Quantum Groups and Knot Invariants 2856290558, 9782856290552

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` PANORAMAS ET SYNTHESES 5

QUANTUM GROUPS AND KNOT INVARIANTS Christian Kassel, Marc Rosso, Vladimir Turaev

Soci´et´e Math´ematique de France 1997

QUANTUM GROUPS AND KNOT INVARIANTS Christian Kassel1 , Marc Rosso2 , Vladimir Turaev3

Ce livre propose une introduction aux groupes quantiques, aux cat´egories mono¨ıdales tress´ees et aux invariants quantiques de nœuds et de vari´et´es de dimension trois. Nous mettons l’accent sur les relations profondes qui existent entre ces domaines et qui ont ´et´e d´ecouvertes au cours de la derni`ere d´ecennie. Dieses Buch bietet eine Einleitung in die Quantengruppen, die verzopften monoidalen Kategorien und die Quanteninvarianten von Knoten und von dreidimensionalen Mannigfaltigkeiten. Die tiefliegenden Beziehungen, die neuerdings zwischen diesen Bereichen entdeckt wurden, werden hier unterstrichen. This book provides a concise introduction to quantum groups, braided monoidal categories, and quantum invariants of knots and of three-dimensional manifolds. The exposition emphasizes the newly discovered deep relationships between these areas.

V knige daets s atoe vvedenie v teori kvantovyh grupp, kosovyh kate˘ i kvantovyh invariantov uzlov i trehmernyh mnogoobrazii ˘ . Osoboe gorii vnimanie udelets nedavno otkrytym glubokim vzaimosvzm me du timi oblastmi. Classification AMS : 16 W 30, 17 B 37, 18 D 10, 20 F 36, 57 M 25, 57 N 10, 81 R 50.

1,2,3

Institut de Recherche Math´ematique Avanc´ee, Universit´e Louis Pasteur, CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg CEDEX (France).

CONTENTS

INTRODUCTION

1

1. THE YANG-BAXTER EQUATION AND BRAID GROUP REPRESENTATIONS

5

1.1. 1.2. 1.3.

5 7 8

The Yang-Baxter Equation Artin’s Braid Groups Alternative description of Bn

2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

11

2.1. 2.2. 2.3. 2.4.

11 15 18 19

Hopf Algebras Monoidal Categories Braidings Braided Bialgebras

3. DRINFELD’S QUANTUM DOUBLE

23

3.1. 3.2. 3.3.

23 26 30

The Dual Double Construction The Quantum Double and its Properties Hopf Pairings and a Generalized Double

4. THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N + 1) 4.1. 4.2. 4.3. 4.4.

The Lie Algebra sl(N + 1) Construction of Uq sl(N + 1) A Poincar´e-Birkhoff-Witt-Type Basis in U+ Specializations and the Universal R-Matrix

35 35 36 39 42

vi

CONTENTS

5. THE JONES POLYNOMIAL AND SKEIN CATEGORIES

45

5.1. 5.2. 5.3. 5.4.

45 48 52 55

Knots, Links, and Link Diagrams The Jones Polynomial of Links Skein Modules of Tangles Categories of Tangles

6. FROM RIBBON CATEGORIES TO TOPOLOGICAL INVARIANTS OF LINKS AND 3-MANIFOLDS

59

6.1. 6.2. 6.3. 6.4.

59 62 65 67

Ribbon Categories The Functor F Modular Categories Invariants of 3-Manifolds

7. THE REPRESENTATION THEORY OF Uq sl(N + 1)

71

7.1. 7.2. 7.3. 7.4.

71 74 76 79

Highest Weight Modules Quantum Theory of Invariants The Case of Roots of Unity Quantum Groups with a Formal Parameter

8. VASSILIEV INVARIANTS OF LINKS

81

8.1. 8.2. 8.3. 8.4.

81 84 85 87

Definition and Examples Chord Diagrams and Kontsevich’s Theorem The Pro -Unipotent Completion of a Braided Category
0 (R) Another description of T

9. ADVANCED TOPICS 9.1. 9.2. 9.3. 9.4.

Infinitesimal Symmetric Categories Quantization of an Infinitesimal Symmetric Category The Kontsevich Universal Invariant An Action of Gal(Q/Q)

91 91 95 100 102

GUIDE TO THE LITERATURE

105

INDEX

113

` PANORAMAS ET SYNTHESES 5

INTRODUCTION

In this book we survey recent spectacular developments in the theory of Lie algebras and low-dimensional topology. These developments center around quantum groups and braided categories on the algebraic side and new invariants of knots, links, and three-manifolds on the topological side. This new theory has been, to a great extent, inspired by ideas that arose in theoretical physics. This strongly emphasizes a remarkable unity between the theory of Lie algebras, low-dimensional topology, and mathematical physics. Quantum groups were introduced around 1983–1985 by V. Drinfeld and M. Jimbo. They may be roughly described as one-parameter deformations of the enveloping algebras of semisimple Lie algebras. Quantum groups appeared as an algebraic formulation of the work of physicists, especially from the Leningrad school of L. Faddeev, on the Yang-Baxter equation. An important feature of quantum groups is that the category of their representations has a so-called braiding. The notion of a braided monoidal category, formulated by A. Joyal and R. Street, plays a fundamental rˆole in this theory. An independent breakthrough was made in knot theory in 1984 by V. Jones. He used von Neumann algebras to define a new polynomial invariant of links. The study of the Jones polynomial rapidly involved ideas of statistical mechanics including the Yang-Baxter equation. It was observed by N. Reshetikhin and V. Turaev that the braided categories derived from quantum groups were the right algebraic objects needed to define representations of the braid groups and link invariants. This leads to a vast set of polynomial invariants of links whose components are coloured with finitedimensional representations of a complex semisimple Lie algebra. This generalizes the Jones polynomial, which arises when all components of a link are coloured with the fundamental two-dimensional representation of sl2 (C). Further study proceeded in several different, albeit related, directions. In 1988 E. Witten invented the notion of a topological quantum field theory and outlined a fascinating picture of such a theory in three dimensions. This picture includes a path integral definition of numerical invariants of three-manifolds and links in threemanifolds generalizing the values of the Jones polynomial at the roots of unity. A rigourous mathematical definition of such three-manifold invariants was given by N. Reshetikhin and V. Turaev in 1988 on the basis of the theory of quantum groups at roots of unity.

2

INTRODUCTION

At about the same time (1989-90), in the quite different context of singularity theory, V. Vassiliev introduced the notion of a knot invariant of finite degree. The invariants of degree 0, 1, 2, . . . form an increasing filtration on the vector space of all knot invariants. The associated graded vector space can be described in terms of chord diagrams which may be viewed as a mathematical version of Feynman diagrams. In 1992 M. Kontsevich constructed a universal Vassiliev-type invariant of knots with values in the algebra of chord diagrams. This invariant dominates all finite degree invariants. The polynomial invariants of knots derived from quantum groups can also be computed from the Kontsevich invariant. Indeed, these polynomials may be expanded as formal series whose coefficients are invariants of finite degree. This survey is intended to introduce the reader in the world of quantum groups, braided categories, knots, three-manifolds, and their invariants. We have not tried to give a complete picture of the theory, but rather to highlight its main features. Unfortunately, we had to leave out of the scope of this book a number of important aspects of the theory, including Woronowicz’s approach to quantum groups in the framework of operator algebras, the dual construction by Faddeev, Reshetikhin, and Takhtadjian, the connections with state sum models of statistical mechanics, the quantization of Poisson structures and the theory of Poisson-Lie groups. The book is organized as follows. We start in Chapter I with the Yang-Baxter equation. We show how solutions of this equation lead to representations of the braid groups. In Chapter II we introduce the concept of a braided bialgebra and show that the category of representations of such a bialgebra is a braided monoidal category. Examples of braided bialgebras are provided by the quantum groups defined and studied in Chapters III and IV. In Chapter III we present Drinfeld’s quantum double construction, which is a general method to produce braided bialgebras. Quantum groups appear in Chapter IV with the quantization Uq sl(N + 1) of the Lie algebra sl(N + 1). We give an overview of their main properties. In Chapter V we enter the world of low-dimensional topology. We start with a definition of the Jones polynomial using the Kauffman bracket. We show that the study of knots, links, and more general objects called tangles naturally leads to braided monoidal categories. This yields geometric constructions of such categories. In Chapter VI we introduce an important class of braided monoidal categories, namely the ribbon categories. Monoidal categories derived from the representations of quantum groups or from tangles are ribbon categories. We explain how to construct isotopy invariants of knots, links, and tangles, whose components are coloured with objects of a ribbon category. Then we introduce the more restricted class of modular categories and show how to derive from each modular category the corresponding Reshetikhin-Turaev invariant of three-manifolds and links in three-manifolds. In Chapter VII we survey the representation theory of Uq sl(N + 1) and show that it leads to ribbon and modular categories, hence to the construction of “quantum invariants” of links and three-manifolds. Chapter VIII is devoted to the theory of Vassiliev invariants of links. Examples of ` PANORAMAS ET SYNTHESES 5

INTRODUCTION

3

Vassiliev invariants are provided by the quantum invariants of the previous chapter. We formulate an important theorem due to Kontsevich. In Chapter IX we present more advanced topics based on work of Drinfeld. In particular, we give a construction of Kontsevich’s universal link invariant and show how to recover the quantum invariants from it. In the very last section we explain how the Galois group Gal(Q/Q) acts on the space of Vassiliev invariants. We close with a guide to the literature for the reader wishing to get more information on the subject. This book grew out of notes we wrote for the “Journ´ees Quantiques” that took place at the Department of Mathematics of the Universit´e Louis Pasteur in Strasbourg ´ on April 2–4, 1993. This was the first of a series of semi-annual meetings “Etat de la Recherche” initiated by the Soci´et´e Math´ematique de France (SMF) and sponsored by the SMF, the Minist`ere de la Recherche et de la Technologie, the Minist`ere de ´ l’Education Nationale (DRED), and the Institut de Recherche Math´ematique Avanc´ee (Strasbourg).

´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

Chapter 1 THE YANG-BAXTER EQUATION AND BRAID GROUP REPRESENTATIONS

In this chapter we introduce the Yang-Baxter equation and show how any solution of this equation gives rise to representations of the braid groups.

1. The Yang-Baxter Equation 1.1. R-matrices. — Consider a vector space V over a field k. The Yang-Baxter equation is the following equation for a linear automorphism c of V ⊗ V : (1.1)

(c ⊗ idV )(idV ⊗c)(c ⊗ idV ) = (idV ⊗c)(c ⊗ idV )(idV ⊗c).

Equation (1.1) holds in the automorphism group of V ⊗ V ⊗ V . A solution is called an R-matrix. Let (vi )i be a basis of the vector space V . An automorphism c of V ⊗ V is defined by the family (ck ij )i,j,k, of scalars determined by c(vi ⊗ vj ) =



ck ij vk ⊗ v .

k,

Then c is a solution of the Yang-Baxter equation if and only if, for all i, j, k, , m, n,  pq yn  qr y (1.2) cij cqk cm cjk ciq cmn py = yr . p,q,y

y,q,r

Solving the non-linear equations (1.2) is a highly non-trivial problem. Nevertheless, numerous solutions of the Yang-Baxter equation have been discovered in the 1980’s. Let us list a few simple ones. 1.2. Example. — For any vector space V we denote by τ V,V ∈ Aut(V ⊗ V ) the flip switching the two copies of V . It is defined by τV,V (v1 ⊗ v2 ) = v2 ⊗ v1

for any v1 , v2 ∈ V .

6

CHAPTER 1. THE YANG-BAXTER EQUATION

The flip satisfies the Yang-Baxter equation because of the Coxeter relation (12)(23)(12) = (23)(12)(23) holding in the symmetric group S3 , where (ij) denotes the transposition exchanging i and j. 1.3. Example. — Let V be a finite-dimensional vector space with a basis (e 1 , . . . , eN ). For any invertible scalar q, we define an automorphism c of V ⊗ V by  if i = j,  q ei ⊗ ei e ⊗ e if i < j, (1.3) c(ei ⊗ ej ) = j i  e ⊗ e + (q − q −1 ) e ⊗ e if i > j. j i i j 1.4. Proposition. — The automorphism c is a solution of the Yang-Baxter equation. Moreover, we have (c − q idV ⊗V )(c + q −1 idV ⊗V ) = 0. We leave the proof as an exercise. Observe that Example 1.3 is a one-parameter-deformation of the automorphism τV,V of Example 1.2. To recover the latter, set q = 1 in (1.3). Note that, when N = 2, the matrix of the automorphism c in the basis formed by the vectors (v1 ⊗ v1 , v2 ⊗ v2 , v1 ⊗ v2 , v2 ⊗ v1 ) is   q 0 0 0 0  0 q 0 (1.4) .  0 0 0 1 0 0 1 q − q −1 1.5. Exercises. (a) Resume the hypotheses and the notations of Example 1.3, which we generalize using two invertible scalars p, q and a family {rij }1≤i,j≤N of scalars such that rii = q and rij rji = p when i = j. Define an automorphism c of V ⊗ V by  if i = j,  qei ⊗ ei e ⊗ e if i < j, r c(ei ⊗ ej ) = ji j i  rji ej ⊗ ei + (q − pq −1 )ei ⊗ ej if i > j. Check that the automorphism c is an R-matrix such that (c − q idV ⊗V )(c + pq −1 idV ⊗V ) = 0. (b) Consider a matrix of the form  p 0 0 a  0 c 0 0 ` PANORAMAS ET SYNTHESES 5

0 b d 0

 0 0 . 0 q

2. ARTIN’S BRAID GROUPS

7

Prove that it is an R-matrix if and only if the following relations hold: adb = adc = ad(a − d) = 0, p2 a = pa2 + abc,

q 2 a = qa2 + abc,

p2 d = pd2 + dbc,

q 2 d = qd2 + dbc.

2. Artin’s Braid Groups 2.1. Definition. — Fix an integer n ≥ 3. We define the braid group with n strands as the group Bn generated by n − 1 generators σ1 , . . . ,σn−1 and the relations (2.1)

σi σj = σj σi

if |i − j| > 1,

(2.2)

σi σi+1 σi = σi+1 σi σi+1

for 1 ≤ i,j ≤ n − 1.

When n = 2, we define B2 as the free group on one generator σ1 . It is useful to set B0 = B1 = {1}. There is a natural surjection of groups from Bn to the symmetric group Sn , that is the group of all permutations of the set {1, . . . , n}. Indeed, consider the (n − 1) transpositions si = (i, i + 1)

(i = 1, . . . , n − 1).

They clearly satisfy Relations (2.1–2.2). It follows that there exists a unique group morphism π : Bn → Sn such that π(σi ) = si for all i. This morphism is surjective because the transpositions si form a generating set for Sn . Actually, to pass from a presentation of the group Bn to a presentation of Sn , it suffices to add the relations σi2 = 1

(i = 1, . . . , n − 1).

One big difference between symmetric groups and braid groups is that the former are finite groups while the latter are infinite groups when n > 1. Moreover, the group Bn is torsion-free, that is to say, all elements = 1 have infinite order. 2.2. From the Yang-Baxter equation to representations of the braid groups. — Let V be a vector space and c an automorphism of V ⊗ V that is an R-matrix as defined in Section 1. For 1 ≤ i ≤ n − 1, define a linear automorphism ci of the n-fold tensor power V ⊗n by  if i = 1,  c ⊗ idV ⊗(n−2) (2.3) ci = idV ⊗(i−1) ⊗ c ⊗ idV ⊗(n−i−1) if 1 < i < n − 1,  if i = n − 1. idV ⊗(n−2) ⊗ c We claim that Relations (2.1–2.2) hold for the automorphisms c1 , . . . , cn−1 . This is immediate for (2.1). As for (2.2), observe that it suffices to check that c1 c2 c1 = c2 c1 c2 is satisfied in Aut(V ⊗ V ⊗ V ), but this is another way to write the Yang-Baxter equation. This proves the following. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

8

CHAPTER 1. THE YANG-BAXTER EQUATION

2.3. Proposition. — Let c ∈ Aut(V ⊗ V ) be a solution of the Yang-Baxter equation. Then, for any n > 0, there exists a unique homomorphism ρcn : Bn → Aut(V ⊗n ) such that ρcn (σi ) = ci for i = 1, . . . ,n − 1.

3. Alternative description of B n In Section 2 we gave an algebraic definition of the braid group. We now present Artin’s original geometric definition and explain the term “braid”. Consider the product Cn of n copies of the complex line. Inside it we define Yn as the subset of all n-tuples (z1 , . . . , zn ) of points of C such that zi = zj when i = j. We distinguish a point p = (1, . . . , n) in Yn . The symmetric group Sn acts on Yn by permutations of the coordinates. The quotient space Xn = Yn /Sn is the configuration space of n points in C. 3.1. Theorem. — The fundamental group π1 (Xn ,p) of the configuration space Xn is isomorphic to the braid group Bn . This theorem is due to E. Artin. We shall content ourselves with exhibiting a homomorphism from Bn to π1 (Xn , p). To the generator σi of Bn we assign the continuous map f = (f1 , . . . , fn ) : [0, 1] → Cn defined for s ∈ [0, 1] and all j by √

fi (s) = 12 2i + 1 − exp( −1 πs) , √

fi+1 (s) = 12 2i + 1 + exp( −1 πs) , fj (s) = j

if j = i, i + 1.

i It is easy to check that f is a loop at the point p in the configuration space Xn . Let σ be its class in π1 (Xn , p). The elements σ 1 , . . . , σ n−1 satisfy Relations (2.1) and (2.2). Thus, by definition of Bn , there exists a unique homomorphism Bn → π1 (Xn , p) sending σi to σ i for all i = 1, . . . , n−1. This homomorphism is in fact an isomorphism. For details, see [Bir74] or [BZ85]. We now wish to give a more familiar image of the braid group. Let f = (f1 , . . . , fn ) : [0, 1] −→ Yn ⊂ Cn be a continuous map representing an element of π1 (Xn , p), hence of Bn . Consider the subset Lf of C × [0, 1] n    Lf = (fi (s), s) | s ∈ [0, 1] . i=1

It is the disjoint union of n intervals continuously embedded in C × [0, 1]. We call it a braid with n strands. Note that (i) the boundary of Lf is the set {1, . . . , n} × {0, 1} and (ii) for all s ∈ [0, 1] the intersection of Lf with C × {s} consists of exactly n points. Conversely, any disjoint union of n intervals continuously embedded in C × [0, 1] such that Conditions (i) and (ii) hold is a subset Lf for some loop f of Xn . ` PANORAMAS ET SYNTHESES 5

3. ALTERNATIVE DESCRIPTION OF

Bn

9

It follows that there is an equivalence relation on braids with n strands such that Bn ∼ = π1 (Xn , p) is in bijection with the set of equivalence classes of braids with n strands. This equivalence is called isotopy. We shall encounter isotopy again in Section 1 of Chapter 5 when we introduce tangles, which generalize braids as well as knots. 3.2 Exercise. — Describe the group structure induced by Bn on the set of braids with n strands.

´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

Chapter 2 HOPF ALGEBRAS AND MONOIDAL CATEGORIES

The purpose of this chapter is to define a class of algebras carrying a solution of the Yang-Baxter equation on each of their modules. We need to introduce the concepts of Hopf algebras and of monoidal categories.

1. Hopf Algebras Let k be a field. Recall that an algebra is a triple (A, µ, η) where A is a vector space over k and µ : A ⊗ A → A and η : k → A are k-linear maps satisfying the conditions (i) (Associativity) µ(µ ⊗ idA ) = µ(idA ⊗µ), and (ii) (Unitality) µ(η ⊗ idA ) = µ(idA ⊗η) = idA . Here we identified k ⊗ A and A ⊗ k with A. The first condition expresses that the multiplication µ is associative whereas the second one implies that the element η(1) of A is a left and a right unit for µ. The algebra A is commutative if, in addition, µ = µop where µop = µτA,A and τA,A switches the factors: τA,A (a ⊗ a ) = a ⊗ a. 1.1. Definition. — A bialgebra is an algebra equipped with two algebra morphisms ∆: A → A ⊗ A and ε: A → k satisfying the following two conditions: (i) (Coassociativity) (1.1)

(∆ ⊗ idA )∆ = (idA ⊗∆)∆,

(ii) (Counitality) (1.2)

(ε ⊗ idA )∆ = (idA ⊗ε)∆ = idA .

If, furthermore, ∆op = ∆ where ∆op = τA,A ∆, we say that the bialgebra A is cocommutative. The map ∆ is called the comultiplication or the coproduct of the bialgebra A. The map ε is called the counit. Let A, A be bialgebras. A morphism of algebras f : A → A is a morphism of bialgebras if (1.3)

∆ f = (f ⊗ f )∆ and ε f = ε.

12

CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

1.2 Example (Dual bialgebra). — Let A = (A, µ, η, ∆, ε) be a bialgebra. Consider the dual vector space A∗ = Hom(A, k). By duality, ∆ and ε give rise to linear maps ∆∗

µ : A∗ ⊗ A∗ −−→ (A ⊗ A)∗ −−→ A∗ λ

and η  = ε∗ : k = k ∗ → A∗ where λ is the map determined by   λ(α ⊗ β), a ⊗ b = α, aβ, b for α, β ∈ A∗ and a, b ∈ A. Conditions (1.2)–(1.3) imply that µ is an associative product on the dual vector space A∗ with unit equal to ε. If A∗ is finite-dimensional, then the map λ : A∗ ⊗A∗ → (A⊗A)∗ is an isomorphism, which allows us to define the map ∆ = λ−1 µ∗ from A∗ to A∗ ⊗ A∗ . It is easy to check that (A∗ , µ , η  , ∆ , ε = η ∗ ) is a bialgebra. 1.3. Example (Opposite bialgebras). — For any bialgebra A = (A, µ, η, ∆, ε), we may define three other bialgebras, namely Aop = (A, µop , η, ∆, ε), Acop = (A, µ, η, ∆op , ε), and Aop, cop = (A, µop , η, ∆op , ε). 1.4. Example (Bialgebra of a group). — Let G be a group with unit e and let A = k[G] be the group algebra. We put a bialgebra structure on A by defining (1.4)

∆(x) = x ⊗ x

and ε(x) = 1

for all x ∈ G. The dual algebra A∗ is the algebra of k-valued functions on G, the unit being the constant function ε. If, furthermore, G is finite, we may endow it with a bialgebra structure, as explained above. Identifying A∗ ⊗ A∗ with the space of functions on the product G × G, we see that the comultiplication and the counit of A∗ are given by ∆(f )(g ⊗ h) = f (gh) and ε(f ) = f (e) for any function f on G. Elements of a bialgebra satisfying Equation (1.4) are called grouplike elements. Such elements form a monoid for the product. 1.5. Example. — Consider the tensor algebra T (V ) and the symmetric algebra S(V ) on a vector space V . There exists a unique bialgebra structure on T (V ) and on S(V ) such that (1.5)

∆(v) = 1 ⊗ v + v ⊗ 1 and ε(v) = 0

for any element v of V . This bialgebra is cocommutative and for all v1 , . . . , vn ∈ V we have n−1  vσ(1) · · · vσ(p) ⊗ vσ(p+1) · · · vσ(n) (1.6) ∆(v1 . . . vn ) = 1 ⊗ v1 · · · vn + p=1 σ + v1 · · · vn ⊗ 1 ` PANORAMAS ET SYNTHESES 5

1. HOPF ALGEBRAS

13

where σ runs over all (p, n − p)-shuffles of the symmetric group Sn , i.e., all permutations σ such that σ(1) < σ(2) < · · · < σ(p) and σ(p + 1) < σ(p + 2) < · · · < σ(n). Elements of a bialgebra satisfying Equation (1.5) are called primitive elements. The subspace Prim(A) of primitive elements of A is a Lie subalgebra of A, where A is equipped with the Lie bracket given by the commutators [a, a ] = aa − a a (for all a, a ∈ A). 1.6. Example. — The enveloping algebra U (g) of a Lie algebra g carries a bialgebra structure which is determined by the requirement that the canonical projection from the tensor algebra T (g) to U (g) is a morphism of bialgebras. Equivalently, the comultiplication ∆, which is given by (1.6), can be defined as the composition of the morphism from U (g) to U (g ⊕ g) induced by the diagonal map x → (x, x) on g, and of the canonical isomorphism U (g ⊕ g) ∼ = U (g) ⊗ U (g). 1.7. Exercises. (a) Show that the subspace of primitive elements of the enveloping bialgebra of a Lie algebra g coincides with g when the field k is of characteristic zero. (b) Let M (n) = k[x11 , . . . , xnn ] be a polynomial algebra in n2 variables xij with 1 ≤ i, j ≤ n. For all i, j, set ∆(xij ) =

n 

xik ⊗ xkj

and ε(xij ) = δij .

k=1

Check that these formulas define morphisms of algebras ∆ : M (n) → M (n) ⊗ M (n) and ε : M (n) → k equipping M (n) with a bialgebra structure. 1.8. Sweedler’s sigma notation. — Let A = (A, µ, η, ∆,  ε) be a bialgebra. For any x ∈ A, the element ∆(x) of A ⊗ A is of the form ∆(x) = i xi ⊗ xi . We formally rewrite this sum as  x(1) ⊗ x(2) . (1.7) ∆(x) = (x)

The fact that ∆ is a morphism of algebras implies that   (1.8) (xy)(1) ⊗ (xy)(2) = x(1) y(1) ⊗ x(2) y(2) . (xy)

(x)(y)

In this notation the coassociativity axiom (1.1) becomes       (x(1) )(1) ⊗ (x(1) )(2) ⊗ x(2) = x(1) ⊗ (x(2) )(1) ⊗ (x(2) )(2) , (1.9) (x)

(x(1) )

which we simply write reexpressed by (1.10)

(x)

 (x)

 (x)

(x(2) )

x(1) ⊗ x(2) ⊗ x(3) . The counitality axiom (1.2) can be

ε(x(1) )x(2) =



x(1) ε(x(2) ) = x

(x) ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

14

CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

for all x ∈ A. Finally, the bialgebra A is cocommutative if and only if 

(1.11)



x(1) ⊗ x(2) =

(x)

x(2) ⊗ x(1)

(x)

for all x ∈ A. 1.9. Convolution. — There is a natural operation, called the convolution, on the vector space End(A) of endomorphisms of a bialgebra A = (A, µ, η, ∆, ε). By definition, if f, g are endomorphisms, the convolution f 6 g is the composition of the maps f ⊗g

∆

µ

A −− → A ⊗ A −−−→ A ⊗ A −−→ A.

(1.12)

Using Sweedler’s sigma notation, we have (1.13)

(f 6 g)(x) =



f (x(1) )g(x(2) )

(x)

for any element x ∈ A. The convolution is clearly bilinear and by (1.10) we see that the endomorphism ηε is a left and right unit for the convolution. 1.10. Definition. — A Hopf algebra is a bialgebra in which the identity of A has a two-sided inverse S for the convolution. Such an inverse is called an antipode. Being a two-sided inverse, an antipode (if it exists) is necessarily unique. By definition, it satisfies the relation S 6 idA = idA 6S = ηε.

(1.14)

In Sweedler’s notation, this can be rewritten as 

(1.15)

x(1) S(x(2) ) = ε(x)1 =



(x)

S(x(1) )x(2)

(x)

for all x ∈ A. A morphism of Hopf algebras is a morphism between the underlying bialgebras commuting with the antipodes. The antipode S of a Hopf algebra A has the following properties. It is a bialgebra morphism from A to Aop,cop , i.e., we have S(1) = 1 and (1.16)

S(xy) = S(y)S(x),

ε(S(x)) = ε(x), and (1.17)

 (S(x))

` PANORAMAS ET SYNTHESES 5

S(x)(1) ⊗ S(x)(2) =

 (x)

S(x(2) ) ⊗ S(x(1) )

2. MONOIDAL CATEGORIES

15

for all x, y ∈ A. If, furthermore, A is cocommutative,, then the antipode is involutive: S 2 = idA .

(1.18)

For more details and for proofs, see Abe [Abe80] and Sweedler [Swe69]. 1.11. Examples. (a) Let A be a finite-dimensional Hopf algebra with antipode S. Then the dual bialgebra A∗ is a Hopf algebra with antipode S ∗ . (b) The group bialgebra k[G] of a group G is a Hopf algebra with antipode defined by S(x) = x−1 for x ∈ G. (c) The tensor bialgebra T (V ) is a Hopf algebra with an antipode determined by S(1) = 1 and for all v1 , v2 , . . . , vn ∈ V by S(v1 v2 · · · vn ) = (−1)n vn · · · v2 v1 .

(1.19)

Formula (1.19) also defines an antipode on the symmetric bialgebra S(V ) and on the enveloping bialgebra U (g).

2. Monoidal Categories 2.1. Modules over a bialgebra. — One of the important features of a bialgebra is that its category of modules forms a monoidal category. It is the aim of this section to explain this concept. Let A = (A, µ, η, ∆, ε) be a bialgebra. The tensor product U ⊗ V of two A-modules is an A ⊗ A-module by (2.1)

(a ⊗ a )(u ⊗ v) = au ⊗ a v,

where a, a ∈ A, u ∈ U and v ∈ V . The comultiplication ∆ allows us to equip U ⊗ V with an A-module structure by (2.2)

a(u ⊗ v) = ∆(a)(u ⊗ v) =



a(1) u ⊗ a(2) v.

(a)

The counit ε equips the ground field k with a trivial A-module structure by (2.3)

ax = ε(a)x

where a ∈ A and x ∈ k. For three A-modules U , V , W , consider the canonical isomorphisms (2.4)

(U ⊗ V ) ⊗ W ∼ = U ⊗ (V ⊗ W ),

(2.5)

k⊗V ∼ =V ∼ =V ⊗k ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

given respectively for u ∈ U , v ∈ V and w ∈ W by (u ⊗ v) ⊗ w → u ⊗ (v ⊗ w), 1 ⊗ v → v → v ⊗ 1. The reader may easily check that Conditions (1.1)–(1.2) defining a bialgebra imply that the isomorphisms (2.4)–(2.5) are A-linear, hence belong to the category A -Mod of left A-modules. Observe that the canonical isomorphism V ⊗ W ∼ = W ⊗ V given by the flip τV,W is, in general, not A-linear; nevertheless, it is A-linear if A is cocommutative. 2.2. Strict monoidal categories. — We now make a digression on monoidal categories. Let C be a category and ⊗ a functor from C × C to C. This means that (a) we have an object V ⊗ W associated to any pair (V, W ) of objects of the category, (b) we have a morphism f ⊗ g : V ⊗ W → V  ⊗ W  associated to any pair (f : V → V  , g : W → W  ) of morphisms of C, (c) we have (2.6)

(f  ⊗ g  ) ◦ (f ⊗ g) = (f  ◦ f ) ⊗ (g  ◦ g)

whenever the composition is defined, (d) and (2.7)

idV ⊗W = idV ⊗ idW .

A strict monoidal category is a category equipped with a functor ⊗ : C × C and an object 1 , called the unit object, such that (2.8)

(U ⊗ V ) ⊗ W = U ⊗ (V ⊗ W ),

(2.9)

(f ⊗ g) ⊗ h = f ⊗ (g ⊗ h),

(2.10)

V ⊗ 1 = V = 1 ⊗ V,

(2.11)

f ⊗ id1 = f = id1 ⊗f,

for all objects U , V , W and all morphisms f , g, h in C. 2.3. Example. — Let (G n )n∈N be a family of groups indexed by the natural integers such that G0 = {1}. We consider the category G whose objects are the non-negative integers and whose morphisms are given by  ∅ if m = n, HomG (m, n) = Gn if m = n, the composition on HomG (n, n) being given by the group product on Gn . The identity morphism on the object n is the unit of Gn . ` PANORAMAS ET SYNTHESES 5

2. MONOIDAL CATEGORIES

17

Suppose that, in addition, for any pair (m, n) there exists a homomorphism ρm,n : Gm × Gn → Gm+n such that (2.12)

ρm+n,p ◦ (ρm,n × idGp ) = ρm,n+p ◦ (idGm ×ρn,p )

for all m, n, p ∈ N. Then we can define a functor ⊗ : G × G → G on the objects by m ⊗ n = m + n, and on the morphisms by

f ⊗ g = ρm,n (f, g) ∈ Gm+n where f ∈ Gm and g ∈ Gn . Equations (2.6)–(2.7) are satisfied due to the fact that ρm,n is a homomorphism. Relations (2.8) and (2.10) hold trivially with 1 = 0. Relation (2.12) implies (2.9) and (2.11). This proves that G is a strict monoidal category. 2.4. The categories B and S. — We apply the construction of Example 2.3 to the family of braid groups (Bn )n and to the family of symmetric groups (Sn )n . We agree that B0 and S0 are trivial groups. We define homomorphism ρm,n : Bm × Bn → Bm+n on the generators σi of Section I.2 by

(2.13) ρm,n (σi , σj ) = σi σm+j for 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1. This operation on braids is well defined. Observe that ρm,n (f, g) is the braid obtained by placing the braid g to the right of the braid f so that there is no mutual linking or intersection. From this observation, it is clear that (2.12) holds. It is easy to check that ρm,n also induces a homomorphism Sm × Sn → Sm+n with the same properties. Applying the construction of 2.3, we get two strict monoidal categories, which we denote by B and S, respectively. 2.5. The category of vector spaces. — The categories we had in mind at the beginning of Section 2 were not those of Sections 2.3–2.4, but rather those coming up in linear algebra such as the category Vect(k) of vector spaces over a field k together with the usual tensor product ⊗ : Vect(k) × Vect(k) → Vect(k) of vector spaces and of linear maps. This category is not a strict monoidal category because the isomorphisms (2.4)– (2.5) are not identity morphisms. There is a way around this difficulty. It consists in replacing the equal signs in (2.8)–(2.11) by isomorphisms verifying certain conditions. We shall give details about these conditions in Section 2.1 of Chapter 9, where we shall define monoidal categories in full generality. The category Vect(k) is a monoidal category in this broader sense. For the time being, we shall pretend, as is common practice, that the isomorphisms (2.4)–(2.5) are identity morphisms. Actually, there is a theorem by Mac Lane asserting that any monoidal category is equivalent to a strict one. This allows us to ignore the general concept of a monoidal category until we reach the very last sections of this book. As a consequence of the observations made at the beginning of this section, the category A -Mod of left A-modules is a monoidal category with the same tensor product and the same isomorphisms as Vect(k). The unit object is the trivial Amodule k. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

18

CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

3. Braidings 3.1. Commutativity constraint. — We now define the categorical framework that produces solutions of the Yang-Baxter equation. A commutativity constraint c in a monoidal category C is a family of isomorphisms cV,W : V ⊗ W −→ W ⊗ V

(3.1)

defined for all couples (V, W ) of objects of C such that the square cV,W

V ⊗ W −−−−−→ W ⊗ V      g⊗f  f ⊗g  

(3.2)

cV  ,W 

V  ⊗ W  −−−−−→ W  ⊗ V  commutes for all morphisms f, g. 3.2. Definition. — Let C be a strict monoidal category. A braiding is a commutativity constraint in C satisfying the two relations (3.3)

cU⊗V ,W = (cU ,W ⊗ idV )(idU ⊗cV ,W ),

(3.4)

cU ,V ⊗W = (idV ⊗cU ,W )(cU ,V ⊗ idW )

for all objects U ,V ,W. Note that if c is a braiding in C, then so is the inverse c−1 . The concept of braiding can also be defined in a non-strict monoidal category (a precise definition will be given in Section 2.1 of Chapter 9). A braided monoidal category is a monoidal category with a braiding. We now state an important property of a braided monoidal category. It is the categorical version of the Yang-Baxter equation. 3.3. Theorem. — Let U ,V ,W be objects in a strict braided monoidal category. We have (3.5)

(cV ,W ⊗ idU )(idV ⊗cU ,W )(cU ,V ⊗ idW ) = (idW ⊗cU ,V )(cU ,W ⊗ idV )(idU ⊗cV ,W ).

This implies that the automorphism cV,V of V ⊗ V is a solution of the Yang-Baxter equation for any object V of a braided monoidal category. By Proposition 2.3 of Chapter 1, we get a representation of the braid group Bn in the automorphism group of V ⊗n for any object V and any positive integer n. Proof. — We have (cV,W ⊗ idU )(idV ⊗cU,W )(cU,V ⊗ idW ) = (cV,W ⊗ idU ) cU,V ⊗W = cU,W ⊗V (idU ⊗cV,W ) = (idW ⊗cU,V )(cU,W ⊗ idV )(idU ⊗cV,W ). ` PANORAMAS ET SYNTHESES 5

4. BRAIDED BIALGEBRAS

19

The first and last equalities follow from (3.4), the second one from (3.2) with f = idU and g = cV,W . 3.4. Example. — The flip τ of Example 1.2 of Chapter 1 is a braiding in the monoidal category Vect(k). It is also a braiding in the category k[G] -Mod of representations of a group G and, more generally, in the monoidal category A -Mod of modules over a cocommutative bialgebra A. 3.5. Exercises. (a) Show that in any strict braided monoidal category with unit 1 we have cV,11 = idV = c1,V for all objects V . (b) Prove that the strict monoidal category B of Section 2.4 has a braiding c uniquely determined by c1,1 = σ1 : 1 ⊗ 1 → 1 ⊗ 1 where σ1 is the generator of the braid group B2 . Determine cn,m in terms of the generators of Bn+m .

4. Braided Bialgebras We now characterize the bialgebras whose monoidal categories of modules are braided. By Theorem 3.3, this will produce solutions of the Yang-Baxter equation in a systematic way. Let A = (A, µ, η, ∆, ε) be a bialgebra and let A -Mod be the corresponding monoidal category (as defined in Section 2). 4.1. Theorem. — The monoidal category A -Mod is braided if and only if there exists an invertible element R of A ⊗ A such that (4.1)

∆op (x) = R ∆(x)R−1

(4.2)

(∆ ⊗ idA )(R) = R13 R23 ,

(4.3)

( idA ⊗∆)(R) = R13 R12 ,

for all x ∈ A,

where R12 = R ⊗ 1, R23 = 1 ⊗ R, and R13 = (τA,A ⊗ idA )(1 ⊗ R). Condition (4.1) means that A, though not necessarily cocommutative, is not far from being so. An element R satisfying Conditions (4.1)–(4.3) is called a universal R-matrix for A. A bialgebra A with a universal R-matrix is called a braided bialgebra. Drinfeld used the term “quasi-triangular bialgebra” instead of “braided bialgebra”. Observe that any cocommutative bialgebra is braided with the universal R-matrix R = 1 ⊗ 1. Proof. (a) Let A be a braided bialgebra with universal R-matrix R. For all couples (V, W ) of left A-modules we define a natural isomorphism cR V,W : V ⊗ W → W ⊗ V of Amodules by

 (4.4) cR ti w ⊗ s i v V,W (v ⊗ w) = τV,W R(v ⊗ w) = i ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

 for all v ∈ V and w ∈ W , where R = i si ⊗ ti . We claim that the family (cV,W )V,W is a braiding in the monoidal category A -Mod, as defined in Section 3. First, we have to prove that cV,W belongs to the category A -Mod, i.e., it is A-linear. This is a consequence of (4.1), as shows the following computation for all x ∈ A.

cR V,W (x(v ⊗ w)) = τV,W R∆(x)(v ⊗ w)

= τV,W ∆op (x)R(v ⊗ w)

= ∆(x)τV,W R(v ⊗ w)

= x cR V,W (v ⊗ w) . Next, we observe that cV,W is an isomorphism with inverse given by (4.5)

−1 (w ⊗ v) = R−1 (v ⊗ w). (cR V,W )

Finally, an easy computation shows that Conditions (4.2) and (4.3) imply Relations (3.3) and (3.4), respectively. (b) Conversely, suppose there exists a braiding c in the monoidal category A -Mod. Define an element R in A ⊗ A by (4.6)



R = τA,A cA,A (1 ⊗ 1) .

Let us show that R is a universal R-matrix for A. If v, w are elements of A-modules V , W , Condition (3.2) implies that v ⊗ w) ¯ = (w¯ ⊗ v¯) cA,A cV,W (¯ where v¯ : A → V and w ¯ : A → W are the A-linear maps uniquely defined by v¯(1) = v and w(1) ¯ = w. Hence,





v ⊗ w)(R) ¯ = τV,W R(v ⊗ w) . ¯ ⊗ v¯) cA,A (1 ⊗ 1) = τV,W (¯ (4.7) cV,W (v ⊗ w) = (w By A-linearity of cA,A we have

x cA,A (1 ⊗ 1) = cA,A x(1 ⊗ 1) for all x ∈ A. By (4.7) we get ∆(x)τA,A (R) = τA,A (R ∆(x)), which is equivalent to (4.1). Finally, Relations (3.3) and (3.4) imply (4.2) and (4.3). By Theorem 3.3, we conclude that for any A-module V the automorphism cR V,V of V ⊗ V is a solution of the Yang-Baxter equation. This efficient way of producing Rmatrices on all A-modules explains why the element R is called the universal R-matrix of A. Observe that, if R = 1 ⊗ 1, then cR V,W = τV,W is the flip of Example I.1.2. Let us give a few properties of universal R-matrices. ` PANORAMAS ET SYNTHESES 5

4. BRAIDED BIALGEBRAS

21

4.2. Proposition. — Let A be a braided bialgebra with universal R-matrix R. We have (4.8)

R12 R13 R23 = R23 R13 R12 ,

(4.9)

(ε ⊗ idA )(R) = 1 = (idA ⊗ε)(R).

If, moreover, A has an antipode S, then (4.10)

R−1 = (S ⊗ idA )(R).

Proof. (a) Relations (4.1)–(4.2) imply that R12 R13 R23 = R12 (∆ ⊗ id)(R) = (∆op ⊗ id)(R)R12 = (τA,A ⊗ id)(∆ ⊗ id)(R)R12 = (τA,A ⊗ id)(R13 R23 )R12 = R23 R13 R12 . From (ε ⊗ id)∆ = id and from (4.2), we get R = (ε ⊗ id ⊗ id)(∆ ⊗ id)(R) = (ε ⊗ id ⊗ id)(R13 R23 ) = (ε ⊗ id)(R)ε(1)R. Since ε(1) = 1 and R is invertible, we obtain (ε ⊗ id)(R) = 1. Similarly, we may use the relation (id ⊗ε)∆ = id and (4.3) to derive (id ⊗ε)(R) = 1. (b) By (1.15) an antipode S verifies µ(S ⊗ id)∆(x) = ε(x)1 for all x ∈ A. This implies that (µ ⊗ id)(S ⊗ id ⊗ id)(∆ ⊗ id)(R) = (ε ⊗ id)(R) = 1 by (4.9). From this and from (4.2) we get 1 = (µ ⊗ id)(S ⊗ id ⊗ id)(R13 R23 ) = (S ⊗ id)(R) S(1)R. Since S(1) = 1, we have (S ⊗ id)(R) = R−1 . 4.3. The square of the antipode. — As we observed in Section 1, the antipode S of a cocommutative Hopf algebra is an involution. In the braided case, S 2 is in general not equal to the identity. Nevertheless, it is an inner automorphism. Indeed, let (A, µ, η, ∆, ε, S, R) be a braided Hopf algebra with an invertible antipode S and a universal R-matrix R. Consider the element u of A given by (4.11)

u=



S(ti )si

i ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

22

where R =

CHAPTER 2. HOPF ALGEBRAS AND MONOIDAL CATEGORIES

 i

si ⊗ ti . Then u is invertible in A with inverse u−1 =

(4.12)



S −1 (ti ) S(si ),

i

and the square of the antipode is given by S 2 (x) = uxu−1

(4.13) for all x ∈ A. Moreover, we have (4.14)

∆(u) = (R21 R)−1 (u ⊗ u) = (u ⊗ u)(R21 R)−1

and (4.15)

uS(u) = S(u)u.

The element uS(u) belongs to the centre of A. A proof of these properties can be found in [Dri89a] (see also [Kas95, VIII.4]). 4.4. Exercises. (a) Let A be a braided bialgebra with universal R-matrix R. Show that the braiding R cR given by Formula (4.4) satisfies the relation cR W,V cV,W = idV ⊗W for all modules V , W if and only if R−1 = τA,A (R). (b) Let G be a finite abelian group and A be the bialgebra dual of the group bialgebra k[G]. Let (δg )g∈G be the basis of A defined by δg (g) = 1 and δg (h) = 0 if h is an element of G different from g. For any function γ : G × G → k \ {0}, set R=



γ(a, b) δa ⊗ δb .

a,b∈G

Check that R is a universal R-matrix for A if and only if γ is bimultiplicative, i.e., we have γ(aa , b) = γ(a, b)γ(a , b) and γ(a, bb ) = γ(a, b)γ(a, b ) for all a, a , b, b, ∈ G. (c) Let A be the algebra generated by two elements x, y, subject to the relations x2 = 1, y 2 = 0, and yx + xy = 0. Check that there exists a unique Hopf algebra structure on A such that ∆(x) = x ⊗ x,

ε(x) = 1, S(x) = x,

∆(y) = 1 ⊗ y + y ⊗ x,

ε(y) = 0,

S(y) = xy.

Set



Rλ = 12 1 ⊗ 1 + 1 ⊗ x + x ⊗ 1 − x ⊗ x + 12 λ y ⊗ y + y ⊗ xy + xy ⊗ y − xy ⊗ xy where λ is a scalar. Show that Rλ is a universal R-matrix for A. Prove that the corresponding braiding is involutive. ` PANORAMAS ET SYNTHESES 5

Chapter 3 DRINFELD’S QUANTUM DOUBLE

The purpose of this chapter is to describe a general method yielding braided Hopf algebras. This method, due to V. Drinfeld [Dri87], associates to any (finite-dimensional) Hopf algebra A a braided Hopf algebra D(A) called the quantum double of A. We start with a dual construction, which is slightly easier.

1. The Dual Double Construction Let (A, µ, η, ∆, ε, S) be a Hopf algebra with invertible antipode S over a field k. It is easy to see (cf. Example 1.2 of Chapter 2) that (A, µ, η, ∆op , ε, S −1 ) is also a Hopf algebra. We may ask, more generally, whether the algebra (A, µ, η) admit other comultiplications. Motivated by the notion of a braided Hopf algebra, we may try to construct comultiplications in (A, µ, η) as follows. Choose an invertible element F in A ⊗ A and set ∆F (a) = F ∆(a)F −1 ∈ A ⊗ A for all a ∈ A. Clearly, ∆F is an algebra homomorphism; we look for F such that ∆F is coassociative and the augmentation ε is a counit. If these conditions are satisfied, we say that ∆F is a twisted coproduct. 1.1. Lemma. (i) A sufficient condition for ∆F to be coassociative is F12 · (∆ ⊗ id)(F ) = F23 · (id ⊗∆)(F ). (ii) A sufficient condition for ε to be a counit with respect to ∆F is (id ⊗ε)(F ) = (ε ⊗ id)(F ) = 1. Proof. (i) We want





(∆F ⊗ id)∆F (a) = (id ⊗∆F )∆F (a)

for all a ∈ A, i.e., (∆F ⊗ id)(F ∆(a)F −1 ) = (id ⊗∆F )(F ∆(a)F −1 ).

24

CHAPTER 3. DRINFELD’S QUANTUM DOUBLE

This is equivalent to the equality −1 (∆F ⊗ id)(F ) · F12 · (∆ ⊗ id)(∆(a)) · F12 · (∆F ⊗ id)(F −1 ) −1 · (id ⊗∆F )(F −1 ). = (id ⊗∆F )(F ) · F23 · (id ⊗∆)(∆(a)) · F23

So, in view of the coassociativity of ∆, a sufficient condition is given by (∆F ⊗ id)(F )F12 = (id ⊗∆F )(F )F23 or, equivalently, F12 (∆ ⊗ id)(F ) = F23 (id ⊗∆)(F ). (ii) Easy, left to the reader. 1.2. Remark. — If we ask, furthermore, that (∆ ⊗ id)(F ) = F13 F23

and (id ⊗∆)(F ) = F13 F12 ,

then Condition (i) above is nothing but the Yang-Baxter equation for F . 1.3. — We shall point out a simple way to construct an element F satisfying the conditions of Lemma 1.1. Start with a finite-dimensional Hopf algebra A with invertible antipode. Consider the dual Hopf algebra A∗ and the Hopf algebra  = A∗ ⊗ Aop Aop = (A, µop , η, ∆, ε, S −1 ) (cf. Example 1.2 of Chapter 2). Define A ∗ op as the tensor product of the Hopf algebras A and A . Let (ei ) be a basis of A and (e∗i ) be the dual basis in A∗ . The following element of ⊗A  is canonical, i.e., independent of the choice of the basis (ei ): A F =



(1A∗ ⊗ ei ) ⊗ (e∗i ⊗ 1A ).

⊗ A  to twist the coalgebra The next lemma will allow us to use the element F −1 ∈ A  structure in A. 1.4. Lemma. — The element F is invertible and F −1 satisfies the conditions of Lemma 1.1. Proof. — Note that under the canonical identification of Aop ⊗ A∗ with End(Aop ), the op ∗ tensor product of the algebra structures on becomes the convolution product  A ⊗A op ∗ on End(A ). Furthermore, the element ei ⊗ ei is identified with 1 ∈ End(Aop ) and the fact that it is invertible in Aop ⊗ A∗ simply means that Aop has an invertible antipode S −1 . Using this for the middle terms in the formula defining F, one gets that F is invertible with the inverse F−1 = ` PANORAMAS ET SYNTHESES 5



(1A∗ ⊗ S −1 ei ) ⊗ (e∗i ⊗ 1A ).

1. THE DUAL DOUBLE CONSTRUCTION

25

Since the product (resp. the coproduct) in A∗ is the transpose of the coproduct (resp. the product) in A, we have the identities (∆ ⊗ id)(F ) = F13 F23

and (id ⊗∆)(F) = F13 F12 .

It is clear that F12 and F23 commute. Therefore, F13 F23 F12 = F13 F12 F23 , i.e., (∆ ⊗ id)(F ) · F12 = (id ⊗∆)(F) · F23 . Taking the inverse, we obtain Condition (i) of Lemma 1.1. Condition (ii) of Lemma 1.1 concerning the counit ε is easily checked if we take 1A to be one of the basis elements and the remaining ones in Ker ε. 1.5. Theorem (see [RS88]). — Let A be a finite-dimensional Hopf algebra with invertible antipode. Give H = A∗ ⊗ Aop the following structure: (i) As an algebra, H is the tensor product of the algebras A∗ and Aop . (ii) The coproduct ∆F : H → H ⊗ H is given by ∆F (x) = F ∆(x)F −1 where x ∈ H, the map ∆ is the tensor product of the coproducts in A∗ and in Aop , and  F = (1A∗ ⊗ S −1 ei ) ⊗ (e∗i ⊗ 1A ), where (ei ) is a basis of A. (iii) The augmentation of H is the tensor product of the augmentations of A∗ and Aop . Then H is a Hopf algebra with invertible antipode given by

−1 SH ( ⊗ a) = f SA∗ () ⊗ SA (a) f −1 ,  where f = i e∗i ⊗ ei ,  ∈ A∗ , and a ∈ A. Proof. — The only assertion which does not directly follow from Lemmas 1.1 and 1.4 is that SH is indeed the antipode, i.e., that for all  ∈ A∗ and a ∈ A we have



µ ◦ (SH ⊗ id) ∆F ( ⊗ a) = µ ◦ (id ⊗SH ) ∆F ( ⊗ a) = ε()ε(a). Note that ∆F ( ⊗ a) = F



 ((1) ⊗ a(1) ) ⊗ ((2) ⊗ a(2) ) F −1 ,

where we use Sweedler’s convention of Section 3.8 of Chapter 2. We have  ∆F ( ⊗ a) = ((1) ⊗ S −1 ei a(1) ej ) ⊗ (e∗i (2) e∗j ⊗ a(2) ), 

e∗k S(1) e∗r ⊗ ek S −1 ej S −1 a(1) S −2 ei S −1 er ⊗ (e∗i (2) e∗j ⊗ a(2) ), 

e∗k S(1) e∗r e∗i (2) e∗j ⊗ ek S −1 ej S −1 a(1) S −2 ei S −1 er a(2) . µ(SH ⊗ id) ∆F ( ⊗ a) = (SH ⊗ id)∆F ( ⊗ a) =

´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 3. DRINFELD’S QUANTUM DOUBLE

Observe that 

  e∗r e∗i ⊗ S −2 ei S −1 er = (id ⊗S −1 ) e∗r e∗i ⊗ er S −1 ei = (id ⊗S −1 )(1A∗ ⊗ 1A ) = 1 A∗ ⊗ 1 A = 1 H .

Therefore, 

e∗k S(1) (2) e∗j ⊗ ek S −1 ej S −1 a(1) a(2) µ(SH ⊗ id) ∆F ( ⊗ a) =  = ε()ε(a) e∗k e∗j ⊗ ek S −1 ej = ε()ε(a) 1H .

The relation µ ◦ (id ⊗SH ) ∆F ( ⊗ a) = ε()ε(a) is proved in a similar way.

2. The Quantum Double and its Properties The dual Hopf algebra of the Hopf algebra H constructed in Theorem 1.5 is precisely Drinfeld’s quantum double, as described in the following theorem. 2.1. Theorem. — The Hopf algebra structure of the dual Hopf algebra H ∗ = A ⊗ A∗, cop can be described as follows. (i) As a coalgebra, it is the tensor product of the coalgebras A and A∗, cop . (ii) Via the natural inclusions A → H ∗ and A∗, cop → H ∗ given by a → a ⊗ 1 and  → 1 ⊗  respectively, A and A∗, cop become Hopf subalgebras of H ∗ . (iii) For all  ∈ A∗ and a ∈ A, we have (a ⊗ 1)(1 ⊗ ) = a ⊗  and (1 ⊗ )(a ⊗ 1) =



(1) ,S −1 a(1) (3) ,a(3)  a(2) ⊗ (2) ,

where  ,  denotes the natural pairing between A∗ and A. The Hopf algebra H ∗ is called the quantum double of A and is denoted by D(A). As we shall see in Theorem 2.2, the Hopf algebra D(A) is braided in a canonical way. Proof. (i) This part is clear. (ii) Let a, a ∈ A. We have   (a ⊗ 1) · (a ⊗ 1),  ⊗ b = (a ⊗ 1) ⊗ (a ⊗ 1), ∆F ( ⊗ b)    = (a ⊗ 1) ⊗ (a ⊗ 1), ((1) ⊗ S −1 ei b(1) ej ) ⊗ (e∗i (2) e∗j ⊗ b(2) ) = a, (1)  ε(ei )ε(b(1) )ε(ej ) a , e∗i (2) e∗j  ε(b(2) ). ` PANORAMAS ET SYNTHESES 5

2. THE QUANTUM DOUBLE AND ITS PROPERTIES

27

We choose the basis (ei ) of A so that ε(ek ) = 0 if k = 0, and ε(e0 ) = 1. Then the result is ε(b) a, (1)  a , (2)  = aa ⊗ 1,  ⊗ b. It follows that A is a subalgebra of H ∗ . In a similar way, we prove that A∗,cop is a subalgebra of H ∗ . (iii) For a, b ∈ A and , k ∈ A∗ , we have 

   (a ⊗ 1)(1 ⊗ ), k ⊗ b = (a ⊗ 1) ⊗ (1 ⊗ ), ∆F (k ⊗ b)    = (a ⊗ 1) ⊗ (1 ⊗ ), (k(1) ⊗ S −1 ei b(1) ej ) ⊗ (e∗i k(2) e∗j ⊗ b(2) ) = a, k(1)  ε(ei )ε(b(1) )ε(ej )ε(e∗i )ε(k(2) )ε(e∗j ) , b(2)  = a ⊗ , k ⊗ b.

This proves the first relation. As for the second one, we compute (1 ⊗ )(a ⊗ 1), k ⊗ b    = (1 ⊗ ) ⊗ (a ⊗ 1), (k(1) ⊗ S −1 ei b(1) ej ) ⊗ (e∗i k(2) e∗j ⊗ b(2) )  = ε(k(1) ) , S −1 ei b(1) ej  a, e∗i k(2) e∗j  ε(b(2) )  = , S −1 ei bej  a, e∗i ke∗j   = (1) , S −1 ei  (2) , b (3) , ej  a(1) , e∗i  a(2) , k a(3) , e∗j     S −1 (1) ⊗ a(1) , ei ⊗ e∗i (3) ⊗ a(3) , ej ⊗ e∗j (2) , b a(2) , k. = But, for all m ∈ A∗ and c ∈ A, we have 

m ⊗ c, ei ⊗ e∗i  = m, c

i

(to see this write c as





c, e∗i ei ). Thus, we are left with

S −1 (1) , a(1)  (3) , a(3)  (2) ⊗ a(2) , b ⊗ k



and the result follows. 2.2. Theorem. — Let R = with

 i

(ei ⊗ 1) ⊗ (1 ⊗ e∗i ) ∈ D(A) ⊗ D(A). Then R is invertible

R−1 =



(Sei ⊗ 1) ⊗ (1 ⊗ e∗i ),

i

and satisfies Relations (4.1)– (4.3) of Chapter 2. Therefore, R is a universal R-matrix for D(A). ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 3. DRINFELD’S QUANTUM DOUBLE

Proof. — The invertibility of R and the formula for R−1 are proved as in Lemma 1.4. Relations (4.2)–(4.3) of Chapter 2 follow from the fact that the coproduct in A (resp. in A∗,cop ) is the transpose (resp. the opposite of the transpose) of the product in A∗,cop (resp. in A). Let us check Relation (4.1) of Chapter 2. Clearly, the subspace   x ∈ D(A) | R ∆(x) = ∆op (x)R is a subalgebra of D(A). So we have only to check Relation (4.1) for x = a ⊗ 1 with a ∈ A and for x = 1 ⊗  with  ∈ A∗ . We shall use the following lemma due to Radford [Rad93].  2.3. Lemma. — Let H be a Hopf algebra, and R = i αi ⊗ βi ∈ H ⊗ H be an invertible element. The identity R ∆(a) = ∆op (a)R for all a ∈ H is equivalent to any of the following conditions holding for all a ∈ H: R (a ⊗ 1) =

(i)

(a ⊗ 1) R =

(ii) (iii)

R (1 ⊗ a) =

(iv)

(1 ⊗ a) R =

   

a(2) αi ⊗ a(1) βi Sa(3) , αi a(2) ⊗ Sa(1) βi a(3) , a(3) αi S −1 a(1) ⊗ a(2) βi , S −1 a(3) αi a(1) ⊗ βi a(2) .

Proof. — We prove that the identity R ∆(a) = ∆op (a)R is equivalent to (i), the other equivalences being proved in a similar fashion. If (i) holds, then R ∆(a) = R



  (a(2) αi ⊗ a(1) βi Sa(3) ) · (1 ⊗ a(4) ) (a(1) ⊗ 1) · (1 ⊗ a(2) ) =  = a(2) αi ⊗ a(1) βi = ∆op (a)R.

Let us prove the converse implication. Recall that a ⊗ 1 = (id ⊗ε)∆(a). We have

 (R ⊗ 1) (∆ ⊗ id)∆(a) = R ∆(a(1) ) ⊗ a(2)   = ∆op (a(1) ) ⊗ a(2) (R ⊗ 1). Hence,



R(a(1) ⊗ a(2) ) ⊗ a(3) =



a(2) αi ⊗ a(1) βi ⊗ a(3) .

We now apply id ⊗ id ⊗S to both sides of the equality and multiply the last two terms. We get R(a ⊗ 1) = a(1) αi ⊗ a(1) βi Sa(3) , which is (i). ` PANORAMAS ET SYNTHESES 5

2. THE QUANTUM DOUBLE AND ITS PROPERTIES

29

We can now finish the proof of Theorem 2.2. It suffices to check Condition (iii) of Lemma 2.3 for elements a ⊗ 1 with a ∈ A and Condition (ii) of the same lemma for elements 1 ⊗  with  ∈ A∗ . The left-hand side of Condition (iii) is equal to



R (1 ⊗ 1) ⊗ (a ⊗ 1) = (ei ⊗ 1) ⊗ (1 ⊗ e∗i )(a ⊗ 1)  = (ei ⊗ 1) ⊗ (a(2) ⊗ e∗i(2) ) e∗(1) , S −1 a(1)  e∗i(3) , a(3) . The right-hand side is equal to 

(a(3) ei S −1 a(1) ) ⊗ 1 ⊗ (a(2) ⊗ e∗i ), and one computes a(3) ei S −1 a(1) = =

 

e∗j , a(3) ei S −1 a(1)  ej e∗j(3) , a(3)  e∗j(1) , S −1 a(1)  e∗j(2) , ei  ej .

So, the right-hand side is equal to 

(ej ⊗ 1) ⊗ (a(2) ⊗ e∗i ) e∗j(3) , a(3)  e∗j(1) , S −1 a(1)  e∗j(2) , ei 

and we only have to observe that



∗ ∗ i ej(2) , ei  ei

= e∗j(2) to conclude.

In the same way, the left-hand side of Condition (ii) of Lemma 2.3 is 



(1 ⊗ )(ei ⊗ 1) ⊗ (1 ⊗ e∗i ) (1 ⊗ ) ⊗ (1 ⊗ 1) R =  = (1) , S −1 e(1)  (3) , ei(3)  (ei(2) ⊗ (2) ) ⊗ (1 ⊗ e∗i ), and the right-hand side of (ii) is 

(ei ⊗ (2) ) ⊗ (1 ⊗ S −1 (1) e∗i (3) ).

We compute S −1 (1) e∗i (3) = =

 

S −1 (1) e∗i (3) , ej  e∗j S −1 (1) , ej(1)  e∗i , ej(2)  e∗i , ej(2)  (3) , ej(3)  e∗j .

So the right-hand side is 

(ei ⊗ (2) ) ⊗ (1 ⊗ e∗j ) S −1 (1) , ej(1)  (3) , j(3)  e∗i , ej(2) .

We conclude by observing that

 i

ei e∗i , ej(2)  = ej(2) . ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 3. DRINFELD’S QUANTUM DOUBLE

2.4. Remark. — One can prove that the algebra structure in D(A) is the only Hopf algebra structure on  the coalgebra A ⊗ A∗,cop such that A and A∗,cop are Hopf subalgebras and R = i (ei ⊗ 1) ⊗ (1 ⊗ e∗i ) is a universal R-matrix. 2.5. Example. — Let G be a finite group and A = F (G) be the algebra of complexvalued functions on G. Then A∗,cop is the group algebra C[G]. One checks that the double D(F (G)) is isomorphic, as an algebra, to the crossed product algebra of F (G) by G where G acts on F (G) by the adjoint action.

3. Hopf Pairings and a Generalized Double We may observe that the construction of D(A) uses only the fact that we have a pairing between A and A∗,cop satisfying a number of appropriate conditions. This suggests the following definition. 3.1. Definition. — Let A and B be Hopf algebras with invertible antipodes over a field k. A Hopf pairing between A and B is a bilinear form ϕ: A × B → k such that (i) for all a ∈ A, for all b,b ∈ B, ϕ(a,bb ) =



ϕ(a(1) ,b) ϕ(a(2) ,b ),

(ii) for all a,a ∈ A, for all b ∈ B, ϕ(aa ,b) =



ϕ(a,b(2) ) ϕ(a ,b(1) ),

(iii) for all a ∈ A and for all b ∈ B, ϕ(a,1B ) = ε(a)

and

ϕ(1A ,b) = ε(b),

(iv) for all a ∈ A and for all b ∈ B, ϕ(Sa,b) = ϕ(a,S −1 b). We do not require ϕ to be non-degenerate. If ϕ is degenerate, we can derive from it a non-degenerate Hopf pairing by quotienting out the annihilator ideals. More precisely,   IA = a ∈ A | ϕ(a, b) = 0 for all b ∈ B ,   IB = b ∈ B | ϕ(a, b) = 0 for all a ∈ A . It is easy to check that IA and IB are Hopf ideals in A and B, respectively, and that the Hopf pairing ϕ¯ : A/IA × B/IB → k obtained by passing to the quotients is non-degenerate. The next theorem generalises Theorem 2.1. ` PANORAMAS ET SYNTHESES 5

3. HOPF PAIRINGS AND A GENERALIZED DOUBLE

31

3.2. Theorem. — Let A and B be Hopf algebras over a field k and ϕ: A × B → k be a Hopf pairing between them. Then there is a unique Hopf algebra structure on A ⊗ B satisfying Conditions (i), (ii), and (iii) of Theorem 2.1, where B replaces A∗, cop and ϕ replaces the duality bracket. The Hopf algebra provided by Theorem 3.2 is called the generalized double of A with respect to B and ϕ. It is denoted by Dϕ (A, B), or D(A, B) when ϕ is clear. Generally speaking, the Hopf algebra Dϕ (A, B) is not braided. However, if A and B are finite-dimensional and ϕ is non-degenerate, then the same construction as in Theorem 2.2 makes Dϕ (A, B) into a braided Hopf algebra. More generally, assume that A and B are graded Hopf algebras with finite-dimensional homogenous components and that ϕ is compatible with the gradings. Then the quotient Hopf algebras A/IA and B/IB are also graded and can be identified via ϕ with the duals of each other. In this case the Hopf algebra Dϕ¯ (A/IA , B/IB ) is braided by the same construction as in Theorem 2.2. 3.3. — Most Hopf algebras we shall be considering in the sequel will be defined by generators and relations. The following lemma provides us with a method of constructing Hopf pairings in this setting (see [Dae93]).  (resp. B)  be a free algebra generated by elements a1 , . . . ,ap (resp. 3.4. Lemma. — Let A  and B  have Hopf algebra structures such b1 , . . . ,bq ) over a field k. Suppose that A that each ∆(ai ) for 1 ≤ i ≤ p (resp. ∆(bj ) for 1 ≤ j ≤ q) is a linear combination of tensors ar ⊗ as (resp. of tensors br ⊗ bs ). Given pq scalars λij ∈ k with 1 ≤ i ≤ p and ×B  → k such that ϕ(ai ,bj ) = λij . 1 ≤ j ≤ q, there is a unique Hopf pairing ϕ: A Proof. — We have to define ϕ(ai1 . . . air , bjs . . . bjs ) for all i1 , . . . ir ∈ {1, . . . , p} and j1 , . . . , js ∈ {1, . . . , q}. Observe first that Axioms (i) and (ii) of Definition 3.1 and the assumptions on ∆(ai ) or ∆(bj ) allow us to define ϕ(ai1 · · · air , b) = where ∆(r) (b) =





ϕ(ai1 , b(r) ) · · · ϕ(air , b(1) ),

b(1) ⊗ · · · ⊗ b(r) is the iterated coproduct, and

ϕ(a, bj1 · · · bjs ) = where ∆(s) (a) =





ϕ(a(1) , bj1 ) · · · ϕ(a(s) , bjs ),

a(1) ⊗ · · · ⊗ a(s) . We then define

ϕ(ai1 · · · air , bj1 · · · bjs ) =



ϕ(ai1 , bj1 (r) · · · bjs (r) ) · · · ϕ(air , bj1 (1) · · · bjs (1) ),

expanding the right-hand side by the previous remarks. It is not difficult to check  and B.  that this gives a well-defined Hopf pairing between A  (resp. B)  Suppose now that A (resp. B) is the algebra obtained as the quotient of A   by the ideal generated by elements r1 , . . . , rn ∈ A (resp. s1 , . . . , sm ∈ B). Suppose also ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 3. DRINFELD’S QUANTUM DOUBLE

 (resp. B)  induces a Hopf algebra structure in A that the Hopf algebra structure in A   (resp. in B). Then a Hopf pairing ϕ : A × B → k induces a Hopf pairing A × B → k if and only if ϕ(ri , bj ) = 0 for all i = 1, . . . , n and j = 1, . . . , q, and ϕ(ai , sj ) = 0 for all i = 1, . . . , p and j = 1, . . . , m. 3.5. — A Hopf algebra obtained by the generalized double construction carries a canonical bilinear form which is invariant under the adjoint action. To describe this form, recall first the definition of the adjoint action of a Hopf algebra H. The (left) adjoint representation ad of H onto itself is defined by ad(x)(y) =



x(1) y S(x(2) )

for x, y ∈ H. This generalizes the usual notions of a group acting on itself by conjugation, and of the adjoint action of a Lie algebra. Another action may be obtained by using S −1 instead of S. One can also define two right adjoint actions. Using the comultiplication in H, we can extend the adjoint action to H ⊗ H. By duality, we obtain the adjoint action of H onto the bilinear forms on H. Given a representation ρ of a Hopf algebra H in a vector space V , an element v ∈ V is said to be ρ-invariant if ρ(h)v = ε(h)v for all h ∈ V . In particular, we may speak of bilinear forms on H invariant under the adjoint action, or, briefly, of ad-invariant forms. It is easy to check that a bilinear form K on H is ad-invariant if and only if it satisfies one of the following two equivalent conditions holding for all h, h , h ∈ H: (i)



K ad(h(1) )h , ad(h(2) )h = ε(h)K(h , h ),

(ii)





K ad(h)h , h = K h , ad(S(h))h .

3.6. Example. — Let H be a braided Hopf algebra and let u ∈ H be such that S 2 (x) = u x u−1

for all x ∈ H.

If ρ : H → End(V ) is a finite-dimensional representation, then the following formula defines an ad-invariant bilinear form on H:

K(x, y) = Tr ρ(u)ρ(x)ρ(y) where Tr is the usual trace. In the framework of Hopf algebras, ad-invariant bilinear forms are the natural generalization of the Killing form on a Lie algebra. So, it is important to know whether there exists such a form and, if so, whether it is degenerate or not. For generalized doubles we give the following simple answer. ` PANORAMAS ET SYNTHESES 5

3. HOPF PAIRINGS AND A GENERALIZED DOUBLE

33

3.7. Theorem. — Let A and B be Hopf algebras over a field k and ϕ: A × B → k be a Hopf pairing. Let Dϕ (A,B) be the corresponding generalized double. Then the following formula defines an ad-invariant bilinear form K on D ϕ (A,B):

K(a ⊗ b,a ⊗ b ) = ϕ(a,b ) ϕ S −2 (a ),b , where a,a ∈ A and b,b ∈ B. Moreover, the form K is non-degenerate if and only if ϕ is non-degenerate. This theorem will be used in the next chapter to construct ad-invariant bilinear forms on the quantized enveloping algebras.

´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

Chapter 4 THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N+1)

In this chapter we construct the quantum groups associated to the Lie algebras sl(N + 1), using the generalized double of Section 3 of Chapter 3.

1. The Lie Algebra sl(N + 1) 1.1. — Let sl(N + 1) be the Lie algebra of complex (N + 1) × (N + 1)-matrices with trace 0. Let Eij be the elementary matrix whose entries are all 0, except the entry in position (i, j) which is equal to 1. It is well known that the elements Eij with i = j and Eii − Ei+1,i+1 form a basis of sl(N + 1). For 1 ≤ i ≤ N , set Ei = Ei,i+1 ,

Fi = Ei+1,i ,

Hi = Eii − Ei+1,i+1 .

Let h denote the (commutative) Lie subalgebra generated by H1 , . . . , HN . Consider the linear forms ε1 , . . . , εN +1 on the space of (N + 1) × (N + 1)-matrices defined by  δik if k = , εi (Ek ) = 0 if k = . Then the linear forms α1 = ε1 − ε2 , α2 = ε2 − ε3 , . . . , αN = εN − εN +1 form a basis of h∗ , and for all H ∈ h and all 1 ≤ i < j ≤ N + 1 we have [H, Eij ] = (αi + αi+1 + · · · + αj−1 )(H) Eij , [H, Eji ] = −(αi + αi+1 + · · · + αj−1 )(H) Eji . The linear forms αi + αi+1 + · · · + αj with 1 ≤ i < j ≤ N are called the positive roots of sl(N + 1) and their set is denoted by ∆+ . The set of roots of sl(N + 1) is ∆ = ∆+ ∪ (−∆+ ).

36

CHAPTER 4. THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N + 1)

We may observe that ∆+ (resp. −∆+ ) parametrizes exactly the basis elements Eij (resp. Eji ) with 1 ≤ i < j ≤ N + 1; if α = αi + · · · + αj−1 , then the matrix Eij (resp. Eji ) will be denoted by Eα (resp. by E−α , or equivalently, by Fα ). 1.2. — It is well known (see [Ser87]) that sl(N + 1) is generated by Ei , Fi , Hi where 1 ≤ i ≤ N , subject to the relations [Hi , Hj ] = 0

for i, j = 1, . . . , N ;

[Hi , Ej ] = αj (Hi )Ej ,

for 1 ≤ i, j ≤ N ;

[Hi , Fj ] = −αj (Hi )Fj

for 1 ≤ i, j ≤ N ;

[Ei , Fj ] = δij Hi

for 1 ≤ i, j ≤ N ;

[Ei , Ej ] = 0,   Ei , [Ei , Ej ] = 0,

[Fi , Fj ] = 0   Fi , [Fi , Fj ] = 0

if |i − j| ≥ 2; if |i − j| = 1.

The same generators and relations define the enveloping algebra U sl(N + 1). We observe that the adjoint action of h on U sl(N + 1) makes it into a Q-graded algebra, where Q is the free abelian group with basis (α1 , . . . , αN ). The degree of Ei is αi , the degree of Fi is −αi , and the degree of Hi is 0. In the sequel, Q will often be denoted multiplicatively, in particular, when we consider its group algebra U0 = C[Q]. 1.3. — By the Poincar´e-Birkhoff-Witt theorem the subalgebras U b+ and U b− of U sl(N + 1) respectively generated by Ei , Hi and Fi , Hi for 1 ≤ i ≤ N , are dual to each other as Q-graded linear spaces. However, this duality is not a Hopf pairing because both algebras are cocommutative but not commutative. The constructions of the next section may be seen as a way to circumvent this fact.

2. Construction of Uq sl(N + 1) In the sequel, q is an indeterminate and the ground field is the field of fractions C(q). + × U  − → C(q). 2.1. A Hopf pairing U + be the C(q)-algebra generated by Ei , K ±1 where 1 ≤ i ≤ N , subject to • Let U i the relations Ki Kj = Kj Ki ,

Ki Ki−1 = Ki−1 Ki = 1,

Ki Ej = q αj (Hi ) Ej Ki

for 1 ≤ i, j ≤ N . − be the C(q)-algebra generated by Fi , K ±1 where 1 ≤ i ≤ N , subject to • Let U i the relations Ki Kj = Kj Ki , for 1 ≤ i, j ≤ N . ` PANORAMAS ET SYNTHESES 5

Ki−1 Ki = Ki Ki−1 = 1,

Ki Fj = q −αj (Hi ) Fj Ki

2. CONSTRUCTION OF Uq sl(N + 1)

37

+ and U − have a Hopf algebra structure given by The algebras U ∆Ki = Ki ⊗ Ki ,

∆Ei = Ei ⊗ 1 + Ki ⊗ Ei ,

∆Ki = Ki ⊗ Ki ,

∆Fi = Fi ⊗ Ki−1 + 1 ⊗ Fi .

They are Q-graded with Ei of degree αi , Fi of degree −αi , and Ki and Ki of degree 0. + × U − → C(q) such that for 2.2. Theorem. — There exists a unique Hopf pairing ϕ: U 1 ≤ i,j ≤ N δij , q − q −1

(i)

ϕ(Ei ,Fj ) = −

(ii)

ϕ(Ei ,Kj ) = ϕ(Ki ,Fj ) = 0,

(iii)

ϕ(Ki ,Kj ) = q −αi (Hj ) = q −αj (Hi ) .

Proof. — It follows from the results of Section 3.3 of Chapter 3. 2.3. Remarks. (a) The choice of the scalar −1/(q − q −1 ) in Theorem 2.2 (i) gives the right normalization for the commutation relations appearing in the literature. It also yields the correct relation when one specializes at q = 1. (b) The pairing ϕ is Q-graded. + ⊗ U − a Hopf algebra By Theorem 3.2 of Chapter 3 the pairing ϕ induces on U  structure denoted D(U+ ). We deduce the following fact from Theorem 2.1 (iii) of Chapter 3. + ), for 1 ≤ i,j ≤ N we have 2.4. Proposition. — In D( U Ki Fj = q −αj (Hi ) Fj Ki , [Ei ,Fj ] = δij

Ki Ej = q αj (Hi ) Ej Ki , Ki − Ki−1 · q − q −1

The Hopf pairing ϕ is degenerate, as shown by the following lemma. 2.5. Lemma. (a) The elements Ei Ej − Ej Ei

when |i − j| ≥ 2 and

Ei2 Ej − (q + q −1 )Ei Ej Ei + Ej Ei2

when |i − j| = 1

+ generated by these are in the annihilator ideal IU . The two-sided ideal I+ of U + elements is a Hopf subideal of IU . +

+ replaced by Fi and U − . (b) A similar statement holds with Ei and U ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 4. THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N + 1)

Actually, we shall see in Corollary 3.10 that I+ = IU and I− = IU . + − Proof of (a). — One first proves that, if x = Ei Ej − Ej Ei with |i − j| ≥ 2, then ∆x = x ⊗ 1 + Ki Kj ⊗ x, and that, if y = Ei2 Ej − (q + q −1 )Ei Ej Ei + Ej Ei2 with |i − j| = 1, then ∆y = y ⊗ 1 + Ki2 Kj ⊗ y. − . Consequently, these elements are orthogonal to all decomposable elements of U  Next, we observe that the pairing is graded and that all elements of U− of degree −(αi + αj ) in the first case or of degree −(2αi + αj ) in the second case are decomposable. It follows that the elements x and y above lie in IU . The formulas for + ∆x and ∆y show, in addition, that I+ is a Hopf ideal. Put + /I+ U+ = U

− /I− . and U− = U

These are Hopf algebras and ϕ induces a Hopf pairing between them, still denoted by ϕ. According to Theorem 3.2 of Chapter 3, U+ ⊗U− has a Hopf algebra structure given by the generalized double construction. This Hopf algebra will be denoted by D(U+ ). 2.6. Definition. — The Hopf algebra Uq sl(N + 1) is the quotient of D(U+ ) by the two-sided ideal generated by the elements Ki − Ki , 1 ≤ i ≤ N . A presentation of Uq sl(N + 1) by generators and relations is obtained as follows. The generators are Ei , Fi , Ki±1 with 1 ≤ i ≤ N . The relations consist of those defining + and U − (but replacing K  by Ki ), those of Proposition 2.4 (after replacing K  U i i by Ki ), and the following relations: Ei Ej − Ej Ei = 0  Fi Fj − Fj Fi = 0 Ei2 Ej − (q + q −1 )Ei Ej Ei + Ej Ei2 = 0  Fi2 Fj − (q + q −1 )Fi Fj Fi + Fj Fi2 = 0

if |i − j| ≥ 2,

if |i − j| = 1.

We still denote by U+ and U− the Hopf subalgebras of Uq sl(N + 1) generated by Ei , Ki±1 for 1 ≤ i ≤ N , and Fi , Ki±1 for 1 ≤ i ≤ N , respectively. Let U0 be the Hopf subalgebra of Uq sl(N + 1) generated by Ki±1 with 1 ≤ i ≤ N , and V+ , V− the subalgebras generated by the elements E1 , . . . , EN , and by F1 , . . . , FN , respectively. (These are not Hopf subalgebras!) ` PANORAMAS ET SYNTHESES 5

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39

2.7. Lemma. (a) The Hopf algebra U0 is isomorphic to the group algebra C(q)[ZN ]. (b) There are linear isomorphisms U+ ∼ = V+ ⊗ U0 ,

U− ∼ = V− ⊗ U0 ,

Uq sl(N + 1) ∼ = V− ⊗ U0 ⊗ V+ .

3. A Poincar´e-Birkhoff-Witt-Type Basis in U+ In view of Lemma 2.7 and of the fact that the algebras V+ and V− are isomorphic, it suffices to construct a basis of V+ to get a basis of Uq sl(N + 1). Recall that the classical theorem of Poincar´e-Birkhoff-Witt uses a basis of sl(N +1), for instance, the basis consisting of the vectors Hi for 1 ≤ i ≤ N and Eα for α ∈ ∆. Observe that, if α ∈ ∆+ is given by α = αi + αi+1 + · · · + αj , then, putting β = αi+1 + · · · + αj , one has Eα = [Ei , Eβ ]. One constructs analogues in V+ of the root vectors Eα (α ∈ ∆+ ), using the adjoint action introduced in Section 3.4 of Chapter 3. 3.1. Definition. — Let α = αi + · · · + αj ∈ ∆+ with i ≤ j. The element Eα ∈ V+ is defined inductively on the length of α by Eαi = Ei if 1 ≤ i ≤ N , and by Eα = ad Ei (Eβ ) where β = αi+1 + · · · + αj . One can determine the commutation relations between these elements. 3.2.. Proposition. — For α = α i + · · · + αj and β = αk + · · · + α with 1 ≤ i ≤ j ≤ N and 1 ≤ k ≤  ≤ N we have (i)

Eα Eβ = Eβ Eα −1

if k > j + 1,

(ii)

Eα Eβ − q

(iii)

Eα Eβ − q −1 Eβ Eα = 0

if i = k,

(iv)

Eα Eβ − qEβ Eα = 0

if j = ,

(v)

Eα Eβ = Eβ Eα

(vi)

Eβ Eα = Eα+β

Eα Eβ − Eβ Eα = (q − q

if k = j + 1,

if i < k <  < j , −1

)Eγ Eγ 

if i < k < j < ,

where γ = αi + · · · + α and γ  = αk + · · · + αj . Let us consider the following lexicographic linear ordering on ∆+ : α1 > α1 + α2 > α1 + α2 + α3 > · · · > α1 + α2 + · · · + αN > α2 > α2 + α3 > · · · > α2 + · · · + αN > α3 > · · · > α3 + · · · + αN > ··· αN −1 > αN −1 + αN > αN . ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 4. THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N + 1)

Observe that, writing roots as differences (εi −εj ) this is nothing but the lexicographic ordering. To simplify notation, we write β1 = αN < β2 = αN −1 + αN < · · · < βs = α1 where s = card(∆+ ). From the commutation relations of Proposition 3.2, we deduce the following. 3.3. Proposition. — The ordered monomials Eβm11 Eβm22 · · · Eβmss where (m1 , . . . ,ms ) ∈ Ns span the vector space V+ . In order to prove that these ordered monomials actually form a basis of V+ , and to investigate the Hopf pairing, we shall need the following q-multinomial formula. 3.4. Lemma. — Let u 1 , . . . ,ur be elements in an algebra such that uk u = q 2 u uk for all k > . Then for n ≥ 1 (u1 + u2 + · · · + ur )n =



n1 +···+nr =n n1 ,...,nr ≥0

(n)! un1 · · · unr r (n1 )! . . . (nr )! 1

where (0)! = 1 and, for n ≥ 1, (n) =

qn − 1 , q−1

(n)! = (n)(n − 1) · · · (2)(1).

The following is proved by induction on the lengths of the roots. 3.5. Proposition. (a) For α = αi + · · · + αj with 1 ≤ i < j ≤ N we have ∆Eα = Eα ⊗ 1 + (1 − q −2 )



Eαi +···+αk Kk+1 · · · Kj ⊗ Eαk+1 +···+αj

i≤k≤j−1

+ Ki Ki+1 · · · Kj ⊗ Eα .

(b) For α as above and n ∈ N, ∆Eαn =



n1 +···+nj−i+2

(n)! nj−i+2 (1 − q −2 )n2 +···+nj−i+1 un1 1 un2 2 · · · uj−i+2 (n1 )!...(nj−i+2 )!

where u1 = Eα ⊗ 1, uj−i+2 = Ki · · · Kj ⊗ Eα , and for 1 < k < j − i + 2 uk = Eαi +···+αj−k+1 Kj−k+2 · · · Kj ⊗ Eαj−k+2 +···+αj . The relations of Proposition 3.2 have the following consequence. ` PANORAMAS ET SYNTHESES 5

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41

3.6. Proposition. — Let (V + )j be the linear subspace of V+ spanned by the ordered m monomials Eβm11 Eβm22 · · · Eβjj where (m1 , . . . ,mj ) ∈ Nj . Then (i) (V+ )j is a subalgebra of V+ and (ii) (V+ )j is a left U+ -subcomodule of V+ for the left coaction induced by the coproduct on U+ , that is, ∆(V+ )j ⊂ U+ ⊗ (V+ )j . 3.7. The Hopf pairing restricted to V + × V− . — We define root vectors Fα for α ∈ ∆+ in V− , proceeding as in the case of V+ , but using the adjoint action ad associated with the opposite comultiplication on U− and given by ad (x)(y) =



x(2) yS −1 (x(1) ) for all x, y ∈ U− .

For α = αi + · · · + αj in ∆+ , we define Fα by induction by Fαi = Fi and by Fα = ad Fi (Fβ ) where β = αi+1 + · · · + αj . Using Proposition 3.6 and computing ϕ(Eβi , Fβi ), we get the following. 3.8.. Proposition. — Let K ,K  ∈ U0 , and (m1 , . . . ,ms ) and (m1 , . . . ,ms ) ∈ Ns . Then m

m

ϕ(KEβm11 · · · Eβmss ,K  Fβ1 1 · · · Fβs s ) =

s " ! 1   (mi )! ϕ(K ,K ) δ m ,m i i (q −1 − q)M k=1

where M =

s 

mk (βk ) and (β) = j − i + 1 for β = αi + · · · + αj .

k=1

3.9. Corollary. — The ordered monomials Eβm11 · · · Eβmss (resp. Fβm1 1 · · · Fβms s ) with (m1 , . . . ,ms ) ∈ Ns form a basis of V+ (resp. of V− ). Proof. — The ordered monomials Fβm1 1 · · · Fβms s span V− . This is proved using a version of Proposition 3.2 for the root vectors Fα . Proposition 3.8 shows that (Eβm11 · · · Eβmss )β∈Ns is in duality with a generating set, which implies that it is linearly independent. Reversing the argument, we see that (Fβm1 1 · · · Fβms s )β∈Ns is linearly independent as well. 3.10. Corollary. — The Hopf pairing ϕ: U + × U− → C(q) is non-degenerate. In particular, IU = I± . ±

Since Uq sl(N + 1) is a quotient of a generalized double, we can apply Theorem 3.7 1 of Chapter 3. To this end it is convenient to extend the ground field C(q) to C(q 2 ). This leads to the following. 3.11. Theorem. — There is a unique ad-invariant bilinear form B on U q sl(N + 1) such that ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 4. THE QUANTIZED ENVELOPING ALGEBRA Uq sl(N + 1)

(i) for all E ,E  ∈ V+ , all F ,F  ∈ V− , and all K ,K  ∈ U0 , B(EKF ,E  K  F  ) = B(E ,F  )B(K ,K  )B(S −2 (E  ),F ); (ii) for all E ∈ V+ and F ∈ V− , B(E ,F ) = ϕ(E ,F ), where ϕ is the Hopf pairing between V+ and V− ;   (iii) for α = i ni αi and β = j mj αj in the root lattice, B(Kα ,Kβ ) = q − 2 Σi,j ni mj αi (Hj ) , 1

where Kα =

# i

Kini and Kβ =

# j

m

Kj j . Moreover, the form B is non-degenerate.

The form B was first constructed in [Ros90] by another method. For the interpretation in terms of the Hopf pairings, see [Ros92].

4. Specializations and the Universal R-Matrix Let A be the subring C[q, q −1 ] of C(q). For 1 ≤ i ≤ N , set [Ki ;0] =

Ki − Ki−1 ∈ Uq sl(N + 1). q − q −1

Let UA be the A-subalgebra of Uq sl(N + 1) generated by the elements Ei , Fi , Ki±1 , [Ki ;0] with 1 ≤ i ≤ N . It is easy to see that it is a Hopf A-subalgebra of Uq sl(N + 1). For ε ∈ C∗ , we consider the “specialization” Uε = UA /(q − ε)UA which is a Hopf algebra over C. For ε = ±1, a presentation of Uε by generators and relations is obtained from that of Uq sl(N + 1) by putting q = ε in the relations. The following result shows that the Hopf algebra U1 is closely connected to the enveloping algebra U sl(N + 1). 4.1. Proposition. — The algebra U 1 is generated by Ei , Fi , Ki , and Hi (1 ≤ i ≤ N ), subject to the condition that all elements Ki are central and the relations [Hi ,Ej ] = αj (Hi ) Ki Ej ,

[Hi ,Fj ] = −αj (Hi ) Ki Fj ,

[Ei ,Fj ] = δij Hi , Ki2 = 1,     Ei , . . . [Ei ,Ej ] = Fi , . . . [Fi ,Fj ] = 0 for i = j. %& ' $ %& ' $ 1−αi (Hj )

In particular, U1 /

N 

1−αi (Hj )

(Ki − 1) is isomorphic to U sl(N + 1).

i=1

` PANORAMAS ET SYNTHESES 5

4. SPECIALIZATIONS AND THE UNIVERSAL R-MATRIX

43

Let ε ∈ C∗ such that ε = ±1. One can construct elements Eα , Fα ∈ Uε which are the specializations of the elements of Uq sl(N + 1) denoted by the same symbols. Clearly, we have the same formula as in Proposition 3.8 for the Hopf pairing. One concludes that if ε is not a root of unity, then the ordered monomials provide a basis and the Hopf pairing is non-degenerate. In case when ε is a primitive root of unity, say of order , one immediately sees that Eα , Fα , Ki − 1 are in the kernel of the Hopf pairing. In fact, they are in the center of Uε and the two-sided ideal they generate is a Hopf coideal. Let uε be the quotient of Uε by this ideal. This is a finite-dimensional Hopf algebra, for which one − 0 can define Hopf subalgebras u+ ε , uε , uε in an obvious way. − From the remarks above, the Hopf pairing descends to u+ ε × uε . From Proposition 3.8 we deduce the following result (see [Ros92]). 4.2. Proposition. — Let ε be a primitive -th root of unity of order  prime to N + 1. − Then the Hopf pairing between u+ ε and uε is non-degenerate. 4.3. The universal R-matrix. — The quantum double construction of Section 2 of Chapter 3 allows us to write down a formula for the universal R-matrix. In the case of Uq sl(N + 1) or Uε with ε not a root of unity one has to introduce a completion involving infinite sums, cf. [Ros89], [LS90], [KR90], and Section 4 of Chapter 7. The next theorem treats the case where ε is a root of unity. 4.4. Theorem (see [Ros93]). — Let ε be a primitive -th root of unity with  odd and prime to N + 1. Then the finite-dimensional Hopf algebra uε is braided with universal R-matrix R=

1 N

−1 " !  (1 − ε2α )k − k(k−1)/2 k ε Eα ⊗ Fαk [k]! +

α∈∆

k=0



ε(β ,γ) Kβ ⊗ Kγ

"

β ,γ∈Q

where the product is taken in the order defined after Proposition 3.2, Q =

N (

) mi αi ;0 ≤ mi <  ,

i=1 mN for β = Kβ = K1m1 · · · KN

[k] =

N 

mi αi , and where

i=1

εk − ε−k , ε − ε−1

[k]! = [k][k − 1] · · · [2][1].

´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

Chapter 5 THE JONES POLYNOMIAL AND SKEIN CATEGORIES

In this chapter we enter the world of low-dimensional topology. Our aim is twofold. First, we define a remarkable isotopy invariant of links in Euclidean 3-space, the Jones polynomial. The discovery of this polynomial in 1984 gave the original impetus to the theory presented in this and the next chapters. Secondly, we explain how the knotted topological objects in Euclidean 3-space, the so-called tangles, give rise to braided monoidal categories. It was historically one of the first constructions of braided categories. In a more elaborate form (not discussed here), it provides a geometric construction of the representation categories of quantum groups. Here is the plan of the chapter. In Section 1 we define knots and links in Euclidean 3-space, as well as link diagrams and the Reidemeister moves. In Section 2 we use the technique of link diagrams to define the Kauffman bracket polynomial of framed links and the Jones polynomial of oriented links. In Section 3 we introduce tangles and define skein modules of tangles. In Section 4 we define the categories of tangles and provide them with braiding.

1. Knots, Links, and Link Diagrams 1.1. Framed links. — A link in Euclidean 3-space is a finite collection of disjoint circles smoothly embedded in R3 . These circles are called the components of the link. Unless explicitly stated to the contrary, the link components are not assumed to be parametrized or oriented. If a link L ⊂ R3 has only one component it is called a knot. A few examples of knots and links are given in Figure 1.1.

Figure 1.1.

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CHAPTER 5. THE JONES POLYNOMIAL AND SKEIN CATEGORIES

An isotopy of a link L ⊂ R3 is a smooth deformation of L in the class of links in R3 . Such a deformation slowly moves the components of L in R3 without creating intersections or self-intersections. Two links L1 , L2 ⊂ R3 are said to be isotopic if there is an isotopy transforming L1 into L2 . It is clear that isotopy is an equivalence relation in the class of links in R3 . By abuse of language, the isotopy classes of links are also called links. The main goal of knot theory is to classify knots and links up to isotopy. To this end, topologists construct various isotopy invariants of knots and links. In the next section, we shall define one such invariant, the Jones polynomial of links. We need to consider normal vector fields on links in R3 . A link L ⊂ R3 provided with a non-singular normal vector field is said to be framed. Two vector fields on L homotopic in the class of non-singular normal vector fields determine the same framing of L. In other words, a framing of L is a homotopy class of non-singular normal vector fields on L. Note that the homotopy class of a non-singular normal vector field on a knot K ⊂ R3 is completely determined by an integer, called the rotation number or the framing number. It is defined as follows: orient the circle K in an arbitrary way, consider the oriented circle K  obtained by pushing K along the given vector field into R3 \ K, and take the linking number of K and K  . (A definition of the linking number is recalled in Section 1.4.) Therefore, in order to specify a framing of a link it suffices to assign an integer to each component. By isotopy of framed links, we mean an isotopy of the underlying links accompanied by a homotopy of the normal vector fields. (Such an isotopy preserves the framing numbers.) By abuse of language, the isotopy classes of framed links are also called framed links. 1.2. Link diagrams. — To present knots and links in R 3 one usually uses the pictures as in Figure 1.1. Such pictures are called link diagrams. Here we give a formal definition of link diagrams. Consider the projection of a link L ⊂ R3 into the horizontal plane R2 = R2 × 0. Deforming L slightly in R3 , we may assume that its projection is generic, i.e., consists of immersed loops with only double transversal crossings. Over each such crossing lie exactly two points of L. One of these two points lie higher than the second one which allows to provide the crossing with an additional information: one of the two intersecting branches is “higher” and the second one is “lower”, cf. Figure 1.1. These branches are called overcrossing and undercrossing, respectively. A generic system of loops in R2 provided with the over/undercrossing data in each crossing point is a link diagram. The construction above transforms every generic projection of a link L ⊂ R3 into a link diagram. Looking at this diagram we can reconstruct L, at least up to isotopy. It is clear that a diagram of a link L may be regarded simply as a picture of L. For instance, the diagrams in Figure 1.1 present the trefoil, the figure eight knot, the Hopf link, and the Borromean link, respectively. The link presented by the diagram consisting of m disjoint embedded circles is called the trivial m-component link. A more careful analysis of definitions shows that every link diagram in R2 actually ` PANORAMAS ET SYNTHESES 5

1. KNOTS, LINKS, AND LINK DIAGRAMS

47

presents a framed link. We should simply regard this diagram as a picture of a link L ⊂ R3 lying very close to R2 and provide L with a vertical vector field orthogonal to R2 and directed “up”, i.e., towards the reader. The corresponding framing numbers of the components may be computed as follows. Orient all components and assign ±1 to each crossing point in the way indicated in Figure 1.2. For each component, sum up the signs associated to its self-crossings. This gives the framing number in question.

--1

+1 Figure 1.2.

The technique of link diagrams is sufficiently general: any framed link L is isotopic to a framed link presented by a link diagram. To see this, we first deform L so that its projection in R2 is generic. Providing this projection with the over/undercrossing data as above we get a diagram of L, although the framing of L determined by this diagram may differ from the given framing of L. The remedy is to change the diagram by introducing in every component a certain number of small curls as in Figure 1.3. This does not change the isotopy type of the link but changes its framing. It is easy to compute that the first (resp. second) transformation in Figure 1.3 increases (resp. decreases) the framing number of the corresponding component by 1. Therefore, applying these transformations we can obtain a link diagram presenting L with the given framing.

−→

−→

(a)

(b) Figure 1.3.

It is convenient to consider the empty subspace of R3 as a framed link with 0 components. This link is presented by an empty diagram. 1.3. Reidemeister moves. — A natural question is when two link diagrams present isotopic framed links. It is clear that if two diagrams are ambient isotopic in the plane R2 then the corresponding framed links are isotopic in R3 . (An ambient isotopy of a link diagram is induced by an isotopy of the plane). In Figure 1.4 we depict three local transformations of diagrams which preserve the isotopy type of the associated framed link. These transformations are denoted −1 −1 by Ω0 , Ω2 , Ω3 . These and the inverse transformations Ω−1 0 , Ω2 , Ω3 are called Reidemeister moves. By definition, each Reidemeister move changes a link diagram inside ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 5. THE JONES POLYNOMIAL AND SKEIN CATEGORIES

a small disk keeping intact the part of the diagram outside this disk.

− → Ω0

− →

− →

Ω2

Ω3

Figure 1.4. 1.4. Theorem. — Two link diagrams in R 2 present isotopic framed links in R3 if and only if they may be related by an ambient isotopy and a finite sequence of moves ±1 ±1 Ω±1 0 ,Ω2 ,Ω3 . This theorem shows that to define invariants of framed links, it is enough to define invariants of link diagrams preserved under ambient isotopy and the moves ±1 ±1 Ω±1 0 , Ω2 , Ω3 . We shall use this approach in Section 2 to define the Jones polynomial. To treat non-framed links in a similar fashion, we need another move on link diagrams, specifically the first transformation in Figure 1.3. It is denoted by Ω1 . A theorem, due to K. Reidemeister, affirms that two link diagrams in R2 present isotopic links in R3 if and only if they may be related by an ambient isotopy and a ±1 ±1 finite sequence of moves Ω±1 1 , Ω2 , Ω3 (see [BZ85, Chapter 1]). It is not difficult to deduce Theorem 1.4 from this classical fact. To conclude this section, we recall the definition of the linking number of oriented (disjoint) knots K, K  ⊂ R3 . Let us present the link K ∪ K  by a diagram and sum the numbers ±1 associated to the crossing points where the loop representing K crosses the loop representing K  from above (cf. Figure 1.2). It is easy to see that the resulting integer is preserved under the Reidemeister moves and therefore yields an isotopy invariant of the link K ∪ K  . This integer is called the linking number of K, K  and denoted by lk(K, K  ). 1.5. Exercises. (i) Compute the framing numbers for the framed links presented by the diagrams in Figure 1.1. Orient the components of the Hopf link and compute their linking number. The same for the Borromean link. (ii) Check that lk(K, K  ) = lk(K  , K).

2. The Jones Polynomial of Links 2.1. Skein classes of diagrams. — Fix a non-zero complex number a. Let E(a) be the complex vector space generated by all link diagrams quotiented by (i) ambient isotopy in the plane; (ii) the relation D ∪ O = −(a2 + a−2 )D, where D is an arbitrary link diagram and O is a simple closed curve in R2 bounding a disk in the complement of D; (iii) the identity in Figure 2.1. ` PANORAMAS ET SYNTHESES 5

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49

The formula in Figure 2.1 is Kauffman’s skein relation. This relation involves three link diagrams identical except in a small 2-disk where they are as shown.

=a

+ a−1 Figure 2.1.

Two diagrams on the right-hand side are obtained by smoothing a crossing of the diagram on the left-hand side. These two smoothed diagrams acquire coefficients a and a−1 . In order to determine which coefficient corresponds to which diagram we use the following rule. Let us move towards the crossing point along the upper branch. Just before reaching the crossing point turn to the left (in the clockwise direction) and move until attaining the lower branch, then move along this lower branch away from the crossing. In this way we get one of the two smoothings. It acquires the coefficient a and the second possible smoothing acquires the coefficient a−1 . We call E(a) the skein module corresponding to a. Every link diagram D represents an element of E(a) denoted by D(a) or simply by D and called the skein class of D. For instance, if D consists of m disjoint embedded circles then by (ii), D(a) = (−1)m (a2 + a−2 )m ∅

(2.1)

where ∅ is an empty link diagram. The following theorem shows that E(a) is a onedimensional vector space generated by ∅. 2.2. Theorem. — For any a ∈ C, we have dim E(a) = 1. Proof. — Let D be a link diagram. Applying the skein relation to all crossing points of D we may expand D as a linear combination of diagrams without crossing points. This expansion does not depend on the order in which we apply the skein relation to the crossing points of D. Applying formula (2.1) to the latter diagrams, we obtain D = c∅ where c = c(D, a) ∈ C. The number c(D, a) is determined by D and a uniquely and therefore E(a) is a one-dimensional vector space generated by ∅. 2.3. Theorem. — The skein class of any link diagram is invariant under the moves ±1 ±1 Ω±1 0 ,Ω2 ,Ω3 . Proof. — Let D be a link diagram obtained from a link diagram D by inserting a small positive curl, as in Figure 1.3.a. The computations in Figure 2.2 show that . D  = −a3 D. The symbol = in the figures denotes equality of the corresponding skein classes.

. a =

+ a−1

. (−a3 − a−1 + a−1 ) =

. − a3 =

Figure 2.2. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 5. THE JONES POLYNOMIAL AND SKEIN CATEGORIES

Similarly, if D is a link diagram obtained from D by inserting a small negative curl, as in Figure 1.3.b, then D  = −a−3 D. This implies the invariance of D under Ω0 . The invariance of D under the move Ω2 is verified in Figure 2.3. Here we apply the Kauffman relation twice: first in the lower crossing and then in the higher crossing. We also use the result of the previous paragraph concerning positive curls.

. a =

+ a−1

. a2 =

+

− a2

. =

Figure 2.3. In Figure 2.4 we check the invariance of D under the move Ω3 . The second and third equalities in Figure 2.4 follow from the invariance of D under Ω2 .

. a =

+ a−1

. a =

+ a−1

. a =

+ a−1

. =

Figure 2.4. 2.4. The bracket polynomial. — We can use Theorem 2.3 to define for every non-empty framed link L ⊂ R3 a Laurent polynomial L(a) in one variable a. This Laurent polynomial has integer coefficients and depends only on the isotopy type of L. It is called the bracket polynomial of L. The polynomial L(a) is defined as follows. Present L by a diagram D and take a non-zero complex number a such that a2 + a−2 = 0. Theorems 1.4 and 2.3 imply that the skein class D(a) ∈ E(a) is an isotopy invariant of L independent of the choice of D. We identify E(a) = C∅ with C via ∅ = 1 ∈ C. It follows from Kauffman’s recursive formula that the function D(a) ∈ E(a) = C is a Laurent polynomial in a divisible by a2 + a−2 . Set L(a) = −(a2 + a−2 )−1 D(a) ∈ C. The bracket polynomial L is normalized so that its value for the trivial knot with framing 0 is equal to 1. More generally, the value of the bracket polynomial for an m-component link with zero framing of the components is equal to (−a2 − a−2 )m−1 . In Figures 2.5 and 2.6 we compute the bracket polynomial of the Hopf link (with zero framing numbers) and the trefoil with framing +3. These computations show, in ` PANORAMAS ET SYNTHESES 5

2. THE JONES POLYNOMIAL OF LINKS

51

particular, that the trefoil is not isotopic to the trivial knot.

. a =

+ a−1

. = a(−a3 ) + a−1 (−a−3 ) = −a4 − a−4 Figure 2.5.

. a =

+ a−1

. = a(−a4 − a−4 ) + a−1 (−a−3 )2 = −a5 − a−3 + a−7 Figure 2.6. The bracket polynomial of a framed link does depend on the choice of the framing. Twisting the framing n times around a component leads to multiplication of the bracket polynomial by (−a3 )n . 2.5. The Jones polynomial. — The Jones polynomial is a version of the bracket polynomial for oriented (non-framed) links in R3 . A link is said to be oriented if all its components are oriented. Let L ⊂ R3 be an oriented link with components L1 , . . . , Lm where m ≥ 1. Provide every component Li with the framing −



lk(Li , Lj )

j =i

where lk is the linking number. Denote the resulting framed link by Lf . (For instance, if L is a knot then Lf is L with framing 0.) It is easy to see that Lf (a) is a Laurent polynomial in a2 . To define the Jones polynomial VL (q) of L we substitute a = q −1/2 in Lf (a). Thus, VL (q) = Lf (q −1/2 ). This is a Laurent polynomial in q. The polynomial VL (q) may be computed from any diagram D of L as follows. The orientation of L determines signs ±1 of the crossing points of D as in Figure 1.2. Let w(D) ∈ Z be the sum of these signs over all crossing points of D. (The number w(D) is called the writhe of D.) Let |D| be the number of crossing points of D. Then (2.2)

VL (q) = (−1)|D|+1 q 3w(D)/2

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The Jones polynomial VL (q) admits the following useful characterization. We say that three oriented links L+ , L− , L0 form a Conway triple if they are identical except within a ball where they are as shown in Figure 2.7.

L+

L−

L0

Figure 2.7. 2.6. Theorem. — There exists a unique function V : {non-empty oriented links in R3 } −→ Z[q,q −1 ] such that (i) if L is isotopic to L , then V (L) = V (L ), (ii) V (a trivial knot) = 1, (iii) for any Conway triple L+ ,L− ,L0 , q −2 V (L+ ) − q 2 V (L− ) = (q − q −1 )V (L0 ). Sketch of proof. — The existence is now easy: the Jones polynomial obviously satisfies (i) and (ii); it is a pleasant exercise for the reader to deduce (iii) from Kauffman’s skein relation for link diagrams and formula (2.2). The uniqueness of the function V is also straightforward: one uses induction on the number of crossing points of a diagram and the fact that any link diagram may be transformed into a diagram of a trivial link by changing certain overcrossings into undercrossings. 2.7. Exercises. (i) If L is the mirror image of an oriented link L ⊂ R3 then VL (q) = VL (q −1 ). Deduce that the trefoil is not isotopic to its mirror image. (ii) If an oriented link L ⊂ R3 is a disjoint union of oriented links L1 and L2 then VL (q) = (−q − q −1 ) VL1 (q) VL2 (q). (iii) Verify that the Jones polynomial of the figure eight knot is equal to q 4 − q 2 + 1 − q −2 + q −4 .

3. Skein Modules of Tangles 3.1. Tangles. — The theory developed in Sections 1 and 2 may be generalized to tangles. Tangles are similar to links but besides circles may contain arcs with prescribed ends. As we shall see in Section 4, a study of tangles directly leads to braided monoidal categories. ` PANORAMAS ET SYNTHESES 5

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53

Let k,  be non-negative integers. A tangle with k inputs and  outputs (or, briefly, a (k, )-tangle) is a finite system of disjoint smoothly embedded arcs and circles in the strip R2 × [0, 1] such that: the endpoints of the arcs are the points (1, 0, 0), (2, 0, 0), . . . , (k, 0, 0) and (1, 0, 1), (2, 0, 1), . . . , (, 0, 1); the circles lie in R2 × (0, 1). These arcs and circles in R2 × [0, 1] are called the components of the tangle. Examples of a (3,1)-tangle and a (2,2)-tangle are shown in Figure 3.1.

(3, 1)-tangle

(2, 2)-tangle

Figure 3.1. A tangle L ⊂ R2 × [0, 1] is framed if it is equipped with a non-singular normal vector field equal in the endpoints of the arcs to the vector (0, −1, 0). Normal vector fields on L homotopic in the class of non-singular normal vector fields (fixed in the endpoints of the arcs) determine a framing of L. Two (framed) (k, )-tangles L1 , L2 are said to be isotopic if L1 may be smoothly deformed into L2 in the class of (framed) (k, )-tangles. Of course, during the deformation there should be no intersections and no self-intersections of the tangle components. It is clear that the isotopy is an equivalence relation in the class of tangles. The main goal of the theory of tangles is to classify them up to isotopy. The study of tangles generalizes the study of links which are just (0, 0)-tangles. The fact that (0, 0)-tangles lie in R2 × (0, 1) rather than in R3 does not lead to a loss of generality because any link in R3 may be deformed into R2 × (0, 1). 3.2. Tangle diagrams. — The definition of link diagrams given in Section 1 directly generalizes to tangles. A tangle (k, )-diagram consists of a finite number of arcs and loops in R × [0, 1], the end points of the arcs being the points (1, 0), (2, 0), . . . , (k, 0) and (1, 1), (2, 1), . . . , (, 1). At each self-crossing point of the diagram one of the two crossing branches should be distinguished and said to be the lower one, the second branch being the upper one. We always assume that the arcs and loops of a tangle diagram lie in general position. Unless explicitly stated to the contrary, we do not assume them to be oriented. As in Section 1, every tangle (k, )-diagram in R × [0, 1] represents a framed (k, )-tangle in R2 × [0, 1]. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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The Reidemeister moves on link diagrams Ω0 − Ω3 defined in Section 1 apply to tangle diagrams as well. It is understood that these moves proceed far away from the endpoints of the arcs and do not move these endpoints. 3.3. Theorem. — Two tangle diagrams present isotopic framed tangles if and only if they may be related by an ambient isotopy in R × [0,1] and a finite sequence of moves ±1 ±1 Ω±1 0 ,Ω2 ,Ω3 . This is a direct generalization of Theorem 1.4 to tangles. 3.4. The skein module E k, . — Fix a non-zero complex number a. Let k,  be nonnegative integers. Let Ek, = Ek, (a) be the complex vector space generated by all tangle (k, )-diagrams quotiented by (i) ambient isotopy in R × [0, 1] constant on the boundary of R × [0, 1]; (ii) the relation D ∪ O = −(a2 + a−2 )D, where D is an arbitrary tangle (k, )diagram and O is a simple closed curve in R×(0, 1) bounding a disk in the complement of D; (iii) the identity in Figure 2.1 which involves three tangle (k, )-diagrams identical except in a small 2-disk where they are as shown. The vector space Ek, is called the skein (k, )-module corresponding to a. Each tangle (k, )-diagram D represents an element of Ek, denoted by D and called the skein class of D. Note that if k +  is odd then there are no tangle (k, )-diagrams and Ek, = 0. Applying the identity in Figure 2.1 to all crossing points of a tangle (k, )diagram D we can expand D as a formal linear combination of the classes of diagrams without crossing points (with complex coefficients). Further, using the relation (ii) we may get rid of loops. This gives a canonical expansion of D as a linear combination of tangle diagrams consisting of 12 (k + ) disjoint simple arcs. We call such (k, )-diagrams simple. This argument shows that Ek, has a basis represented by simple (k, The number of simple (k, )-diagrams is equal to the Catalan

)-diagrams. /(n + 1) where n = 12 (k + ). number 2n n 3.5. Theorem. — The skein class of a tangle (k,)-diagram is invariant under the moves ±1 ±1 Ω±1 0 ,Ω2 ,Ω3 . This theorem is a direct generalization of Theorem 2.3 to tangles. Theorems 3.3 and 3.5 imply that the skein class D ∈ Ek, of a tangle (k, )diagram D is an isotopy invariant of the framed (k, )-tangle represented by D. It is called the skein class of the tangle. This is a quite powerful invariant of tangles suitable for recursive computations. Moreover, the coefficients of the skein class of a tangle with respect to the basis of Ek, (a) described above are Laurent polynomials in a. This yields 2n n /(n + 1) polynomial invariants of tangles. In the case k =  = 0, we recover the bracket polynomial of framed links (up to the factor −a2 − a−2 ). ` PANORAMAS ET SYNTHESES 5

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55

4. Categories of Tangles 4.1. The category of framed tangles. — We can organize framed tangles into a monoidal category denoted T . As we shall see, this category is braided in a natural way. The objects of T are non-negative integers 0, 1, 2, . . . A morphism k →  in T is an isotopy class of a framed (k, )-tangle. The composition f g of morphisms g : k →  and f :  → m is represented by the tangle obtained by attaching f on the top of g and compressing the result into R × [0, 1], cf. Figure 4.1.

f ◦ g =

f g

L ⊗ L =

L

L

Figure 4.1. The tangle consisting of k disjoint vertical arcs with constant framing represents the identity idk : k → k. The identity id0 : 0 → 0 is represented by an empty tangle. We provide the category T with a tensor product. The tensor product of the objects k,  ∈ {0, 1, 2, . . .} is the object k + . The tensor product of two morphisms represented by tangles L, L is the juxtaposition obtained by placing L to the right of L without any intersection or linking, cf. Figure 4.1. It is clear that T is a strict monoidal category with unit object 0. Note that T contains the category of braids B (defined in Chapter 2) as a subcategory. Let k,  ∈ {0, 1, 2, . . .}. To define the braiding morphism ck, : k ⊗  → l ⊗ k we take a bunch of k vertical arcs representing idk and place it from above across a bunch of  vertical arcs representing id . This results in a tangle (k + , k + )-diagram with k crossing points, see Figure 4.2 where k = 3,  = 2.

Figure 4.2. We define ck, : k ⊗  →  ⊗ k to be the tangle represented by this diagram. Invertibility of ck, is obvious: the inverse morphism is presented by the tangle diagram obtained from the one of c,k by taking the mirror image with respect to the plane of the picture. 4.2. Theorem. — T is a braided monoidal category. Proof. — Both sides of the braiding identity (4.1)

ck,⊗m = (id ⊗ck,m )(ck, ⊗ idm ), ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 5. THE JONES POLYNOMIAL AND SKEIN CATEGORIES

where k, , m = 0, 1, . . . are framed tangles. It is enough to draw their diagrams in order to see that these tangles are isotopic. This yields formula (4.1). The identity ck⊗,m = (ck,m ⊗ id )(idk ⊗c,m ) is verified similarly. To prove the naturality of the braiding we should check that for any morphisms f :  →  and g : m → m in T (4.2)

(g ⊗ f ) c,m = c ,m (f ⊗ g).

As above, the tangle diagrams appearing on the left-hand and right-hand sides represent isotopic framed tangles. This yields formula (4.2). 4.3. Digression on functors. — In the next subsection we shall need the notion of a functor. A functor P of a category X into a category Y assigns to each object V of X an object P (V ) of Y and to each morphism f : V → W in X a morphism P (f ) : P (V ) → P (W ) in Y so that P (idV ) = idP (V ) for any object V of X and P (f g) = P (f )P (g) for any two composable morphisms f, g in X . If X and Y are strict monoidal categories then the functor P : X → Y preserves the tensor product if for any objects V, W of X we have P (V ⊗ W ) = P (V ) ⊗ P (W ), for any morphisms f, g in X we have P (f ⊗ g) = P (f ) ⊗ P (g), and P (1 1X ) = 1 Y . 4.4. Skein category and skein functor. — For every non-zero complex number a, we define a monoidal category S = S(a) formed by the skein modules. The definition of S is similar to the definition of the category of framed tangles T except that we replace everywhere the isotopy classes of tangles with the skein classes. Here are the details. The objects of S are non-negative integers 0, 1, 2, 3, . . . A morphism k →  of S is an element of Ek, (a). Thus, HomS (k, ) = Ek, . The composition f g of morphisms g : k → , f :  → m represented by tangle diagrams is defined by attaching a diagram of f on the top of a diagram of g and compressing the result into R × [0, 1]. The composition extends to arbitrary morphisms by linearity. The diagram consisting of k disjoint vertical arcs represents the identity morphism idk : k → k. The tensor product of the objects k,  ∈ {0, 1, 2, . . .} of S is the object k + . The tensor product of two morphisms represented by tangle diagrams D, D is obtained by placing D to the right of D. The tensor product extends to arbitrary morphisms by linearity. It is clear that S is a strict monoidal category with the unit object 0. By Theorems 3.3 and 3.5, there is a functor P : T → S(a) which is the identity on the objects and transforms the isotopy class of a framed (k, )-tangle into its skein class in Ek, (a). It is obvious that P preserves the tensor product. We call P the skein functor. For framed links in R3 , considered as morphisms 0 → 0 in T , this functor yields essentially the value of the bracket polynomial in a. Therefore we can regard P as a categorical generalization of the bracket polynomial. We provide S with the braiding {P (ck, ) : k ⊗  →  ⊗ k}k, where ck, : k ⊗  →  ⊗ k is the braiding in T defined above. Hence P (ck, ) is the skein class of the framed ` PANORAMAS ET SYNTHESES 5

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57

(k + , k + )-tangle presented by the diagram defined in Section 4.1. That the morphisms {P (ck, )}k, are invertible and satisfy the braiding identities follows directly from the corresponding properties of {ck, }k, . Naturality of {P (ck, )}k, follows from the naturality of {ck, }k, and the fact that the skein modules are linearly generated by the skein classes of framed tangles. This yields the following theorem. 4.5. Theorem. — The category S is a braided monoidal category. The category S is simpler and more tangible then the category of framed tangles T . The fact that the morphisms k →  in S form a finite-dimensional vector space for any k,  considerably simplifies the study of S. Combining the construction of S with the theory of Jones-Wenzl idempotents in the Temperley-Lieb algebras one can define more subtle braided categories and, in particular, reconstruct the category of representations of the quantum group Uq (sl2 (C)) (see [Tur94, Chapter XII]).

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Chapter 6 FROM RIBBON CATEGORIES TO TOPOLOGICAL INVARIANTS OF LINKS AND 3-MANIFOLDS

In this chapter we introduce an important class of braided monoidal categories, namely, the ribbon categories (Section 1). The categories of tangles and the categories of representations of quantum groups, considered in Chapters 5 and 7, are ribbon categories. For an arbitrary category C, we introduce the category TC formed by the isotopy classes of C-colored tangles, i.e., tangles whose components are endowed with objects of C. The main tool, relating topology of tangles with algebra of categories is a canonical functor FC : TC → C defined when C is a ribbon category (Section 2). This functor is a far-reaching generalization of the skein functor considered in Chapter 5. In Section 3 we introduce a narrower class of modular categories. These are ribbon categories containing a finite set of simple objects in a sense generating the whole category. Finally, in Section 4 we show how to derive from each modular category the corresponding Reshetikhin-Turaev invariant of 3-manifolds and links in 3-manifolds.

1. Ribbon Categories Ribbon categories are braided monoidal categories provided with two additional stuctures, duality and twist. We first discuss duality and twist and then give a precise definition of ribbon categories. 1.1. Duality in monoidal categories. — Duality in monoidal categories is meant to axiomatize duality for modules usually formulated in terms of non-degenerate bilinear forms. Of course, a definition of duality in the framework of categories should avoid the term “linear”. Let C be a monoidal category. Assume that to each object V of C there are associated an object V ∗ of C and two morphisms bV : 1 −→ V ⊗ V ∗ ,

dV : V ∗ ⊗ V −→ 1 .

60

CHAPTER 6. FROM RIBBON CATEGORIES TO TOPOLOGICAL INVARIANTS

The rule V → (V ∗ , bV , dV ) is called a duality in C if the following identities are satisfied: (1.1)

(idV ⊗dV )(bV ⊗ idV ) = idV ,

(1.2)

(dV ⊗ idV ∗ )(idV ∗ ⊗bV ) = idV ∗ .

In formula (1.1) we implicitly assume that (V ⊗ V ∗ ) ⊗ V = V ⊗ (V ∗ ⊗ V ). This equality holds if C is strict monoidal, otherwise one has to involve the associativity isomorphism (V ⊗ V ∗ ) ⊗ V → V ⊗ (V ∗ ⊗ V ), cf. Section 2.1 of Chapter 9. We leave a precise formulation of (1.1) and (1.2) in the non-strict case to the reader. As an exercise, the reader may verify that the standard evaluation and co-evaluation in the theory of modules satisfy (1.1) and (1.2). 1.2. Twist in braided monoidal categories. — Let C be a monoidal category with braiding c = {cV,W : V ⊗ W → W ⊗ V }, where V, W run over objects of C. It would be too restrictive to require the composition cW,V cV,W to be always equal to idV ⊗W . What suits our aims better is to require this composition to be a kind of coboundary. This suggests the notion of a twist as follows. A twist in C is a natural family of isomorphisms θ = {θV : V → V }, where V runs over objects of C, such that for any two objects V, W of C, we have θV ⊗W = cW,V cV,W (θV ⊗ θW ).

(1.3)

The naturality of θ means that for any morphism f : U → V in C, we have θV f = f θU . Using the naturality of the braiding, we can rewrite (1.3) as follows: θV ⊗W = cW,V (θW ⊗ θV ) cV,W = (θV ⊗ θW ) cW,V cV,W . Note that θ1 = id1 . This follows from the invertibility of θ1 and the formula (θ1 )2 = (θ1 ⊗ id1 )(id1 ⊗θ1 ) = θ1 ⊗ θ1 = θ1 . 1.3. Definition of ribbon categories. — A ribbon category is a monoidal category C equipped with braiding, twist θ, and duality (∗, b, d) compatible in the following sense: for any object V of C, (θV ⊗ idV ∗ ) bV = (idV ⊗θV ∗ ) bV . Note that this condition does not involve the braiding; the only axiom relating braiding to twist and duality is (1.3). The axioms of a ribbon category imply the involutivity of duality: the object V ∗∗ = (V ∗ )∗ is canonically isomorphic to V for any object V of C. ` PANORAMAS ET SYNTHESES 5

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61

A ribbon category is said to be strict if its underlying monoidal category is strict. The coherence theorem of Mac Lane establishing equivalence of any monoidal category to a certain strict monoidal category applies in the setting of ribbon categories as well. This enables us to focus attention on strict ribbon categories: all results obtained below for these categories directly extend to arbitrary ribbon categories. (The same remarks hold for the modular categories introduced in Section 3.) For examples of ribbon categories, see Section 3–4 of Chapter 7 and Exercise 1.5. 1.4. Traces and dimensions. — Ribbon categories admit a consistent theory of traces of morphisms and dimensions of objects. This is one of the most important features of ribbon categories, sharply distinguishing them from arbitrary monoidal categories. 1) with Let C be a ribbon category. Denote by K = KC the semigroup End(1 multiplication induced by the composition of morphisms and the unit element id1 . The semigroup K is commutative because for any morphisms k, k  : 1 → 1 , we have kk  = (k ⊗ id1 )(id1 ⊗k  ) = k ⊗ k  = (id1 ⊗k  )(k ⊗ id1 ) = k  k. The traces of morphisms and the dimensions of objects which we define below take values in K. For an endomorphism f : V → V of an object V , we define its trace tr(f ) ∈ K to be the following composition:

tr(f ) = dV cV,V ∗ (θV f ) ⊗ idV ∗ bV : 1 −→ 1 . The main properties of the trace are given in the following lemma (for a proof, see [Tur94, Chapter I] or [Kas95, Chapter XIV]). 1.4.1. Lemma. (i) For any morphisms f : V → W , g : W → V , we have tr(f g) = tr(gf ). (ii) For any endomorphisms f ,g of objects of C, we have tr(f ⊗ g) = tr(f ) tr(g). (iii) For any morphism k : 1 → 1 , we have tr(k) = k. For an object V of C, we define its dimension dim(V ) by the formula dim(V ) = tr(idV ) = dV cV,V ∗ (θV ⊗ idV ∗ ) bV ∈ K. Lemma 1.4.1 implies fundamental properties of the dimension: (i ) isomorphic objects have equal dimensions, (ii ) for any objects V, W , we have dim(V ⊗ W ) = dim(V ) dim(W ), and 1) = 1. (iii ) dim(1 One can show that dim(V ∗ ) = dim(V ) for any object V of C (cf. Exercise 2, 3 (iii) below). ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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CHAPTER 6. FROM RIBBON CATEGORIES TO TOPOLOGICAL INVARIANTS

1.5. Exercise. — Verify that the braided categories T and S constructed in Section 4 of Chapter 5 are ribbon categories with duality and twist shown in Figure 1.1. $

k

%&

...

' $

k

%&

...

'

...

... θk : k → k

$

bk : 0 → 2k

...

%&

k

'

...

dk : 2k → 0

Figure 1.1.

2. The Functor F 2.1. Colored framed tangles. — Fix a category C. A framed tangle is said to be Ccolored (or just colored if it is clear which category C is implied) if each of its components is oriented and equipped with an object of C. This object is called the color of the component. By isotopy of colored framed tangles, we mean an orientationpreserving and color-preserving isotopy. The technique of tangle diagrams readily extends to colored framed tangles, it suffices to specify the orientations and colors of the components. Similarly to the category of framed tangles T in Section 4 of Chapter 5, we can organize the C-colored framed tangles into a monoidal category denoted TC . The objects of TC are finite sequences ((V1 , ε1 ), . . . , (Vm , εm )) where V1 , . . . , Vm are objects of C and ε1 , . . . , εm ∈ {+, −}. The empty sequence is also considered as an object of TC . A morphism η → η  in TC is an isotopy type of a C-colored framed tangle such that η (resp. η  ) is the sequence of colors and directions of those tangle components which hit the bottom (resp. top) boundary endpoints. We agree that ε = + corresponds to the downward direction near the corresponding endpoint and ε = − corresponds to the string directed up. It should be emphasized that isotopic C-colored framed tangles present the same morphism in TC . The composition of two morphisms f : η → η  and g : η  → η  is obtained by putting a colored framed tangle representing g on the top of a colored framed tangle representing f , gluing the corresponding ends, and compressing the result into R2 × [0, 1]. The identity morphisms are represented by framed tangles consisting of disjoint vertical arcs with constant framing. The identity endomorphism of the empty sequence is represented by an empty tangle. The tensor product of objects η and η  of TC is their concatenation ηη  . The tensor product of morphisms f, g is obtained by placing a colored framed tangle representing f to the left of a colored framed tangle representing g so that there is no mutual linking or intersection. This makes TC into a strict monoidal category. ` PANORAMAS ET SYNTHESES 5

2 THE FUNCTOR F

63

We shall need certain specific morphisms in TC presented by tangle diagrams in Figure 2.1 where we also specify the notation for these morphisms. Here V, W run over objects of C. W

V

W

V

V

W

V

W

W

V

− XV,W

+ XV,W

V

W

V

W

V

W − YV,W

+ YV,W

W

V

W

V

W

V

W

V

V

W

V

W

V

W

V

W

− ZV,W

+ ZV,W

− TV,W

+ TV,W

Figure 2.1. The morphisms in TC presented by the diagrams in Figure 2.2 will be denoted by − ↓V , ↑V , ϕV , ϕV , ∩V , ∩− V , ∪V , ∪V , respectively. V

V

V

V

V

V

V

V

Figure 2.2. As the reader may have guessed, the category TC admits a natural braiding, twist, and duality and becomes in this way a ribbon category (cf. Exercise 1.5). We shall not use these structures in TC . 2.2. Theorem. — Let C be a strict ribbon category with braiding c, twist θ, and duality (∗,b,d). There exists a unique functor F = FC : TC → C preserving the tensor product (cf. Section 4.3 of Chapter 5) and satisfying the following conditions: (i) F transforms any object (V ,+) into V and any object (V ,−) into V ∗ ; (ii) for any objects V ,W of C, we have F (XV+,W ) = cV ,W ,

F (ϕV ) = θV ,

F (∪V ) = bV ,

F (∩V ) = dV .

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The functor F has the following properties: F (XV−,W ) = (cW ,V )−1 ,

F (YV+,W ) = (cW ,V ∗ )−1 ,

F (ZV+,W ) = (cW ∗ ,V )−1 ,

F (ZV−,W ) = cV ,W ∗ ,

F (TV+,W ) = cV ∗ ,W ∗ ,

F (TV−,W ) = (cW ∗ ,V ∗ )−1 ,

F (YV−,W ) = cV ∗ ,W , F (ϕV ) = (θV )−1 .

Note the following obvious properties of F . If L and L are isotopic colored framed tangles then L and L represent the same morphism in TC and therefore F (L) = F (L ). Since F is a functor, F (↓V ) = idV ,

F (↑V ) = idV ∗ ,

and F (L L ) = F (L) F (L )

for any two composable colored framed tangles L and L . Since F preserves the tensor product, F (L ⊗ L ) = F (L) ⊗ F (L ) for any two colored framed tangles L and L . Note also that for any colored framed link L, we have F (L) ∈ End(1 1). and ∩− The values of F on ∪− V V may be computed from the formulas +  ∪− V = (↑V ⊗ ϕV ) ◦ ZV,V ◦ ∪V , − ∩− V = ∩V ◦ ZV,V ◦ (ϕV ⊗ ↑V ).

Theorem 2.2 yields isotopy invariants of colored framed tangles and, in particular, invariants of colored framed links. This theorem plays a fundamental role in the theory of quantum invariants of links and 3-manifolds. For a proof of Theorem 2.2, see [Tur94, Chapter I]. The idea of the proof is as follows. One can use tensor product and composition in TC in order to express any colored framed tangle via the tangles mentioned in the item (ii) of the theorem. This allows us to define F for all tangles. Although every tangle admits different expressions of this kind, they may be obtained from each other by the Reidemeister moves and a few other local transformations. To show that F is correctly defined, one verifies the invariance of F under these moves. 2.3. Exercises. (i) Let L be a disjoint union of two C-colored framed links L1 and L2 . Show that F (L) = F (L1 )F (L2 ). (ii) Let L be the trivial knot with framing 0 colored with an object V of C. Show that F (L) = dim(V ). (iii) Let L be a C-colored framed link and let  be a component of L. Let L be the C-colored framed link obtained from L by reversing the orientation of  and replacing the color of  with the dual object. Show that F (L ) = F (L). Deduce that dim(V ∗ ) = dim(V ) for any object V of C. ` PANORAMAS ET SYNTHESES 5

3. MODULAR CATEGORIES

65

3. Modular Categories Ribbon categories are far too general to yield invariants of 3-manifolds. With this view we introduce here the concept of a modular category. Examples of modular categories are provided by quantum groups at roots of unity, see Section 3 of Chapter 7. In Section 4 we shall see that modular categories give rise to invariants of closed oriented 3-manifolds and links in such manifolds. 3.1. Additive categories and simple objects. Let K be a commutative ring. We say that a monoidal category C is additive with ground ring K if the Hom-sets in C are K-modules, the composition and the tensor product of morphisms are K-bilinear, and the formula k → k id1 defines a Kmodule isomorphism K → End(1 1). It is easy to check that under this isomorphism, multiplication in K corresponds to composition of morphisms. In the sequel we shall identify End(1 1) with K via this isomorphism. An object V of an additive monoidal category C with ground ring K is simple if the formula k → k idV defines a K-module isomorphism K → End(V ). For example, the unit object 1 is simple. The simple objects play the same role as the irreducible modules in the theory of modules. It is easy to see that an object isomorphic or dual to a simple object is also simple. The tensor product of simple objects may be non-simple; this is usually the case in the categories of representations of quantum groups. In analogy with the theory of modules, one can try to decompose objects of an additive monoidal category C into direct sums of simple objects. However, in general, C may not admit direct sums. What one can always do is to decompose the endomorphisms of objects into sums of morphisms passing through simple objects. This leads to the following notion of domination. Let {Vi }i∈I be a family of simple objects of C. An object V of C is dominated by {Vi }i∈I if the images of the pairings { (g, f ) → f g : Hom(V, Vi ) ⊗ Hom(Vi , V ) → End(V ) }i∈I additively generate End(V ). For instance, if C admits direct sums and V splits as a sum of a finite number of simple objects {Vi }i∈I , then V is dominated by {Vi }i∈I . 3.2. Modular categories. A modular category is a pair (C, {Vi }i∈I ) consisting of an additive ribbon monoidal category C with ground ring K and a finite family {Vi }i∈I of simple objects of C satisfying the following four axioms. (3.2.1) (Normalization axiom.) There exists 0 ∈ I such that V0 = 1 . (3.2.2) (Duality axiom.) For any i ∈ I, there exists i∗ ∈ I such that the object Vi∗ is isomorphic to (Vi )∗ . (3.2.3) (Axiom of domination.) All objects of C are dominated by {Vi }i∈I . To formulate the last axiom we need more notation. For i, j ∈ I, set Si,j = tr(cVj ,Vi ◦ cVi ,Vj ) ∈ End(1 1) = K. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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By Lemma 1.4.1, we have Si,j = Sj,i . Thus, S = (Si,j )i,j∈I is a symmetric square matrix over K. Note that S0,i = Si,0 = tr(idVi ) = dim(Vi ). There is a geometric interpretation of Si,j as the invariant FC of the colored framed link (Hopf link) presented by the diagram in Figure 3.1.

Vi

Vj

Figure 3.1. (3.2.4) (Non-degeneracy axiom.) The square matrix S = (Si,j )i,j∈I is invertible over K. Since the ring K is commutative, the non-degeneracy axiom amounts to saying that det(S) is invertible in K. This completes the list of axioms for modular categories. Note that if the objects Vi , Vj were isomorphic then the i-th and j-th rows of S would be equal. By the non-degeneracy axiom, the objects Vi , Vj with distinct i, j are not isomorphic. This implies that for any i ∈ I, there exists exactly one i∗ ∈ I such that Vi∗ is isomorphic to (Vi )∗ . The formula i → i∗ defines an involution in I. There is exactly one element 0 ∈ I such that V0 = 1 and 0∗ = 0. To define the invariants of 3-manifolds associated with a modular category (C, {Vi }i∈I ) with ground ring K we shall assume the existence of an element D ∈ K such that 

2 dim(Vi ) . D2 = i∈I

Each such D ∈ K is called a rank of C. The existence of a rank is a minor technical condition which can always be satisfied in an appropriate extension of K. Besides the rank we shall need another element ∆ = ∆C ∈ K defined as follows. Since Vi is a simple object, the twist acts in Vi as vi idVi for a certain vi ∈ K. Since the twist is an isomorphism, vi is invertible in K. In geometric language, the role of vi may be described as follows: when we insert one full right-hand twist in the framing of a framed tangle in R3 colored with Vi the invariant FC of this tangle is multiplied by vi . Set 

2 ∆= vi−1 dim(Vi ) ∈ K. i∈I

One can show that D and ∆C are invertible in K. ` PANORAMAS ET SYNTHESES 5

4. INVARIANTS OF 3-MANIFOLDS

67

4. Invariants of 3-Manifolds Fix a strict modular category (C, {Vi }i∈I ) with ground ring K and rank D ∈ K. For every closed connected oriented 3-manifold M , we shall define a topological invariant τ (M ) = τ(C,D) (M ) ∈ K. 4.1. Surgery on links in the 3-sphere. — The construction of τ is based on a reduction to framed links in Euclidean 3-space. To this end we use the technique of surgery, well known in topology. Let L be a framed link in S 3 with m components L1 , . . . , Lm (cf. Section 1 of Chapter 5); the link L is not assumed to be oriented or colored. Let U be a closed regular neighborhood of L in S 3 . It consists of m disjoint solid tori U1 , . . . , Um whose cores are the components of L. (A solid torus is homeomorphic to S 1 × B 2 where B 2 is a closed 2-disk.) For each n = 1, . . . , m, we identify Un with S 1 × B 2 so that the core Ln of Un is identified with S 1 × 0 where 0 is the center of B 2 and the given framing of Ln is identified with a constant normal vector field on S 1 × 0 in S 1 × B 2 . Let B 4 be a closed 4-ball bounded by S 3 . Let us glue m copies of the 2-handle 2 B × B 2 to B 4 along the solid tori U1 , . . . , Um ⊂ S 3 = ∂B 4 using the identifications Un = S 1 × B 2 = ∂B 2 × B 2 where n = 1, . . . , m. This gluing results in a compact connected orientable 4-manifold denoted WL . A schematic picture for WL is given in Figure 4.1. B2 × B2

→

B2 × B2

→

Lm →

→

B4

→

→

. ..

WL =

L1

Figure 4.1. The 3-manifold ∂WL is formed by S 3 \ Int(U ) and m copies of the solid torus B × ∂B 2 glued to S 3 \ Int(U ) along the boundary. We provide ∂WL with the orientation extending the right-handed orientation in U . We say that the (closed connected) oriented 3-manifold ∂WL is obtained by surgery on S 3 along L. A classical theorem due to Lickorish [Li] and Wallace [Wal] asserts that every closed connected oriented 3-manifold (considered up to orientation preserving homeomorphisms) can be obtained by surgery on S 3 along a certain framed link. Essentially this follows from the elementary theory of handles and the theorem of V.A. Rokhlin asserting that every closed oriented 3-manifold bounds a compact oriented smooth 4-manifold. We shall need an invariant σ(L) ∈ Z of the framed link L ⊂ S 3 defined as the signature of WL , i.e., as the signature of the homological intersection form 2

H2 (WL ;R) × H2 (WL ;R) −→ R. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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This form corresponds to the orientation of WL determined in any point of ∂WL by a tangent vector directed outwards and a positive basis in the tangent space of ∂WL . If L = ∅ then WL = B 4 and σ(L) = 0. 4.2. Invariants of 3-manifolds. — To combine surgery with the technique of ribbon categories we have to pass from links in S 3 = R3 ∪ {∞} to links in R3 . Fortunately, any link in S 3 may be deformed into R3 and isotopic links in S 3 give rise to isotopic links in R3 . Therefore we may apply the invariant F = FC to colored framed links in S 3 . Now we are in a position to define the invariant τ (M ) = τ(C,D) (M ) ∈ K of a closed connected oriented 3-manifold M . We present M as the result of surgery on S 3 along a framed link L ⊂ S 3 . We build up an invariant of M from invariants of L corresponding to different colorings of L by the objects {Vi }i∈I . Let L1 , . . . , Lm be the components of L. Fix an arbitrary orientation of L. Denote by col(L) the set of all mappings {1, 2, . . . , m} → I. This set has (card(I))m elements. For each λ ∈ col(L), the pair (L, λ) is a colored framed link with color of the i-th component being Vλ(i) . By Theorem 2.2, we have an invariant F (L, λ) ∈ K. Set (4.1)



τ (M ) = ∆σ(L) D−σ(L)−m−1

λ∈col(L)

m !

 dim Vλ(n) F (L, λ) ∈ K.

n=1

4.3. Theorem. — The element τ (M ) is a topological invariant of M . For a proof of this theorem, see [Tur94, Chapter II]. The idea of the proof is as follows. First of all, one uses the result of Exercise 2.3.3 and axiom (3.2.2) to show that τ (M ) does not depend on the choice of orientation of L. Then one ought to show that the right-hand side of (4.1) does not depend on the choice of L. It is known that one and the same 3-manifold may be obtained from the 3-sphere by surgery along different framed links. Such links are related by so-called Kirby moves. One verifies the invariance of the right-hand side of (4.1) under these moves. Theorem 4.3 establishes a deep connection between modular categories and the topology of 3-manifolds. The main feature of this connection is that a purely algebraic object (modular category) gives rise to a topological invariant of 3-manifolds. At the moment of writing there is no known way to obtain this invariant via homological or homotopical constructions. The invariants τ derived from the modular categories formed by the representations of the quantum groups at roots of unity are called quantum invariants of 3-manifolds. The definition of the invariant τ is ready for explicit computations. In particular, τ (S 1 × S 2 ) = 1

and τ (S 3 ) = D−1 .

These formulas follow directly from definitions, it suffices to note that S 1 × S 2 is obtained by surgery on S 3 along the trivial knot with framing 0 and S 3 is obtained by surgery on S 3 along the empty link. Another example is provided by the lens spaces ` PANORAMAS ET SYNTHESES 5

4. INVARIANTS OF 3-MANIFOLDS

69

of type (n, 1) with integer n ≥ 2. The lens space L(n, 1) may be obtained from S 3 by surgery along the trivial knot with framing n. It follows from the definitions that 

2

vin dim(Vi ) . τ L(n, 1) = ∆D−3 i∈I

The same manifold with the opposite orientation −L(n, 1) may be obtained from S 3 by surgery along the trivial knot with framing −n. Therefore 

2

vi−n dim(Vi ) . τ −L(n, 1) = ∆−1 D−1 i∈I

4.4. Invariants of links in 3-manifolds. — A link in a 3-manifold M is a finite collection of disjoint circles smoothly embedded in M . The definitions of isotopy, framing, and C-coloring for links in R3 directly extend to links in M . The invariant of 3-manifolds defined above generalizes to closed oriented 3manifolds with C-colored framed (oriented) links sitting inside. Let M be a closed connected oriented 3-manifold. Let Ω ⊂ M be a C-colored framed link. Present M as the result of surgery on S 3 along a framed link L with components L1 , . . . , Lm . Fix an orientation of L. Applying isotopy to Ω we can deform it into S 3 \ U ⊂ M where U is a closed regular neighborhood of L in S 3 . Thus, we may assume that Ω ⊂ S 3 \ U . For any λ ∈ col(L), we form a C-colored framed link (L, λ) ∪ Ω in S 3 . Set τ (M, Ω) = ∆σ(L) D−σ(L)−m−1

 λ∈col(L)

m !



dim Vλ(n) F (L, λ) ∪ Ω ∈ K.

n=1

4.5. Theorem. — The element τ (M ,Ω) is a topological invariant of the pair (M ,Ω). This theorem implies, in particular, that τ (M, Ω) is an isotopy invariant of Ω. It includes Theorem 4.3 as a special case Ω = ∅. The invariant τ (M, Ω) satisfies the following multiplicativity law: τ (M1 # M2 , Ω1 ) Ω2 ) = D τ (M1 , Ω1 ) τ (M2 , Ω2 ) where Ω1 , Ω2 are C-colored framed links in closed connected oriented 3-manifolds M1 , M2 respectively. Note that the invariant M → D τ (M ) is multiplicative with respect to connected sum. If M = S 3 , then τ (S 3 , Ω) = D−1 F (Ω) (we may take L = ∅ to compute τ (S 3 , Ω)). Thus, the invariant τ(C,D) generalizes the invariant FC of colored framed links in Euclidean space, defined in Section 2. The reader should be warned that this generalization applies only when C is a modular category. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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The invariant τ(C,D) admits an extension to 3-cobordisms (possibly, with colored framed tangles inside). The corresponding invariant of a 3-cobordism is not an element of K but rather a K-homomorphism between projective K-modules associated with the bases of the cobordism. These modules and homomorphisms form a so-called topological quantum field theory (TQFT) in dimension 3, see [Tur94].

` PANORAMAS ET SYNTHESES 5

Chapter 7 THE REPRESENTATION THEORY OF Uq sl(N+1)

In this chapter we return to the Hopf algebra Uq sl(N + 1) of Chapter 4 and we show how, when q is a root of 1, the modules over a suitable version of it give rise to a modular category in the sense of Section 3 of Chapter 6. We start by describing the representations of Uq sl(N + 1) when q is generic. We again denote by h the Lie subalgebra of sl(N + 1) generated by H1 , . . . , HN. Let X ⊂ h∗ be the lattice of linear forms that take integral values on H1 , . . . , HN.

1. Highest Weight Modules As in Sections 2, 3 of Chapter 4, we assume q is an indeterminate and the ground field is C(q). Let M be a finite-dimensional Uq sl(N + 1)-module. For λ ∈ X and ε = (ε1 , . . . , εN ) in (Z/2)N , put   Mλ,ε = v ∈ M | Ki v = (−1)εi q λ(Hi ) v for all i = 1, . . . , N . The pairs (λ, ε) for which Mλ,ε = {0} are called weights. It is easy to see that M is the direct sum of the non-zero weight spaces Mλ,ε and that the elements Ei , Fi act nilpotently on M . A non-zero element v of M is said to be a highest weight vector if it belongs to one of the subspaces Mλ,ε and if Ei v = 0 for all i = 1, . . . , N . A module M is said to be a highest weight module if it is generated by a highest weight vector. For λ ∈ X and ε = (ε1 , . . . , εN ) in (Z/2)N , there is a universal highest weight module M (λ, ε) with highest weight (λ, ε), called the Verma module, and constructed as follows. Recall from Section 2 of Chapter 4 the subalgebra U+ of Uq sl(N + 1) generated by Ei and Ki±1 (i = 1, . . . , N ). Denote by 1 λ,ε the one-dimensional U+ module with basis v+ , such that Ei v+ = 0

and Ki v+ = (−1)εi q λ(Hi ) v+

for all i = 1, . . . , N .

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The Verma module is the induced module M (λ, ε) = Uq sl(N + 1) ⊗U+ 1 λ,ε . It is the direct sum of its weight subspaces M (λ, ε)µ,ε , which are finite-dimensional. The module M (λ, ε) has a unique simple quotient L(λ, ε), which is a highest weight module with highest weight (λ, ε). The module L(λ, ε) is finite-dimensional if and only if λ is a dominant weight, i.e., λ(Hi ) ∈ N for all i = 1, . . . , N . For λ = 0, the module L(0, ε) is also denoted Lε . This module is one-dimensional (Ei and Fi act by 0, and Ki by (−1)εi ). For any λ ∈ X, one has an isomorphism L(λ, ε) ∼ = L(λ, 0) ⊗ Lε of Uq sl(N + 1)-modules. This fact allows us to reduce the study of L(λ, ε) to the study of the module L(λ, 0), also denoted L(λ). * Let M = Mλ,0 be a finite-dimensional Uq sl(N + 1)-module whose weights are λ∈X

of the form (λ, 0). Its character is the element of the group algebra of X defined by  Ch(M ) = (dim Mλ,0 ) eλ ∈ Z[X]. We denote by eλ the generator of Z[X] corresponding to λ ∈ X ; in Z[X] we have eλ eµ = eλ+µ . 1.1. Theorem. (i) Any finite-dimensional simple Uq sl(N + 1)-module is of the form L(λ,ε), where λ is a dominant weight and ε ∈ (Z/2)N . (ii) All finite-dimensional Uq sl(N + 1)-modules are semisimple. (iii) The character Ch(L(λ)) is given by the same formula as the character of the simple sl(N +1)-module parametrized by the same highest weight (H. Weyl’s formula). (iv) The multiplicity of a simple module L(ν) in the decomposition of the tensor product L(λ) ⊗ L(µ) of two simple modules is the same as for the decomposition of the corresponding sl(N + 1)-modules. This theorem is due to Lusztig (see [Lus88], [Lus90]) and Rosso (see [Ros88], [Ros90]). Let us make a few comments on the proof. One begins with the case of Uq sl(2), following the proof of the classical sl(2)-case and using the central element Kq + K −1 q −1 + FE (q − q −1 )2 in order to prove complete reducibility. It is interesting to observe that Items (ii) and (iii) are related: considering short exact sequences of the form 0 → L(λ) −→ M −→ L(µ) → 0 with M indecomposable, and the dual exact sequences, one proves that (ii) holds if and only if all finite-dimensional highest weight Uq sl(N + 1)-modules are simple ` PANORAMAS ET SYNTHESES 5

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73

(see [Ros90]). The idea is then to compute the character of a finite-dimensional highest weight module: one finds an explicit formula (Weyl’s formula) and this shows that the highest weight module has to be the unique simple one with the same highest weight. There are several ways to establish the formula for the character of a finitedimensional highest weight module M with highest weight λ. For instance, one may use a Jordan-H¨older series argument to show that Ch(M ) is a linear combination of characters of the form Ch(M (µ)) with integer coefficients. From an analogue of Harish-Chandra’s theorem for infinitesimal characters one deduces that the possible weights µ have to satisfy two conditions: (i) µ ≤ λ, (ii) µ + ρ is in the orbit of λ + ρ under the action of the Weyl group W. (As usual, ρ is the half-sum of the positive roots). One computes Ch(M (µ)) using, for instance, the Poincar´e-Birkhoff-Witt theorem of Section 3 of Chapter 4. Then, using the action of W on the set of weights of M , one gets the desired formula for Ch(M ). This line of arguments is quite classical, see Humphreys’s survey in [Hum78]. The analogue of Harish-Chandra’s theorem quoted above may be proved, using the adinvariant bilinear form on Uq sl(N + 1) discussed in Section 3 of Chapter 4, to compare the center of Uq sl(N + 1) with the ad-invariant linear forms obtained from the “Markov traces” (or quantum traces) of the irreducible finite-dimensional representations x → trM (K2ρ x) where x ∈ Uq sl(N + 1). Another way is to use a generalized Casimir element constructed from the R-matrix (see [Lus93]). Recall from Section 3 of Chapter 4 the subalgebra V− of Uq sl(N + 1) generated by the elements F1 , . . . , FN . For β=

N 

mi αi

where m1 , . . . , mN ∈ N,

i=1

denote by Vβ− the subspace of homogenous elements of degree −β of V− . Let λ ∈ X. As L(λ) is generated by a highest weight vector v+ , one has a surjective linear map V− → L(λ) given by x → xv+ . From the character formula, we get the following proposition.  1.2. Proposition. — Let β = i mi αi with mi ∈ N. Let λ be a dominant weight such that λ(Hi ) ≥ mi for all i = 1, . . . ,N . Then the linear map Vβ− → L(λ) given by x → xv+ is injective. 1.3. Remark. — One can deduce from Proposition 1.2 another proof of the nondegeneracy of the Hopf pairing ϕ : U+ × U− → C(q) of Section 3.7 of Chapter 4. It is enough to show that the restriction to V+ × V− is non-degenerate. Let f ∈ V− be such that ϕ(x, f ) = 0 for all x ∈ V+ . As ϕ is graded, one may assume that f is homogeneous * and that its degree −β is maximal for the standard partial ordering on the lattice Zαi . From the maximality of −β, one deduces ∆f = f ⊗ K−β + 1 ⊗ f ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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 −mN where K−β = K1−m1 . . . KN for β = i mi αi . Using the second formula in Theorem 2.1 (iii) of Chapter 3, one concludes that f commutes with all elements of V+ . Consequently, for any simple module L(λ), the vector f v+ is either zero or a highest weight vector. This second possibility is excluded by the fact that f v+ would then generate a non-trivial proper submodule of L(λ). So f v+ is always zero. Taking λ as in Proposition 1.2, one gets a contradiction.

2. Quantum Theory of Invariants Let (H, R) be a braided Hopf algebra. As explained in Section 4 of Chapter 2, the universal R-matrix R allows us to define a braiding on any H-module. Recall that, if ρ : H → End(V ) is a representation, then cV = τV,V (ρ ⊗ ρ(R)) is the corresponding braiding. For n ∈ N, let ρ(n) be the induced representation of H on V ⊗n. It is defined by ρ(n) = ρ⊗n ◦ ∆(n) where ∆(2) = ∆,

∆(3) = (∆ ⊗ id) ◦ ∆,

etc.

For 1 ≤ i ≤ n − 1 define the automorphism ci = id⊗(i−1) ⊗cV ⊗ id⊗(n−i−1) . Then c1 , . . . , cn−1 define a representation of the braid group Bn on V ⊗n . For all x ∈ H and all i ∈ {1, . . . , n − 1}, we have ci ρ(n) (x) = ρ(n) (x) ci . This means that the image of Bn in End(V ⊗n ) is contained in the commutant of the image of H. Let us now return to Uq sl(N + 1). We know that this Hopf algebra is almost braided in the sense that its universal R-matrix R lives in a suitable completion of Uq sl(N + 1) ⊗ Uq sl(N + 1). However, the image of this R-matrix in any finitedimensional representation makes perfectly good sense. An explicit formula for R can be obtained by introducing suitable logarithms of the generators Ki ; in practice, for a given finite-dimensional representation, as Ei and Fi act nilpotently, it is enough to consider the formula obtained in the case of a root of unity with the order of the root big enough, and to substitute a generic q in it. Let ω1 be the dominant weight given by ω1 (H1 ) = 1 and ω1 (Hi ) = 0 for i > 2. The irreducible representation L(ω1 ) is (N + 1)-dimensional and is a “deformation” of the fundamental representation of sl(N + 1). In fact, one may choose a basis of L(ω1 ) such that the generators Ei and Fi are represented by the elementary matrices Ei,i+1 and Ei+1,i . One easily computes the action of the R-matrix in L(ω1 ). It is given by cL(ω1 ) = q

N +1 

Eii ⊗ Eii +

i=1



Eij ⊗ Eji + (q − q −1 )

1≤i, j≤N +1 i =j

This yields operators c1 , . . . , cn−1 acting on L(ω1 )⊗n . ` PANORAMAS ET SYNTHESES 5



Ejj 1≤i n. We already know that L(ω1 )⊗n is a semisimple Uq sl(N + 1)-module. It splits as L(ω1 )⊗n =

+

L(λ) ⊗ Pλ ,

where Pλ can be seen as the subspace of L(ω1 )⊗n consisting of the highest weight vectors with highest weight λ. Furthermore, the set of weights λ appearing in this decomposition and the multiplicities dim Pλ are the same as in the classical case, i.e., as for U sl(N + 1) acting on (CN +1 )⊗n . The classical theory of invariants asserts that the image of the group algebra of the symmetric group Sn permuting the factors is exactly the commutant of U sl(N + 1) in End((CN +1 )⊗n ). The hypothesis N > n implies that this image is isomorphic to the group algebra, that the multiplicity spaces Pλ are simple Sn -modules and their set is exactly the set of simple modules of Sn . It follows that (dim Pλ )2 = n!. λ

Coming back to the quantum case, we observe that, via the automorphisms ci , the Hecke algebra acts on each multiplicity subspace Pλ . From the double commutant theorem, each Pλ is an irreducible representation of the commutant of Uq sl(N + 1) in L(ω1 )⊗n . If the image of Hn (q 2 ) were strictly smaller, some of the representations Pλ would split into smaller irreducible components and the dimension of the image of Hn (q 2 ), which is the sum of the squares of the dimensions of these irreducible components, would be strictly smaller than n!. So we only have to prove that c1 , . . . , cn−1 generate an n!-dimensional subalgebra of End(L(ω1 )⊗n ). This amounts to show that the “canonical forms” constructed with the automorphisms gi are linearly independent over C(q). This certainly holds since their specializations at q = 1 are linearly independent. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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3. The Case of Roots of Unity There are several points of view on the definition of a quantum group at a root of unity, leading to very different representation theories. Although we shall mainly use the last one below in the applications to 3-dimensional manifolds, we briefly mention the others which have their own interest. First, one may simply consider the Hopf algebra Uε of Section 4 of Chapter 4 with ε a root of unity. This is an infinite-dimensional Hopf algebra with a big center. (Typically, if  is the order of ε, the -th powers of the analogues of root vectors and of the Ki2 are in the center.) Its irreducible finite-dimensional representations have their dimensions bounded by N (N +1)/2 . This algebra has been extensively studied by de Concini, Kac and Procesi (see [CK90], [CKP92]), who found very interesting connections with the theory of Poisson-Lie groups. A second point of view is to work first over the ground ring Z[q, q −1 ] instead of C(q), and to add new generators [Ki ;0] such that Ki − Ki−1 = (q − q −1 )[Ki ;0], (n)

(n)

and q-analogues of the divided powers, i.e., elements Ei , Fi (where 1 ≤ i ≤ N ) Fn En playing the rˆole of i and i . (This approach is analogous to Kostant’s Z-form [n]! [n]! of Usl(N + 1)). The irreducible modules L(λ) also have such a Z[q, q −1 ]-form. One then specializes q to a primitive root ε of 1, say of order . Let us call Uεres the () () resulting Hopf algebra. We have Ei = 0, Fi = 0, but Ei and Fi survive. This approach was introduced by Lusztig (see [Lus88], [Lus90], [Lus93]), who constructed a “Frobenius map in characteristic 0”, with values in the usual enveloping algebra Usl(N + 1), and discovered in the case of a prime number  profound connections with the theory of algebraic groups in characteristic . Note that the modules L(λ) can also be specialized, but in general their specialization is no longer a simple module. The third point of view is the one we shall mainly use here. Namely, we consider the finite-dimensional Hopf algebra uε of Section 4 of Chapter 4. This is nothing but the subalgebra of Uεres generated by Ei , Fi , Ki for 1 ≤ i ≤ N . The representation theory of this algebra is much more complicated than the representation theory in the generic case. It turns out to be an important ingredient for the construction of 3-manifold invariants. Here are two essential facts about the representation theory of uε . (i) The finite-dimensional uε -modules are no longer semisimple. A simple way to see this is to observe that the square of the antipode is a non-trivial automorphism of uε ; but a theorem of Larson and Radford [LR88] states that the antipode of a semisimple finite-dimensional Hopf algebra is involutive. (ii) There are only a finite number of finite-dimensional simple uε -modules. They are parametrized by the dominant weights λ satisfying 0 ≤ λ(Hi ) < 

for all i = 1, . . . , N ,

where  is the order of ε. The corresponding simple module L(λ) is a highest weight module with highest weight λ, but it is smaller than the simple module with the same ` PANORAMAS ET SYNTHESES 5

3. THE CASE OF ROOTS OF UNITY

77

highest weight in the generic case (e.g., the weight multiplicities are smaller). These () () modules are in fact modules over the divided power algebra Uεres , with Ei and Fi acting by 0. As already alluded to before, there is a nice way due to Lusztig [Lus90] to obtain L(λ) from L(λ) by specializing and quotienting. Information about the character of L(λ) in terms of Kazhdan-Lusztig polynomials has been obtained only very recently by Kazhdan and Lusztig [KL93], proving some conjectures of Lusztig’s. In view of the applications to 3-manifold invariants, it is important to discuss the categorical aspects of the representations of uε and of Uεres and, specifically, to derive a modular category in the sense of Section VI.3. In the following, we shall assume that  is prime to N + 1 (this restriction can be avoided by modifying a little bit the definition of uε , but we don’t want to enter into technicalities). 3.1. Proposition. — The category C of finite dimensional representations of u ε is a ribbon category. Proof. — We already know from Section 4 of Chapter 4 that uε is a braided Hopf algebra. So C is a braided monoidal category. The general theory of braided Hopf algebras (see Section 4 of Chapter 2) provides us with an element u ∈ uε such that S 2 (x) = u xu−1 for all x ∈ uε and ∆u = (R21 R)−1 (u ⊗ u) = (u ⊗ u)(R21 R)−1 .   The element u is given by u = i S(ti )si if the universal R-matrix is R = i si ⊗ ti . Furthermore, if ϕ : uε → End(V ) is a finite-dimensional representation, then ρ(u−1 ) : V −→ V satisfies Formula (1.3) of Chapter 6 for a twist. Unfortunately, this homomorphism is not a morphism in the category C because u−1 is not in the center of uε . However, observe that for all x ∈ uε we have S 2 (x) = K −1 xK

and ∆K = K ⊗ K

where K is defined as follows: write the sum of the positive roots as (3.1)

N i=1

mi αi ; then

mN K = K1m1 · · · KN .

Set v = Ku−1 = u−1 K. It is clear that v lies in the center of uε and satisfies ∆v = (R21 , R)(v ⊗ v) = (v ⊗ v)(R21 R). It follows that, for each representation (V, ϕ), the automorphism θV = ϕ(v) defines a twist in C. The compatibility between the twist and duality required in the definition of a ribbon category follows from the equality S(v) = v, which implies that θV ∗ is the usual transpose of θV . To prove that S(v) = v it suffices to check that uS(u−1 ) = K 2 . ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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One computes immediately that uS(u−1 ) is grouplike (i.e., satisfying Formula (1.4) of Chapter 2), and that uS(u−1 )K −2 is in the center. So, it suffices to check that the only grouplike central element is 1. Writing such an element in the Poincar´e-BirkhoffWitt type basis of uε obtained from the triangular decomposition, one sees that such an element lies in the subalgebra u0ε generated by the elements K1 , . . . , KN . As u0ε is isomorphic to the group algebra of (Z/)N , the only grouplike elements are products of Ki ’s, and the only central product is 1. We can now consider traces and dimensions in C, as defined in Section 1 of Chapter 6. One has the following formula. 3.2. Proposition. — Let (V ,ϕ) be a finite dimensional representation of u ε and f : V → V be a morphism in C, i.e., a uε -linear map. Then

tr(f ) = Tr ϕ(K)f , where K is given by Formula (3.1) and Tr : End V → C stands for the standard trace map (normalized by Tr(IdV ) = dimC V ). Proof. — By definition of the trace in C we have

tr(f ) = dV cV,V ∗ (θV f ) ⊗ idV ∗ bV .  Fix a basis (e1 , . . . , er ) of V with dual basis (e∗1 , . . . , e∗r ). Then bV (1) = i ei ⊗ e∗i and dV : V ∗ ⊗ V → C is  the canonical pairing. The standard trace of an endomorphism g is given by Tr(g) = e∗i , gei . We have i



ϕ(S(tj ))e∗i , ϕ(sj )ϕ(v)f ei   e∗j , ϕ(S(tj )sj v)f ei . =

tr(f ) =

t



i,j

But



S(tj )sj v = uv = K and the result follows.

j

Let us now single out the simple modules L(λ) in C such that Denote this (finite) set of dominant weights by I. For λ ∈ I, set

N i=1

λ(Hi ) < .

I(λ) = L(λ). The modules (I(λ))λ∈I have the property that their character is still given by Weyl’s formula. In view of Proposition 3.2, this gives an explicit formula for their dimensions, namely dim I(λ) =

! ε(λ+ρ,α) − ε(−λ+ρ,α) , ε(ρ,α) − ε−(ρ,α) α∈∆ +

where ρ is the half-sum of the positive roots. ` PANORAMAS ET SYNTHESES 5

4. QUANTUM GROUPS WITH A FORMAL PARAMETER

79

One checks immediately that L(0) is the trivial representation and that the dual of L(λ) is isomorphic to L(−w0 (λ)) where w0 is the permutation sending the simple root αi to αN +1−i (where 1 ≤ i ≤ N ). In order to get a modular category as defined in Section 3 of Chapter 6, one has to work a little more. The idea would be to consider the submonoidal category of C generated by the (I(λ))λ∈I and to take a suitable quotient of it (passing to such a quotient is made necessary by the fact that tensor products of I(λ)’s are not direct sums of I(µ)’s: one has to discard a “bad part” in the tensor product). The proper way to do this is to work in a certain category C  of modules over Uεres (namely, modules V such that V and and its dual have filtrations whose successive quotients are specializations of L(λ)’s), and to use the following remarkable result of Andersen [And92]. (For N = 1, it was established by Reshetikhin and Turaev [RT91]). 3.3. Theorem. — For all λ1 , . . . ,λr ∈ I, the tensor product I(λ1 ) ⊗ · · · ⊗ I(λr ) splits as a direct sum of modules I(λ) where λ ∈ I and a module Z such that tr(f ) = 0 for all f ∈ EndC  (Z). , be the submonoidal category of C  generated by the objects I(λ) with Now, let M , are submodules of tensor products of I(λ)’s and morphisms λ ∈ I. The objects of M are morphisms in C  . In order to get around the non-semisimplicity, we now form a quotient category M , by stipulating that a morphism f : A → B between two objects A, B of M , is of M zero in M if tr(gf ) = 0 for all morphisms g : B → A. In particular, if A is an object , is such that f = 0 in M for each object B and each morphism f ∈ Hom (A, B), of M  M then A is isomorphic to the zero object in M. This is so for the objects Z appearing in the statement of Theorem 3.3. The resulting category M is a modular category for the family (I(λ))λ∈I . The axiom of domination (3.2.3) of Chapter 6 is immediate as objects in M are direct sums of (I(λ))λ∈I , and the non-degeneracy axiom (invertibility of the matrix S of Section 3.2 of Chapter 6) follows from character computations due to Kac and Peterson (cf. [TW93]). Applying the constructions of Section 4 of Chapter 6 to M, we get numerical invariants of 3-manifolds and of links in 3-manifolds.

4. Quantum Groups with a Formal Parameter Most of the content of Chapters 4 and 7 holds for an arbitrary complex semisimple Lie algebra g. Such a Lie algebra was quantized by Drinfeld over the algebra C[[h]] of complex formal series (see [Dri85], [Dri87]). All C[[h]]-modules in the sequel are of the form ) ( vm hm | v0 , v1 , v2 , . . . ∈ V V = V [[h]] = m≥0

where V is a complex vector space. We define a topological tensor product on such C[[h]]-modules by , =VW V ⊗ ⊗ W. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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Out of the semisimple Lie algebra g, Drinfeld constructed a C[[h]]-algebra Uh g with the following properties. (i) As a C[[h]]-module, Uh g = U g[[h]].  Uh g, ε : Uh g → C[[h]], and S : Uh g → Uh g (ii) There are maps ∆ : Uh g → Uh g ⊗ turning Uh g into a topological Hopf algebra. (iii) As Hopf algebras, we have Uh g/hUh g = U g.  Uh g such that Rh ≡ 1 ⊗ 1 modulo h. (iv) There is a universal R-matrix Rh ∈ Uh g ⊗ (v) Any finite-dimensional g-module V extends uniquely to a Uh g-module V of the form V [[h]] such that V /hV = V as g-modules. The topological Hopf algebra Uh g is closely related to the quantum groups constructed in Chapter 4 when g = sl(N + 1). Setting q = exp( 12 h), we can consider Uh sl(N + 1) as an A-module where A = C[q, q −1 ]. In Section 4 of Chapter 4 we introduced a Hopf A-subalgebra UA of Uq sl(N + 1) in order to be able to specialize the parameter q. It is easy to check that UA is a Hopf A-subalgebra of Uh sl(N + 1). Let g again be an arbitrary semisimple Lie algebra. The category C h whose objects are the Uh g-modules V , where V is a finite-dimensional g-module, is a braided  defined above category with tensor product given by the topological tensor product ⊗ and with braiding induced by the universal R-matrix Rh . The unit object in C h is the trivial Uh g-module C[[h]]. It turns out that the category C h has a twist and that it is a ribbon category in the sense of Section 1 of Chapter 6. This allows us to construct a “quantum invariant” for framed oriented links for any semisimple Lie algebra g and any finite-dimensional g-module V as follows. We colour the components of a framed oriented tangle with the object V of C h and we proceed as in Section 2 of Chapter 6. We thus get a functor Fg,V : T 0 → C h , where T 0 is the category of framed oriented tangles, cf. Section 2.1 of Chapter 6. Any framed oriented link K can be viewed as an endomorphism of the unit object in the category T 0 . Therefore, its image Fg,V (K) belongs to EndC h (C[[h]]). The latter is clearly isomorphic to C[[h]]. Therefore, Fg,V (K) is an isotopy invariant Fg,V (K) =

 m≥0

with values in C[[h]].

` PANORAMAS ET SYNTHESES 5

Fg,V,m (K) hm

Chapter 8 VASSILIEV INVARIANTS OF LINKS

We now define a class of link invariants, called Vassiliev invariants of finite degree. The quantum invariants constructed in Section 4 of Chapter 7 provide examples of invariants of finite degree. A theorem due to Kontsevich expresses invariants of finite degree in terms of linear forms on certain diagrams, called chord diagrams.

1. Definition and Examples Vassiliev invariants are link invariants with values in an abelian group or, more generally, in a module over a commutative ring R. Let us fix such a ring R and consider the free R-module L(R) generated by the isotopy classes of framed oriented links in R3 . Clearly, any invariant P of framed oriented links in R3 with values in a R-module M extends uniquely to an R-linear map P  : L(R) → M such that for any framed oriented link K P  [K] = P (K), where [K] is the class of K in L(R). As a consequence, there is a bijection between the set of invariants of framed oriented links in R3 with values in M and the R-module HomR (L(R), M ). We now define for each nonnegative integer m a quotient module Lm (R) of L(R) such that HomR (Lm (R), M ) is in bijection with the set of Vassiliev invariants of degree ≤ m. In order to define Lm (R) we need a presentation of the R-module L(R) with more generators. We use the following concepts. A singular link is an immersion in R3 of the disjoint union of a finite number of oriented circles. We assume that the singular points of the immersion are finite in number and are all double points with transversal branches. If a singular link is equipped with a framing (resp. with an orientation), we say that it is framed (resp. oriented). We always assume that at any double point the branches have the same framing, i.e., have a common normal vector field. There is an obvious notion of isotopy of (framed) singular links generalizing the concept of isotopy for (framed) links. Such an isotopy carries double points onto double points.

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In the sequel we shall represent singular framed oriented links by planar diagrams whose framings are given by a vector field perpendicular to the plane and pointing to the reader. We represent a double point as in Figure 1.1.

Figure 1.1. Double point A double point of a singular framed oriented link K can be desingularized in two ways, yielding two singular framed oriented links K+ and K− with one double point less. The links K+ and K− are obtained by replacing the neighbourhood of the double point as in Figure 1.1 by the pictures of Figure 1.2.

K+

K−

Figure 1.2. The links K+ and K− Define L (R) as the R-module generated by the isotopy classes of singular framed oriented links in R3 modulo the relations of the form (1.1)

K = K+ − K−

where K is a singular link, and K+ and K− have been obtained by desingularizing a double point as above. Applying (1.1) repeatedly, we see that the class of any singular link in L (R) is a linear combination with integer coefficients of classes of links without double points, which shows that the natural map L(R) → L (R) is surjective. Actually, it is an isomorphism, which allows us to use the definition of L (R) as a presentation of L(R). The following definition now makes sense. For an integer m ≥ 0, we define Lm (R) as the quotient of L(R) by the submodule generated by all singular links with > m double points. 1.1. Definition. — Let P be an invariant of framed oriented links with values in an Rmodule M . The invariant P is a Vassiliev invariant of degree ≤ m if the induced R-module map L(R) → M factors through Lm (R). In other words, extending P to singular framed oriented links by Rule (1.1), i.e., by P (K) = P (K+ ) − P (K− ), the invariant P is of degree ≤ m if it vanishes on all singular links with > m double points. ` PANORAMAS ET SYNTHESES 5

1. DEFINITION AND EXAMPLES

83

The class in Lm (R) of a link K is the universal Vassiliev invariant of degree ≤ m with values in an R-module. In the next section we shall state a theorem by Kontsevich which gives a completely different description of Lm (R). It will allow us in Section 1.7 of Chapter 9 to construct Vassiliev invariants in a systematic way. Observe that an invariant of degree ≤ m is of degree ≤ k for all k ≥ m. On the level of the R-modules Lm (R), we see that they form a projective system · · · −→ Lm (R) −→ Lm−1 (R) −→ · · · −→ L0 (R).  We denote its limit by L(R). 1.2. Example. — The Jones polynomial V with values in the Laurent polynomial ring Z[q, q −1 ] gives rise to Vassiliev invariants as follows. Recall from Theorem 2.6 of Chapter 5 that (1.2)

q −2 V (K+ ) − q 2 V (K− ) = (q − q −1 )V (K0 )

whenever (K+ , K− , K0 ) is a Conway triple of links, as defined in Section 2 of Chapter 5. Relation (1.2) implies that, if K represents a singular link obtained from K+ or K− by replacing a crossing by a double point as in Figure 1.1, then (1.3)

V (K) = V (K+ ) − V (K− ) = (1 − q −4 )V (K+ ) + q −1 (1 − q −2 )V (K0 ).

We now replace the variable q by a formal series in a variable h, with constant term 1 and with coefficients in a commutative ring R. After this change of variables, the invariant V takes its values in the formal series ring R[[h]]. Then (1.3) implies that (1.4)

V (K) = f (h)V (K+ ) + g(h)V (K0 )

where f (h) and g(h) are formal series divisible by h. By an easy induction on the number of double points, it follows from (1.4) that the value of V on a singular link with > m double points is divisible by hm+1 . Therefore, the class of the formal series V (K) modulo hm+1 is a Vassiliev invariant of degree ≤ m. In the special case q = 1 + h, the previous considerations imply that the value at q = 1 of the m-th derivative of the Jones polynomial is an invariant of degree ≤ m. In Section 4 of Chapter 7 we constructed a “quantum invariant” Fg,V (K) =



Fg,V,m (K) hm

m≥0

of framed oriented links for any complex semisimple Lie algebra g and any finitedimensional g-module V . In Section 3.3 we shall prove the following, which provides a huge set of Vassiliev invariants. 1.3. Proposition. — For all m ≥ 0, the link invariant F g,V ,m is a Vassiliev invariant of degree ≤ m. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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2. Chord Diagrams and Kontsevich’s Theorem  In this section we determine the modules Lm (R) and L(R) in terms of certain elementary diagrams drawn on circles. 2.1. Definition. — Let C1 , . . . ,Cn be oriented circles, each equipped with an element of Z/2, called its residue. A chord diagram is a finite (possibly empty) set of unordered pairs of points on C1 , . . . ,Cn , all points being distinct. By a homeomorphism of chord diagrams, we mean an orientation-preserving homeomorphism of the union of the circles preserving the distinguished pairs of points and the residues. In our pictures, we shall draw a dashed line, called a chord, between the two points of a distinguished pair. Let D be a chord diagram and let a, b, c be three distinct points of the underlying circles, not belonging to the distinguished pairs of points of D. Let a , b be points obtained by slightly pushing a, b along the corresponding circles in the directions given by the orientations. Let D1 and D2 be the chord diagrams obtained from D by adding the pairs of points {a, b}, {a, c} and {a, c}, {a, b} respectively. Let D3 and D4 be the chord diagrams obtained from D by adding the pairs of points {a, b}, {b, c} and {a, b }, {b, c} respectively. The diagrams D1 , D2 , D3 , D4 are schematically presented in Figure 2.1. It is understood that they are identical except for the parts shown in the figure, where the leftmost intervals have equal residues, the middle intervals have equal residues, and the rightmost intervals also have equal residues.

a

b

a

a c

D1

a

c b

D2

a

b b

b c

D3

a

c b

D4

Figure 2.1. The chord diagrams D1 , D2 , D3 , D4 Given a commutative ring R, we define the R-module Am (R) as the R-module generated by all chord diagrams with exactly m chords, subject to all relations of the form (2.1)

D1 − D2 + D3 − D4 = 0

where D is a chord diagram with (m − 2) chords and D1 , D2 , D3 , D4 are obtained from D as above. Relation (2.1) is called the four-term relation between chord diagrams. We may now formulate the main theorem on Vassiliev invariants. It is due to Kontsevich, see [Kon93], [BN95]. (We actually give an improved version, taking care of the framings.) ` PANORAMAS ET SYNTHESES 5

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85

2.2. Theorem. — If the ring R contains the field Q of rational numbers, then for m = 0,1,2, . . . there is an isomorphism of R-modules Lm (R) ∼ =

m +

Ai (R)

i=0

such that the square Lm (R)    

m +

∼ =

i=0 m−1 +

Lm−1 (R) ∼ =

Ai (R)     Ai (R)

i=0

commutes for all m = 0,1,2, . . .. In the above square, the vertical maps are the obvious projections. Theorem 2.2 follows from a more general result, namely Theorem 4.3. As a consequence of the theorem, the Vassiliev invariants of degree ≤ m are in bijection with the linear maps on A0 (R) × · · · Am (R). 2.3. Corollary. — If R ⊃ Q, then there is an isomorphism of R-modules ∼  L(R) =

∞ !

Am (R).

m=0

3. The Pro-Unipotent Completion of a Braided Category We have seen in the previous chapters that the language of monoidal categories is extremely powerful when one wants to construct link invariants. The aim of this section is first to introduce the concept of the pro-unipotent completion of a braided category, then to show how the pro-unipotent completion of the category T0 of framed  oriented tangles naturally leads to the modules Lm (R) and L(R) defined in Section 1. We start with a few definitions. Let R be a commutative ring. We say that a monoidal category C is R-linear if the Hom-sets in C are R-modules and if the composition and the tensor product of morphisms are R-bilinear. By an ideal in an R-linear monoidal category C we mean a class I of morphisms in C satisfying the following condition: for all morphisms f , g in C such that at least one of them belongs to I, the morphisms f ◦ g (when it is defined) and f ⊗ g belong to I. We also assume that the set I(V, W ) of morphisms from V to W belonging to I is an R-submodule of HomC (V, W ). We say that an ideal I of C is generated by a set of morphisms of C if I is the smallest ideal containing that set. Given an integer m ≥ 0, we define I m+1 as the ideal generated by morphisms of the form f1 6 · · · 6 fn where 6 is either the composition or the tensor product of morphisms and at least m + 1 morphisms among the fi ’s belong to I. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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Given an ideal I in C, we define the quotient category C/I as the category whose objects are the same as the objects of C and whose morphisms are given by HomC/I (V, W ) = HomC (V, W )/I(V, W ) for all objects V , W of C. It is clear that C/I is an R-linear monoidal category. We now come to the main concept in this section. 3.1. Definition. — Let C be an R-linear braided category. The augmentation ideal of C is the ideal I generated by the morphisms of the form cW ,V cV ,W − idV ⊗W for all objects V ,W . We call pro-unipotent completion of C the projective limit of the projective system of categories · · · −→ C/I m+1 −→ C/I m −→ · · · −→ C/I. Let us consider the tangle category T0 of framed oriented tangles. We take its Rlinearization T0 (R) which has the same objects as T0 and whose morphisms are freely generated by the morphisms of T0 , i.e.,   HomT0 (R) (s, s ) = R HomT0 (s, s ) for all objects s and s . This is a R-linear braided category and we may consider its augmentation ideal I, the quotient categories T0 (R)/I m+1 , and the pro-unipotent completion T0 (R). Let us relate these categories to the universal Vassiliev modules Lm (R) of Section 1. We know that the empty sequence ∅ is the unit object of the monoidal categories T0 , T0 (R), T0 (R)/I m+1 , and T0 (R). We also know that the endomorphism set of the object ∅ in the category T0 is the set of isotopy classes of framed oriented links in R3 . Linearizing, we see that the endomorphism set of the same object ∅ in the linearized category T0 (R) is the R-module L(R) of Section 1. This generalizes to the categories T0 (R)/I m+1 and T0 (R), as witnessed by the following result, which we leave as an exercise to the reader. 3.2. Proposition. — The endomorphism sets of the unit object ∅ in the categories  respectively: T0 (R)/I m+1 and T0 (R) are the R-modules Lm (R) and L(R), EndT0 (R)/I m+1 (∅) = Lm (R)

and

 EndT (R) (∅) = L(R). 0

3.3. Proof of Proposition 1.3. — Recall from Section 4 of Chapter 7 that the quantum invariant  Fg,V (K) = Fg,V,m (K) hm m≥0

is obtained as the restriction to the endomorphism set of the unit object ∅ of a functor F : T0 → Ch such that F ((+)) = V . Let us show that F factors through T0 (C). It certainly factors through the C-linearization T0 (C) since the category Ch is C-linear. ` PANORAMAS ET SYNTHESES 5

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87

We know that the braiding c in Ch is given by a universal R-matrix Rh such that Rh ≡ 1 ⊗ 1 modulo h, which implies that c − c−1 ≡ 0 modulo h. In other words, the augmentation ideal I of Ch sits in the ideal (h) generated by the morphisms in Ch divisible by h. After completion we get a functor m+1 m+1 Ch = ← lim −→ lim ). − Ch /I ←− Ch /(h m≥0

m≥0

Since the quantum enveloping algebra Uh g is h-adically complete, we have m+1 ∼ lim ) = Ch . ←− Ch /(h

m≥0

By functoriality of the pro-unipotent completion, the diagram T0

F

−−→ T0 (C) −−−−→     F T0 (C) −−−−→

Ch   

========

Ch

m+1 Ch −−−−→ lim ) ←− Ch /(h m≥0

commutes. This proves that F factors through T0 (C). Similarly, the class modulo (hm+1 ) of F factors through T0 (C)/I m+1 . By Proposition 3.2 this implies that the m  truncated sum Fg,V,k (K) hk is a Vassiliev invariant of degree ≤ m. For the same k=0 m−1  Fg,V,k (K) hk is of degree ≤ m − 1. It follows that the m-th coefficient reason, k=0

Fg,V,m is of degree ≤ m.

4. Another description of T
0 (R) 4.1. Chord diagrams on curves. — Theorem 2.2 and Corollary 2.3 have a categorical counterpart which we present in this section. To begin with, we give a categorical version of the chord diagrams of Section 2. Let k,  be non-negative integers. A (k, )-curve Γ is a compact oriented one-dimensional manifold (i.e., a disjoint union of a finite number of oriented intervals and oriented circles) such that each connected component is equipped with an element of the group Z/2, called its residue, and such that the boundary ∂Γ is decomposed as a disjoint union of two totally ordered sets U and V with card(U ) = k and card(V ) = . The elements of U are called bottom free ends or inputs of Γ. The given order in U allows us to enumerate these free ends by the numbers 1, . . . , k. Thus, we may speak of the first, second, etc., inputs of Γ. Similar remarks apply to elements of V which are called top free ends or outputs of Γ. By a homeomorphism of curves, we mean an orientation-preserving homeomorphism that respects the residue, the splitting of the boundary into inputs and outputs, and the order in the sets of inputs and outputs. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1997

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The source s(Γ) of a (k, )-curve Γ is the sequence (ε1 , . . . , εk ) where εi corresponds to the orientation of the interval attached to the i-th input, i.e., εi = + if this interval is oriented towards the i-th input and εi = − otherwise. If k = 0, then the source of Γ is the empty sequence. We define the target t(Γ) of Γ as the sequence (η1 , . . . , η ) where ηj = − if the interval attached to the j-th output is oriented towards it and ηj = + otherwise. If  = 0, then the target of Γ is the empty sequence. A chord diagram on a curve Γ is a finite (possibly empty) set of unordered pairs of points on Γ\∂Γ, all points being distinct. By a homeomorphism of chord diagrams, we mean a homeomorphism of the underlying curves preserving the distinguished pairs of points. We define the source and the target of a chord diagram as the source and the target of the underlying curve, respectively. In our pictures (such as Figure 2.1), (k, )-curves are drawn inside a horizontal strip, the k inputs lying on the bottom boundary line of the strip with the order increasing from left to right, and the  outputs lying on the top boundary line of the strip with the order also increasing from left to right. We draw a dashed line, called a chord, between the two points of a distinguished pair. 4.2. The category A(R). — Fix a commutative ring R. Out of the homeomorphism classes of chord diagrams we build a category A(R) as follows. The objects of A(R) are the same as the objects of the category T0 of framed oriented tangles, namely finite sequences of + and −. A morphism from s to s in A(R) is an element of the R-module generated by the homeomorphism classes of chord diagrams with source s and target s subject to Relation (2.1), which also makes sense in the present context. The identity morphisms in A(R) are represented by chordless diagrams consisting of intervals with residue 0 such that for each interval one boundary point is an input and the other one is an output with the same numbering. The identity endomorphism of the empty sequence is represented by the empty curve. The composition of two chord diagrams D and D is obtained by gluing D with D along the (unique) orderpreserving homeomorphism of the set of inputs of D onto the set of outputs of D. (In the pictures D ◦ D is obtained by placing D on the top of D and gluing the corresponding ends, as in the case of the tangle category.) The residue r ∈ Z/2 of a connected component C of the composition D ◦ D is defined by the formula (4.1)

r=



r(αi ) +

i

 j

r(βj ) +

 i