊗ IB )

(2.24)

(a#b)(c#d) = acR #bR d

(2.25)

or If this new multiplication makes A#R B into an associative algebra with unit 1#1, then we call A#R B a smash product, and (A, B, R) a smash product structure or a factorization structure. Theorem 7. ([44]) (A, B, R) is a smash product structure if and only if R(b ⊗ 1A ) = 1A ⊗ b

(2.26)

R(1B ⊗ a) = a ⊗ 1B R(bd ⊗ a) = aRr ⊗ br dR R(b ⊗ ac) = aR cr ⊗ bRr

(2.27) (2.28) (2.29)

for all a, c ∈ A, b, d ∈ B. Proof. Assume that (A, B, R) is a smash product structure. Then for all b ∈ B, we have 1A #b = (1A #b)(1A #1B ) = 1R #bR and (2.26) follows. (2.27) follows in a similar way. The multiplication is associative, so     (1#b) (a#1)(c#1) = (1#b)(a#1) (c#1) and this implies (2.29). (2.28) follows from     (1#b)(1#d) (a#1) = (1#b) (1#d)(a#1)

the converse is left to the reader.

2.3 Entwined modules and the smash product

51

Remark 3. The smash product is related to the factorization problem. We say that a k-algebra X factorizes through k-algebras A and B if X ∼ = A ⊗ B as k-modules, and, after identifying X and A ⊗ B, the maps ιA : A → A ⊗ B = X

ιA (a) = a ⊗ 1B

ιB : B → A ⊗ B = X

ιB (b) = 1A ⊗ b

are algebra maps. It is not hard to show that there exists a bijective correspondence between the algebra structures on A ⊗ B for which ιA and ιB are algebra maps, and smash product structures of the form (A, B, R): given the multiplication mX on X, we put R(b ⊗ a) = mX (ιB (b) ⊗ ιA (a)) Let (A, B, R), (A′ , B ′ , R′ ) be smash product structures. A morphism (A, B, R) → (A′ , B ′ , R′ ) consists of a pair (α, β), where α : A → A′ and β : B → B ′ are algebra maps such that (α ⊗ β) ◦ R = R′ ◦ (β ⊗ α), or α(aR ) ⊗ β(bR ) = α(a)R′ ⊗ β(b)R′ for all a ∈ A and b ∈ B. S(k) will denote the category of smash product structures over k. Proposition 22. If (A, B, R) ∈ S(k) is a smash product structure, then (B op , Aop , τ ◦R ◦τ ) is also a smash product structure. Furthermore the switch map τ : (A#R B)op → B op #τ ◦R◦τ Aop is an algebra isomorphism. Proof. The first statement is obvious. To prove the second one, we only need to show that τ is anti-multiplicative. Indeed, τ (c#d)τ (a#b) = (d#c)(b#a) = d · bR #cR · a = bR d#RacR = τ (acR #R bR d)   = τ (a#b)(c#d)

Proposition 23. Let (A, B, R) ∈ S(k) be a smash product structure, and assume that R is invertible. Then (B, A, S = R−1 ) is also a smash product structure, and R : B#S A → A#R B is an algebra isomorphism, with inverse S. Proof. We write down conditions (2.26-2.29) as commutative diagrams. Change the direction of the morphisms involving R, and replace R by R−1 = S. We then obtain the diagram telling that (B, A, R−1 ) is a smash product structure. We are left to prove that R is multiplicative. This works as follows: for all b, d ∈ B and a, c ∈ A, we have

52

2 Doi-Koppinen Hopf modules and entwined modules

    R (b#a)(d#c) = R bdS #aS c

= (aS c)R #(bdS )R = (aS c)R1 R2 #bR2 dSR1

(2.28)

= (aSR1 cR3 )R2 #bR2 dSR1 R3 = (acR3 )R2 #bR2 dR3

(2.29) (S = R−1 )

= aR2 cR3 R4 #bR2 R4 dR3

(2.29)

= (aR2 #bR2 )(cR3 #dR3 ) = R(b#a)R(d#c) Theorem 8. Let A be a k-algebra, and C a k-coalgebra which is finitely generated and projective as a k-module. Then there is a bijective correspondence between left-right entwining structures of the form (A, C, ψ) and smash product structures of the form (A, C ∗ , R). If R corresponds to ψ, then the categories A M(ψ)C and A#R C ∗ M are isomorphic. Proof. Let {ci , c∗i | i = 1, · · · , n} be a dual basis for C. For an entwining structure (A, C, ψ) ∈ • E• (k), we define f (ψ) = R : C ∗ ⊗ A → A ⊗ C ∗ by
∗ R(c∗ ⊗ a) = (2.30) c∗ , cψ i aψ ⊗ ci i

∗

We claim that (A, C , R) is a smash product structure. For all c∗ , d∗ ∈ C ∗ and a, b ∈ A, we have
∗ ∗ c∗ , cψ aRr ⊗ c∗r ∗ d∗R = i (AR )ψ ⊗ ci ∗ dR i

=




∗ Ψ ∗ ∗ c∗ , cψ i d , cj aΨ ψ ⊗ ci ∗ cj

i,j

(1.5)

=

(2.3)

=




c∗ , (ci(1) )ψ d∗ , (ci(2) )Ψ aΨ ψ ⊗ c∗i

i




ψ ∗ ∗ c∗ , (cψ i )(1) d , (ci )(2) aψ ⊗ ci

i

=




∗ c∗ ∗ d∗ , cα i aψ ⊗ ci

i

= R(c∗ ∗ d∗ ⊗ a) proving (2.28). aR br ⊗ (c∗ )Rr =




∗ c∗R , cψ i aR bψ ⊗ ci

i

=




∗ Ψ ∗ c∗j , cψ i c , cj aΨ bψ ⊗ ci

i,j

=


i

∗ c∗ , cψΨ i aΨ bψ ⊗ ci

2.3 Entwined modules and the smash product

(2.7)

=




53

∗ c∗ , cψ i (ab)ψ ⊗ ci

i

= R(c∗ ⊗ ab) proving (2.29). (2.26) and (2.27) are left to the reader. Conversely, if (A, C ∗ , R) is a smash product structure, then we define g(R) = ψ : A ⊗ C → A ⊗ C by
(c∗i )R , caR ⊗ ci (2.31) ψ(a ⊗ c) = i

Then aΨ ψ ⊗ (c(1) )ψ ⊗ (c(2) )Ψ = =





(c∗i )R , c(1) aΨ R ⊗ ci ⊗ (c(2) )Ψ i

(c∗i )R , c(1) (c∗j )r , c(2) arR

⊗ ci ⊗ cj

i,j

=




(c∗i )R ∗ (c∗j )r , carR ⊗ ci ⊗ cj

i,j

(2.28)

=




(c∗i ∗ c∗j )R , caR ⊗ ci ⊗ cj

i,j

(1.5)

=




(c∗i )R , caR ⊗ ∆(ci )

i

= aψ ⊗ δ(cψ ) proving (2.3). To prove (2.7): aψ bΨ ⊗ cΨ ψ = (c∗i )R , cΨ aR bΨ ⊗ ci = (c∗i )R , cj (c∗j )r , caR br ⊗ ci    = (c∗i )R , cj c∗j r , c aR br ⊗ ci (2.29)

= (c∗i )Rr , caR br ⊗ ci = (c∗i )R , c(ab)R ⊗ ci

= ψ(ab ⊗ c) Next observe that (g(f (ψ)))(a ⊗ c) =


(c∗i )R , caR ⊗ ci i

=




∗ c∗i , cψ j cj , caψ ⊗ ci

i,j

=




c∗j , caψ ⊗ cψ j

j

= aψ ⊗ cψ = ψ(a ⊗ c)

54

2 Doi-Koppinen Hopf modules and entwined modules

and it follows that (g ◦ f )(ψ) = ψ. In a similar way, we can prove that (f ◦ g)(R) = R, and this finishes the proof of the first part of the Theorem. We will now define an isomorphism F : A M(ψ)C → A#R C ∗ M. For M ∈ C ∗ A M(ψ) , we define F (M ) = M as a k-module, with left A#R C -action defined by (a#c∗ ) · m = c∗ , m[1] a · m[0] (2.32) It is clear that M is an A#R C ∗ -module, since   (a#c∗ )(b#d∗ ) · m = (abR #(c∗R ∗ d∗ )) · m = c∗R ∗ d∗ , m[1] abR m[0]
∗ ∗ c∗ , cψ = i ci , m[1] d , m(2) abψ m[0] i

  = (a#c∗ ) · d∗ , m[1] bm[0]   = (a#c∗ ) · (b#d∗ ) · m

Conversely, if M is a left A#R C ∗ -module, we define G(M ) ∈ G(M ) = M as a k-module, with left A-action

C A M(ψ) :

am = (a#εC ) · m and right C-coaction ρr (m) =




(1#c∗i ) · m ⊗ ci

i

Further details are left to the reader. Theorem 9. Let A be a k-algebra, and C a coalgebra which is finitely generated and projective as a k-module. Then there is a bijection between right-left entwining structures of the form (A, C, ϕ) and smash product structures of the form (C ∗ , A, S). In this situation we have an isomorphism of categories C

M(ψ)A ∼ = MC ∗ #S A

Proof. We use the left-right dictionary. If (A, C, ϕ) ∈ • E• (k), then (Aop , C cop , τ ◦ ϕ ◦ τ ) ∈ • E• (k) (see Proposition 14). Using Theorem 8, we find (Aop , C cop∗ = C ∗op , R) ∈ S(k). Finally Proposition 22 gives the corresponding smash product structure (C ∗ , A, S = τ ◦ R ◦ τ ). From (2.30) and (2.31), it follows that the correspondence between S and ϕ is given by the formulas
∗ S(a ⊗ c∗ ) = c∗ , cϕ (2.33) i ci ⊗ aϕ i


(c∗i )R , cci ⊗ aR ϕ(c ⊗ a) = i

(2.34)

2.3 Entwined modules and the smash product

55

Take an entwining structure (A, C, ψ) ∈ • E• (k). Assume that C is finitely generated and projective, and that ψ is invertible. Then we have a right-left entwining structure (C, A, ϕ = τ ◦ ψ −1 ◦ τ ) ∈ • E• (k) (see Proposition 15). Let (A, C ∗ , R) and (C ∗ , A, S) be the two corresponding smash product structures from Theorems 8 and 9. One is then tempted to conjecture that S = R−1 , and therefore A#R C ∗ ∼ = C ∗ #S A, according to Proposition 23. Surprisingly, this is not true in general! a straightforward computation shows that S = R−1 if and only if
c∗i , cϕ cψ (2.35) c⊗a = i ⊗ aψϕ i


c∗i , cψ cϕ c⊗a = i ⊗ aϕψ

(2.36)

i

for all c ∈ C and a ∈ A. The condition ϕ = τ ◦ ψ −1 ◦ τ amounts to
c∗i , cϕ cψ c⊗a = i ⊗ aϕψ

(2.37)

i

c⊗a =


c∗i , cψ cϕ i ⊗ aψϕ

(2.38)

i

for all c ∈ C and a ∈ A. We will make the difference clear in the Doi-Hopf case. Examples 5. 1) Let A be a right H-comodule algebra, and B a right H-module algebra. Define R : B ⊗ A → A ⊗ B by R(b ⊗ a) = a[0] ⊗ ba[1]

(2.39)

Then (A, B, R) is a smash product structure, and the multiplication on A#R B is given by the formula (a#b)(c#d) = ac[0] #(bc[1] )d

(2.40)

If H has a twisted antipode, then R is invertible, and R−1 is given by the formula R−1 (b ⊗ a) = bS(a[1] ) ⊗ a[0] (2.41) 2) In a similar way, if A is a left H-comodule algebra, and B is a left H-module algebra, then we have a smash product structure (B, A, R), with R(a ⊗ b) = a[−1] b ⊗ a[0] 3) Let (H, A, C) ∈ • aDK• (k), i.e. A is a left H-module algebra, and C is a right H-comodule coalgebra. If C is finitely generated projective, then C ∗ is a left H-comodule algebra, cf. Proposition 7, so we find a smash product structure (A, C ∗ , R). Now (H, A, C) defines an entwining structure (see

56

2 Doi-Koppinen Hopf modules and entwined modules

Proposition 18), and Theorem 8 produces another smash product structure (A, C ∗ , R′ ). As one might expect, R = R′ , since R′ (c ⊗ a) =

n


∗ c∗ , cψ i aψ ⊗ ci

i=1

=

n


c∗ , ci[0] ci[0] a ⊗ c∗i

i=1

= c∗[−1] a ⊗ c∗[0] = R(c ⊗ a) Let (A, B, R) be a smash product structure. Can we find a bialgebra H, an H-coaction on A and an H-action on B such that R is given by (2.39). We have discussed this question already for entwining structures. For smash product structures, the answer is the following: Theorem 10. (Tambara [178]) Let A be a finitely generated and projective algebra, and H = H(A) as in Proposition 19. For every algebra B, we have a bijective correspondence between smash product structures of the form (A, B, R) and right H-module algebra structures on B. A similar result holds if B is finitely generated projective. Proof. The proof is similar to the corresponding proofs for entwining structures (Theorems 5 and 6). We know that A is a right H-comodule algebra. Given R, we define a right H-action on B as follows: b · [a∗ ⊗ a] = a∗ , aR bR We invite the reader to prove that this puts an H-module algebra structure on H. Conversely, if B is a right H-module algebra, then Example 5 1) tells us how to produce a smash product structure. Example 8. If (H, A, C) ∈ • DK• (k), then C ∗ is a right H-module algebra, the right H-action on C ∗ is given by c∗ ↼h, c = c∗ , hc and we obtain a smash product structure (A, C ∗ , R). We have a functor F :

C A M(H)

→ A#R C ∗ M

F (M ) = M as a k-module, with left A#R C ∗ -action (a#c∗ ) · m = c∗ , m[1] am[0] If C is finitely generated and projective, then the map R coincides with the one from Theorem 8. First observe that
c∗ , hci c∗i c∗ ↼h = i

2.3 Entwined modules and the smash product

57

To see this, apply both sides to an arbitrary c ∈ C. Thus, according to (2.30),
c∗ , a[1] ci a[0] ⊗ c∗i R(c∗ ⊗ a) = i

=




a[0] ⊗ c∗ ↼a[1] , ci c∗i

i

= a[0] ⊗ (c∗ ↼a[1] )

(2.42)

If C is projective, but not necessarily finitely generated, then F is fully faithful. Indeed, if f : M → N is a left A#R C ∗ -linear map between two Doi-Hopf modules M and N , then f is left A-linear and left C ∗ -linear, and therefore right C-colinear, by Proposition 3. Consequently, A M(H)C can be viewed as a full subcategory of A#R C ∗ M. Example 9. Now assume that H has an antipode. To (H, A, C) ∈ • DK• (k), we can associate (A, C, ψ) ∈ • E• (k) and (A, C, ϕ) ∈ • E• (k) Recall that ϕ : C ⊗ A → C ⊗ A is given by ϕ(c ⊗ a) = S(a[1] )c ⊗ a[0] We have associated smash product structures (A, C ∗ , R) and (C ∗ , A, S), with R given by (2.42), and S by   (2.43) S(a ⊗ c∗ ) = c∗ ↼S(a[1] ) ⊗ a[0]

Even if C is not necessarily finitely generated and projective, (2.43) defines a smash product structure. In any case, we have a functor F : C M(H)A → MC ∗ #S A . F (M ) = M as a k-module, with action m · (c∗ #a) = c∗ , m[−1] m[0] a

If C is finitely generated and projective, then F is an isomorphism of categories. If H has a twisted antipode, then the inverse of R exists and is given by (see (2.41))   (2.44) R−1 (a ⊗ c∗ ) = c∗ ↼S(a[1] ) ⊗ a[0]

If the antipode S of H is of order 2, then we can conclude from (2.43) and (2.44) that S = R−1 , and we have the following result. Proposition 24. Let (H, A, C) ∈ • DK(k)• , and assume that H has an antipode of order 2. Let (A, C ∗ , R) and (C ∗ , A, S) be defined as in Example 9. Then the smash products A#R C ∗ and C ∗ #S A are isomorphic. Koppinen’s smash product Let (A, C, ψ) ∈ • E• (k) be a left-right entwining structure. The Koppinen smash #ψ (C, A) is equal to Hom(C, A) as a k-module, but with twisted multiplication (f • g)(c) = f (cψ (1) )g(c(2) )ψ for all f, g : C → A and c ∈ C.

58

2 Doi-Koppinen Hopf modules and entwined modules

Proposition 25. If (A, C, ψ) is a left-right entwining structure, then #ψ (C, A) is an associative algebra, with unit ηA ◦ εC . Proof. The proof of the associativity goes as follows: ((f • g) • h)(c) = (f • g)(cψ (1) )h(c(2) )ψ       ψ′ cψ h(c(2) )ψ = f cψ (1) (2) (1) (1) g ψ′     ψ ψ′ h(c(3) )ψΨ (2.3) = f cΨ (1) g c(2) ψ′    ψ   g c(2) h(c(3) )ψ (2.7) = f cΨ (1) Ψ    (g • h)(c ) = f cΨ (2) Ψ (1) = (f • (g • h))(c) From (2.2) and (2.4), it follows easily that ηA ◦ εC is the unit element of #ψ (C, A). Proposition 26. C ∗ and A are subalgebras of #ψ (C, A), via c∗ → ηA ◦ c∗ and a → aη ◦ εC Proof. Obvious. Proposition 27. For (A, B, ψ) ∈ • E• (k), we have a functor F :

C A M(ψ)

→ #ψ (C,A) M

For an entwined module M , F (M ) = M as a k-module, with left #ψ (C, A)action given by f · m = f (m[1] )m[0] Proof. We will prove that F (M ) is a #ψ (C, A)-module, leaving further details to the reader. For f, g : C → A, we have f · (g · m) = f · (g(m[1] )m[0] ) = f (mψ [1] )g(m(2) )ψ m[0] ) = (f • g)(m[1] )m[0] ) = (f • g) · m Proposition 28. Let (A, B, ψ) ∈ • E• (k), and assume that C is finitely generated and projective as a k-module. Let (A, C ∗ , R) be the corresponding smash product structure (cf. Theorem 8). Then we have an algebra isomorphism s : A#R C ∗ → #ψ (C, A) given by s(a#c∗ )(c) = c∗ , ca for all a ∈ A, c ∈ C and c∗ ∈ C ∗ .

2.4 Entwined modules and the smash coproduct

59

Proof. It is well-known that s is a k-module isomorphism if C is finitely generated and projective. So we only need to show that s is an algebra map. Let {ci , c∗i | i = 1, · · · , n} be a finite dual basis of C. For all a, b ∈ A, c ∈ C and c∗ , d∗ ∈ C ∗ , we have     ∗ ∗ s (a#c∗ )(b#d∗ ) (c) = s c∗ , cψ i abψ #ci ∗ d (c) ∗ ∗ = c∗ , cψ i ci ∗ d , cabψ

∗ ∗ = c∗ , cψ i ci , c(1) d , c(2) abψ ∗ = c∗ , cψ (1) d , c(2) abψ   ∗ = (aηA ◦ c∗ )(cψ (1) ) (bηA ◦ d )(c(2) ) ψ   = (aηA ◦ c∗ ) • (bηA ◦ d∗ ) (c)   = s(a#c∗ ) • s(b#d∗ ) (c)

Example 10. Let (H, A, C) ∈ • DK• (k), and let (A, C, ψ) be the associated entwining structure. We will write #H (C, A) = #ψ (C, A). The product on #H (C, A) is given by the formula   (f • g)(c) = f g(c(2) )(1) c(1) g(c(2) )[0]

This multiplication appeared first in [111, Def. 2.2]. In the situation where C = H, it appears already in [68] and [110].

2.4 Entwined modules and the smash coproduct The result in this Section may be viewed as the duals of the ones in Section 2.3. Let C and D be coalgebras, and consider a linear map V : C ⊗ D → D ⊗ C. We will use the notation V (c ⊗ d) = dV ⊗ cV C ⊲