Quantum groups

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IRMA Lectures in Mathematics and Theoretical Physics 12 Edited by Vladimir G. Turaev

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

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IRMA Lectures in Mathematics and Theoretical Physics Edited by Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other materials arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 1 2 3 4 5 6 7 8 9 10 11

Deformation Quantization, Gilles Halbout (Ed.) Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi (Eds.) Three courses on Partial Differential Equations, Eric Sonnendrücker (Ed.) Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher (Ed.) Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schaefke (Eds.) Physics and Number Theory, Louise Nyssen (Ed.) Handbook on Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.)

Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

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Quantum Groups Benjamin Enriquez Editor

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Editor: Benjamin Enriquez Institut de Recherche Mathématique Avancée Université Louis Pasteur et CNRS (UMR 7501) 7, rue René Descartes 67084 Strasbourg Cedex France E-mail: [email protected]

2000 Mathematics Subject Classification (primary; secondary): 81R50; 81R12 Key words: Tensor categories, integrable systems, knot invariants

ISBN 978-3-03719-047-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2008 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

Roughly twenty years after their appearance on the mathematical scene, quantum groups have played a significant role in various branches of mathematics. This volume contains one lecture course and three research expositions devoted to various aspects of this subject. We briefly introduce these contributions and relate them to some of the main themes of the theory. The first contribution, “Lectures on tensor categories” provides an introduction to tensor categories. It is based on the lecture notes (by D. Calaque) of a course taught by P. Etingof in Strasbourg in June 2003. While the origins of the subject of tensor categories lie in the Tannaka–Krein duality for compact groups, it was after the construction by Reshetikhin and Turaev of invariants of 3-manifolds using a tensor category obtained from the quantum group Uq .sl2 / for q a root of unity, and the interpretation by Drinfeld of Kohno’s computation of the monodromy of the Knizhnik–Zamolodchikov differential system in terms of quasi-Hopf algebras that its relations with the circle of problems called “the theory of quantum groups” were firmly established. Some of the main subsequent results in this theory – notably the Kazhdan–Lusztig theorem identifying tensor categories arising from rational conformal field theories and from quantum groups, and the solution by Etingof and Kazhdan of the quantization problem of Lie bialgebras – were based on these relations. The lectures require only minimal prerequisites in category theory and in Hopf algebras, and lead the reader to topics of recent research, like the program of classification of tensor categories. The second contribution, “The Drinfeld associator of gl.1j1/”, by J. Lieberum, deals with associators and Vassiliev invariants of knots. The subject of associators started in 1989–90 with their introduction by Drinfeld as universal objects for the construction of quasitriangular quasi-Hopf algebras; it was related to Vassiliev invariants by the Kontsevich integral. While the set of associators .; ˆ/ 2 C  exp.f2 / is “large” (in bijection with the prounipotent version of the Grothendieck–Teichmüller group; here f2 is the topologically free complex Lie algebra with two generators), its image under “standard” specialization maps is usually small; e.g., it follows from Drinfeld’s theorems on quasi-Hopf algebras that the images of associators (with  ¤ 0 fixed) in U.g/˝3 ŒŒ„ (where g is a simple Lie algebra) are all related by twists by elements of .U.g/˝2 /g ŒŒ„ . In his contribution (a part of his Habilitationsschrift), Lieberum studies the image of associators in U.g/˝3 ŒŒ„, where g is the Lie superalgebra gl.1j1/. It is shown that the images of the even associators (again with  ¤ 0 fixed; even means ˆ.A; B/ D ˆ.A; B/) are all the same. For this, the specialization map to gl.1j1/ is shown to factor through certain quotients of the algebras of trivalent diagrams, in which the uniqueness of even associators is proved. This result is supplemented with an explicit formula for this even associator. In the second part of the paper, the author relates the associator of gl.1j1/ with Viro’s generalization of the multivariable Alexander polynomial.

vi The third contribution is “Integrable systems associated with elliptic algebras”, by Odesskii and Rubtsov. The elliptic algebras here are generalizations, developed by Feigin–Odesskii, of Sklyanin’s elliptic algebras. The main purpose of this contribution is the construction of new integrable systems in elliptic algebras. The authors use a general method of construction, which was rediscovered several times in various contexts, and is related to separation of variables; its main input is a family of elements in an algebra satisfying certain commutation condition; the commuting hamiltonians are then ratios of noncommutative (“Cartier–Foata”) determinants. While in previous applications of this method the commutation relations were ensured by the fact that the elements belonged to different factors of the tensor power of an algebra (whence the relation to separation of variables), the reasons why the commutation relations are satisfied here are different and are related to the structure of elliptic algebras. The resulting integrable systems generalize the antiperiodic solid-on-solid (SOS) models, which are based on Felder’s elliptic quantum group E; .sl2 /. The fourth contribution “On the automorphisms of UqC .g/”, by N.Andruskiewitsch and F. Dumas, deals with a problem of pure algebra: to compute the group of algebra automorphisms of the “nilpotent part” UqC .g/ of a quantum enveloping algebra Uq .g/. While a complete solution had been obtained by Alev and Dumas for g of type A2 , the authors obtain significant partial results in the case where g is of type B2 . Two of these contributions (the second and the third) were presented during the 72th RCP meeting “Quantum Groups” in Strasbourg (June 2003). I wish to end this preface by thanking Vladimir Turaev for the very pleasant atmosphere during our joint organization of the meeting in Strasbourg, and all the referees for their invaluable help with this volume. Strasbourg, May 2008

Benjamin Enriquez

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Damien Calaque and Pavel Etingof Lectures on tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Jens Lieberum The Drinfeld associator of gl(1|1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Alexander Odesskii and Vladimir Rubtsov Integrable systems associated with elliptic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Nicolás Andruskiewitsch and François Dumas On the automorphisms of Uq+ (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Lectures on tensor categories Damien Calaque and Pavel Etingof IRMA (CNRS ) 7 Rue René Descartes, 67084 Strasbourg, France email: [email protected] Department of Mathematics, MIT Cambridge, MA 02138, USA email: [email protected]

Abstract. We give a review of some recent developments in the theory of tensor categories. The topics include realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius– Perron dimensions, and classification of tensor categories.

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1 Finite tensor and fusion categories . . . . . . . . . . 1.1 Basic notions . . . . . . . . . . . . . . . . . . 1.2 Finite tensor and fusion categories . . . . . . . 1.3 Fusion rings . . . . . . . . . . . . . . . . . . 2 Ocneanu rigidity . . . . . . . . . . . . . . . . . . . 2.1 Main results . . . . . . . . . . . . . . . . . . 2.2 Module categories . . . . . . . . . . . . . . . 2.3 Weak Hopf algebras . . . . . . . . . . . . . . 2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . 3 Morita theory, modular categories, and lifting theory 3.1 Morita theory in the categorical context . . . . 3.2 Modular categories and the Verlinde formula . 3.3 Lifting theory . . . . . . . . . . . . . . . . . . 4 Frobenius–Perron dimension . . . . . . . . . . . . . 4.1 Definition and properties . . . . . . . . . . . . 4.2 FP-dimension of the category . . . . . . . . . 4.3 Global and FP-dimensions . . . . . . . . . . . 4.4 Classification . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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2 3 3 7 10 12 12 13 16 19 21 21 25 27 29 29 32 33 35 37

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Damien Calaque and Pavel Etingof

Introduction This paper is based on the first author’s lecture notes of a series of four lectures given by the second author in June 2003 for the Quantum Groups seminar at the Institut de Recherche Mathématique Avancée in Strasbourg. It is about the structure and classification of tensor categories. We always work over an algebraically closed field k. By a tensor category over k, we mean an abelian rigid monoidal category in which the neutral object 1 is simple (i.e., does not contain any proper subobject), the vector spaces Hom.X; Y / are finite dimensional and all objects are of finite length. The category of finite dimensional vector spaces Vectk , the categories of finite dimensional representations of a group G, a Lie algebra g, or a (quasi-)Hopf algebra H (respectively denoted Rep G, Rep g and Rep H ), or the category of integrable modules over an affine Lie algebra gO with fusion product (which can also be obtained from quantum groups) are all tensor categories in this sense. Tensor categories appear in many areas of mathematics such as representation theory, quantum groups, conformal field theory (CFT) and logarithmic CFT, operator algebras, and topology (invariants of knots and 3-manifolds). The goal of this paper is to give an introduction to some recent developments in this subject. The paper is subdivided into four sections, each representing a single lecture. Section 1 introduces the main objects of the paper. We recall basic categorical definitions and results, fix the vocabulary, and give examples (for more details, we recommend the monographies [BaKi], [Mac], [K], [Tu]). The end of the section is devoted to the problem of realizability of fusion rings: examples are given, and the Ocneanu rigidity conjecture is formulated. The goal of Section 2 is to prove the Ocneanu rigidity for fusion categories in characteristic zero. To do this, we introduce and discuss the notions of module categories and weak Hopf algebras. The more technical part of the proof is done at the end of the section. Section 3 is about three distinct subjects. We start with a closer look at module categories, discussing the notion of Morita equivalence for them, and applying general results to the representation theory of groups. Then we recall well-known facts about braided, ribbon and modular categories. Finally, the lifting theory is presented: it allows us to extend some results from characteristic zero to the positive characteristic case. Section 4 covers the theory of Frobenius–Perron dimension, and its applications to classification results for fusion categories. We end this paper with two interesting open problems.

Remarks. 1. Being a set of lecture notes, this paper does not contain original results. Most of the results are taken from the papers [ENO], [EO1], [O1], [O2], [O3] and references therein, including the standard texts on the theory of tensor categories.

3

Lectures on tensor categories

2. To keep this paper within bounds, we had to refrain from a thorough review of the history of the subject and of the original references, as well as from a detailed discussion of the preliminaries. We also often had to omit complete proofs. For all this material we refer the reader to books and papers listed in the bibliography. Acknowledgements. The authors are grateful to the participants of the lectures: P. Baumann, B. Enriquez, F. Fauvet, C. Grunspan, G. Halbout, C. Kassel, V. Turaev, and B. Vallette. Their interest and excitement made this paper possible. The work of the first author was partially supported by the NSF grant DMS-9988786. The second author is greatly indebted to D. Nikshych and V. Ostrik for teaching him much of the material given in these lectures. He is also grateful to IRMA (Strasbourg) for hospitality. His work was partially supported by the NSF grant DMS-9988786.

1 Finite tensor and fusion categories 1.1 Basic notions 1.1.1 Definitions. Let C be a category. Recall that C is additive over k if (i) Hom.X; Y / is a (finite dimensional) k-vector space for all X; Y 2 Obj.C /; (ii) the map Hom.Y; Z/  Hom.X; Y / ! Hom.X; Z/; .'; / 7! ' B for all X; Y; Z 2 Obj.C/;

is k-linear

(iii) there exists an object 0 2 Obj.C / such that Hom.0; X/ D Hom.X; 0/ D 0 for all X 2 Obj.C/; (iv) finite direct sums exist. Remark. When we deal with functors between additive categories, we always assume they are also additive. Further, recall that an additive category C is abelian if (i) every morphism  W X ! Y has a kernel Ker  (an object K together with a monomorphism K ! X ) and a cokernel Coker  (an object C together with an epimorphism Y ! C ); (ii) every morphism is the composition of an epimorphism followed by a monomorphism; (iii) for every morphism ' one has Ker ' D 0 Coker ' D 0 H) ' D Coker.Ker '/.

H)

' D Ker.Coker '/ and

It is known that C is abelian if and only if it is equivalent to a category of (right) comodules over a coalgebra. Recall also that C is monoidal if there exists

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Damien Calaque and Pavel Etingof

(i) a bifunctor ˝ W C  C ! C , (ii) a functorial isomorphism ˆ W . ˝ / ˝  !  ˝ . ˝ /, (iii) an object 1 (called the neutral object) and two functorial isomorphisms W 1 ˝  !  ;

 W  ˝1 ! 

.the unit morphisms/;

such that for any two functors obtained from ˝  ˝ by inserting 1’s and parentheses, all functorial isomorphisms between them composed of ˆ˙1 ’s, ˙1 ’s and ˙1 ’s are equal. Remark. In the spirit of the previous remark, for additive monoidal categories we assume that ˝ is biadditive. Theorem 1.1 (MacLane coherence, [Mac]). The data .C ; ˝; ˆ; ; / with (i), (ii) and (iii) is a monoidal category if and only if the following properties are satisfied: (1) Pentagon axiom. The following diagram is commutative: .. ˝ / ˝ / ˝  ˆ12;3;4

ˆ1;2;3 ˝ id /

. ˝ . ˝ // ˝ 

ˆ1;23;4



. ˝ / ˝ . ˝ /

/  ˝ .. ˝ / ˝ / 

ˆ1;2;34

id˝ˆ2;3;4

/  ˝ . ˝ . ˝ // .

(2) Triangle axiom. The following diagram commutes: ˆ;1;

/  ˝ .1 ˝ / . ˝ 1/ ˝ R RRR RRR RR id˝ ˝ id RRRR )  ˝.

A monoidal category is called strict if .X ˝ Y / ˝ Z D X ˝ .Y ˝ Z/, 1 ˝ X D X ˝ 1 D X, and the associativity and unit isomorphisms are equal to the identity. A theorem also due to Maclane (see [K]) says that any monoidal category is equivalent to a strict one. In view of this theorem, we will always assume that the categories we are working with are strict, unless otherwise specified. Recall that a right dual for X 2 Obj.C / is an object X  with two morphisms eX W X  ˝ X ! 1 and iX W 1 ! X ˝ X  (called the evaluation and coevaluation morphisms) satisfying the following two equations: (i) .idX ˝ eX / B .iX ˝ idX / D idX and (ii) .eX ˝ idX  / B .idX  ˝ iX / D idX  . A left dual X with maps eX0 W X ˝ X ! 1 and iX0 W 1 ! X ˝ X is defined in the same way.

Lectures on tensor categories

5

One can show that if it exists, the right (left) dual is unique up to a unique isomorphism compatible with evaluation and coevaluation maps. A monoidal category is called rigid if any object has left and right duals. Definition 1.2. A tensor category is a rigid abelian monoidal category in which the object 1 is simple and all objects have finite length. Example 1.3. The category Rep H of finite dimensional representations of a quasiHopf algebra H is a tensor category [Dr]. This category is, in general, not strict (although it is equivalent to a strict one): its associativity isomorphism is given by the associator of H . Proposition 1.4 ([BaKi]). In a tensor category, the tensor product functor ˝ is (bi)exact. 1.1.2 The Grothendieck ring of a tensor category. Let C be a tensor category over k. Definition 1.5. The Grothendieck ring Gr.C / of C is the ring whose basis over Z is the set of isomorphism classes of simple objects, with multiplication given by X Z NXY Z; X Y D Z simple

where NXZY D ŒX ˝ Y W Z is the multiplicity (the number of occurrences) of Z in X ˝ Y (which is well defined by the Jordan–Hölder theorem). Examples 1.6. (i) C D RepC SL.2/. Simple objects are highest weight representations Vj (of highest weight j 2 Z), and the structure constants of the Grothendieck ring are given by the Clebsch–Gordan formula Vi ˝ Vj D

iCj X

Vk

kDjij j kiCj mod 2

(ii) C is the category of integrable modules (from category O) over the affine c2 at level l with the fusion product algebra sl Vi ˝ Vj D

ljiCj Xlj

Vk

kDjij j kiCj mod 2

In this case Gr.C/ is a Verlinde algebra. (iii) C D Repk Fun.G/ for a finite group G. Simple objects are evaluation modules Vg , g 2 G, and Vg ˝ Vh D Vgh . So Gr.C / D ZŒG.

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Damien Calaque and Pavel Etingof

More generally, pick a 3-cocycle ! 2 Z 3 .G; C /. To this cocycle we can attach a twisted version C .G; !/ of C: all the structures are the same, except the associativity isomorphism which is given by ˆVg ;Vh ;Vk D !.g; h; k/id (and the morphisms ,  are modified to satisfy the triangle axiom). The cocycle condition !.h; k; l/ !.g; hk; l/ !.g; h; k/ D !.gh; k; l/ !.g; h; kl/ is equivalent to the pentagon axiom. Again we have Gr.C .G; !// D ZŒG. (iv) C D RepC S3 . The basis elements (simple objects) are 1, , V , with product given by  ˝  D 1,  ˝ V D V ˝  D V and V ˝ V D V ˚ 1 ˚ . (v) If C D Rep G for G a unipotent algebraic group over C, then the unique simple object is 1, hence Gr.C/ D Z. In this case, the Grothendieck ring does not give a lot of information about the category because the category is not semisimple. (vi) C D Rep H for the 4-dimensional Sweedler Hopf algebra H , which is generated by g and x, with relations gx D xg, g 2 D 1, x 2 D 0, and the coproduct  given by g D g ˝ g and x D x ˝ g C 1 ˝ x. In this case the only simple objects are 1 and , with  ˝  D 1. 1.1.3 Tensor functors. Let C and D be two tensor categories. A functor F W C ! D is called quasi-tensor if it is exact and equipped with a functorial isomorphism J W F . ˝ / ! F ./ ˝ F ./ and an isomorphism u W F .1/ ! 1. Such a functor defines a morphism of unital rings Gr.C / ! Gr.D/. A quasi-tensor functor F W C ! D is tensor if the diagrams F .. ˝ / ˝ / F .ˆC /

J 12;3



F . ˝ . ˝ //

J 1;23

/ F . ˝ / ˝ F ./ / F ./ ˝ F . ˝ /

F .1 ˝ / F .C /

J1;



F ./ o

J ˝ id



id˝J

ˆD

/ F ./ ˝ .F ./ ˝ F .//,

/ F .1/ ˝ F ./ u˝ id



D

/ .F ./ ˝ F .// ˝ F ./

1 ˝ F ./,

and F . ˝ 1/ F .C /

J;1



F ./ o

/ F ./ ˝ F .1/ 

D

id˝ u

F ./ ˝ 1

are commutative. An equivalence of tensor categories is a tensor functor which is also an equivalence of categories.

Lectures on tensor categories

7

Example 1.7. Let !; ! 0 2 Z 3 .G; k / and ! 0 =! D d is a coboundary. Then  defines a tensor structure on the identity functor C .G; ! 0 / ! C .G; !/: the coboundary condition ! 0 .g; h; k/ .h; k/ .g; hk/ D .gh; k/ .g; h/ !.g; h; k/ is equivalent to the commutativity of the previous diagram. Moreover, it is not difficult to see that this tensor functor is in fact an equivalence of tensor categories. Thus the fusion category C .G; !/, up to equivalence, depends only on the cohomology class of !. In particular, we may use the notation C .G; !/ when ! is not a cocycle but a cohomology class.

1.2 Finite tensor and fusion categories 1.2.1 Definitions and examples Definition 1.8. An abelian category C over k is said to be finite if (i) C has finitely many (isomorphism classes of) simple objects, (ii) any object has finite length, and (iii) any simple object admits a projective cover. This is equivalent to the requirement that C D Rep A as an abelian category for a finite dimensional k-algebra A. Definition 1.9. A fusion category is a semisimple finite tensor category. Examples 1.10. In examples 1.6, (i) is semisimple but not finite, (ii), (iii) and (iv) are fusion, (v) is neither finite nor semisimple, and (vi) is finite but not semisimple. Recall that if C and D are two abelian categories over k, then one can define their Deligne external product C  D. Namely, if C D A-Comod and D D B-Comod are the categories of comodules over coalgebras A and B then C  D ´ A ˝ B-Comod. If C and D are semisimple, the Deligne product is simply the category whose simple objects are X  Y for simple X 2 Obj.C / and Y 2 Obj.D/. If C and D are tensor/finite tensor/fusion categories then C  D also has a natural structure of a tensor/finite tensor/fusion category (in the semisimple case it is simply given by .X  Y / ˝ .X 0  Y 0 / ´ .X ˝ Y /  .X 0 ˝ Y 0 /). 1.2.2 Reconstruction theory (Tannakian formalism). Let H be a (quasi-)Hopf algebra and consider C D Rep H , the category of its finite dimensional representations. The forgetful functor F W C ! Vect k has a (quasi-)tensor structure (the identity morphism). In addition, this functor is exact and faithful. A functor C ! Vectk with such properties ((quasi-)tensor, exact, and faithful) is called a (quasi-)fiber functor.

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Reconstruction theory tells us that every finite tensor category equipped with a (quasi-)fiber functor is obtained in this way, i.e., can be realized as the category of finite dimensional representations of a finite dimensional (quasi-)Hopf algebra. Namely, let .C; F / be a finite tensor category equipped with a (quasi-)fiber functor, and set H D End.F /. Then H carries a coproduct  defined as follows:  W H ! H ˝ H D End.F  F /;

T 7! J B T B J 1 :

Moreover, one can define a counit W H ! k by .T / D T jF .1/ and an antipode S W H ! H by S.T /jF .X / D .T jF .X  / / (in the quasi-case this depends on the choice of the identification jX W F .X/ ! F .X  /). This gives H a (quasi-)Hopf algebra structure (the choice of jX has to do with Drinfeld’s special elements ˛; ˇ 2 H ). Thus we have bijections: Finite tensor categories with quasi-fiber functor up to equivalence and changing quasi-tensor structure of the functor

!

Finite dimensional quasi-Hopf algebras up to isomorphism and twisting

Finite tensor categories with fiber functor up to equivalence

!

Finite dimensional Hopf algebras up to isomorphism

1.2.3 Braided and symmetric categories. Let C be a monoidal category with a functorial isomorphism W  ˝ !  ˝op , where X ˝op Y ´ Y ˝ X. For given objects V1 ; : : : ; Vn in C , we consider an expression obtained from Vi1 ˝    ˝ Vin by inserting 1’s and parentheses, and where .i1 ; : : : ; in / is a permutation of f1; : : : ; ng. To any composition ' of ˆ’s, ’s, ’s, ’s and their inverses acting on it, we assign an element of the braid group Bn as follows: assign 1 to ˆ,  and , and the generator k of Bn to Vk VkC1 . Definition 1.11. A braided monoidal category is a monoidal category as above such that the '’s depend only on their images in the braid group. Again, we have a coherence theorem for braided categories: Theorem 1.12 ([JS]). The data .C ; ˝; 1; ˆ; ; ; / define a braided category if and only if .ˆ; ˛/ satisfy the Hexagon axioms: the diagrams .12/3 ˝ id

ˆ



.21/3

ˆ

/ 1.23/ / 2.13/

1;23

/ .23/1 

id˝ 

ˆ

/ 2.31/

9

Lectures on tensor categories

and .12/3  1 ˝ id

ˆ



.21/3

ˆ

/ 1.23/ / 2.13/

1 1;23

/ .23/1 

id˝  1

ˆ

/ 2.31/

are commutative. Remark. “2(31)” is short notation for the 3-functor .V1 ; V2 ; V3 / 7! V2 ˝ .V3 ˝ V1 /. To get the definition of a symmetric monoidal category, the reader just has to replace the braid group Bn by the symmetric group Sn in the definition. To say it in another way, a symmetric monoidal category is a braided one for which satisfies

V W B W V D idV ˝W . Example 1.13. Let H be a quasi-triangular bialgebra (resp. Hopf algebra), i.e., a bialgebra (resp. Hopf algebra) with an invertible element R 2 H ˝ H satisfying op .x/ D R.x/R1 , .id ˝ /.R/ D R13 R12 and . ˝ id/.R/ D R13 R23 . Then Rep H is a braided monoidal (resp. rigid monoidal, i.e., tensor) category with braiding

V W W a˝b 7! R21 .b ˝a/. Moreover, axioms for R are equivalent to the requirement that Rep H is braided (it is not difficult to show that the first equation satisfied by R is equivalent to the functoriality of , and the two others are equivalent to the Hexagon relations). If R is triangular, i.e., RR21 D 1 ˝ 1 (in particular if H is cocommutative), then Rep H becomes a symmetric monoidal (resp. tensor) category. 1.2.4 The Drinfeld center. Tannakian formalism tells us that there is a strong link between finite tensor categories and Hopf algebras. So it is natural to ask for a categorification of the notion of the Drinfeld double for Hopf algebras. Definition 1.14. The Drinfeld center Z.C / of a tensor category C is a new tensor category whose objects are pairs .X; ˆ/, where X 2 Obj.C / and ˆ W X ˝ ! ˝X is a functorial isomorphism such that ˆY ˝Z D .id ˝ ˆZ / B .ˆY ˝ id/, and with morphisms defined by Hom..X; ˆ/; .Y; ‰// ´ ff 2 Hom.X; Y / j .f ˝id/BˆZ D ‰Z B.id˝f / for all Zg: Proposition 1.15. Z.C / is a braided tensor category, which is finite if C is. Proof. See [K] for the proof (the finiteness statement can be found for example in [EO1]). Let us just note that the tensor product of objects is given by .X; ˆ/˝.Y; ‰/ D .X ˝ Y; ƒ/, where ƒ.Z/ D .ˆ.Z/ ˝ idY / B .idX ˝ ‰.Z//, the neutral object by .1; id/, and the braiding by .X;ˆ/;.Y;‰/ D ˆY .

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Theorem 1.16. If C is a fusion category over C, then Z.C / is also fusion. Proof. This will be a consequence of a more general statement given in Section 3.1.1. Remark. In positive characteristic, Z.C / is, in general, not fusion. For example, if C D C .G; 1/ over k D Fxp , then Z.C / D Rep.kŒG Ë Fun.G// which is not semisimple if jGj is divisible by p.

1.3 Fusion rings 1.3.1 Realizability of fusion rings. Broadly speaking, fusion rings are rings which have the basic properties of Grothendieck rings of fusion categories. So let us consider a tensor category C . First, we have seen that if C is a tensor category, then A D Gr.C / is a ring which is a free Z-module with a distinguished basis fXi gi2I such that X0 D 1 and multiplication P (D fusion) rule Xi  Xj D k Nijk Xk , Nijk  0 (property 1). Second, from the semisimplicity condition we have Proposition 1.17 (see, e.g., [ENO]). If C is a semisimple tensor category, then for every simple object V one has V  Š V (so V Š V  ). Proof. The coevaluation map provides an embedding 1 ,! V ˝ V  . Since the category is semisimple, it implies that V ˝ V  Š 1 ˚ W . Then there exists a projection p W V ˝ V   1. But in a rigid category, the only simple object Y such that V ˝ Y projects on 1 is V . Thus there exists an involution  W i 7! i  of I , defining an antiautomorphism of A D Gr.C /, and such that Nij0 D ıij  (property 2). Definition 1.18. A finite dimensional ring with a basis satisfying properties 1 and 2 is called a based ring or a fusion ring. One of the basic questions of the theory of fusion categories is Problem 1.19. Given a fusion ring A, can it be realized as the Grothendieck ring of a fusion category? If yes, in how many ways? This problem is quite nontrivial, so let us start with a series of examples to illustrate it. 1.3.2 Some important examples. In this section we work over C unless stated otherwise.

Lectures on tensor categories

11

Example 1.20. Consider A D ZŒG for a finite group G, with involution  W g 7! g 1 being the inversion, and the fusion rule being the group law. Proposition 1.21. The set of realizations of A is H 3 .G; k /=Out.G/. Proof. Indeed, it is easy to see that the only realizations of A are C .G; !/, and two realizations corresponding to 3-cocycles !, ! 0 are equivalent iff the cohomology classes of !, ! 0 are linked by an automorphism of G. Since G acts trivially on its cohomology, we get the result. Example 1.22. Consider fusion ring structures on A D Z2 (as a Z-module). All such rings are of the form An D h1; X i with X 2 D 1 C nX and X  D X. Theorem 1.23 ([O2]). (i) A0 has two realizations: C .Z2 ; 1/ and C .Z2 ; !/, where ! is the nontrivial element in H 3 .Z2 ; k / D Z2 . c2 -modules (ii) A1 has two realizations: the fusion category of even highest weight sl at level 3, and its Galois image. (iii) For all n > 1, An has no realization. Remark. The categories in part (ii) of Theorem 1.23 are called the Yang–Lee categories and can also be obtained as quotients of the categories of tilting modules over the quantum group Uq .sl2 /, respectively with q D e ˙i=10 and q D e ˙3i =10 . by X0 ; : : : ; Xn1 and Y , satisfying the Example 1.24. Let Bn be the ring generated Pn1 2 following relations: Y D .n  1/Y C iD0 Xi , X Y D YX D Y , Y  D Y , Xi Xj D XiCj and X  D X1 (indices are taken mod n). Theorem 1.25 ([EGO], Corollary 7.4). Bn is realizable if and only if q ´ n C 1 is a prime power. More precisely, it has three realizations for q D 3, two when q D 4 or 8, and only one for other prime powers. One of the realizations is always Rep.Z q ËZq /, the others being obtained by 3-cocycle deformation. Example 1.26 (Tambara–Yamagami categories, [TY]). Let .G; / be a finite group. Consider P RG Š ZŒG ˚ ZX, with fusion product defined by the following relations: X 2 D g2G g, gX D Xg D X, gh D g  h, g  D g 1 and X  D X. Theorem 1.27 ([TY]). RG is realizable if and only if G is abelian. Realizations are parametrized by a choice of a sign ˙ and a symmetric isomorphism G ! G  (such an isomorphism always exists for abelian groups since it exists for cyclic ones). If G D Z2 , we obtain the fusion ring corresponding to the Ising model: R D h1; g; Xi with fusion rules g 2 D 1, gX D Xg D X and X 2 D 1 C g. In this case R c2 at corresponds to the Grothendieck ring of the category of integrable modules of sl level 2 (V0 D 1, V1 D X and V2 D g).

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1.3.3 The rigidity conjecture Conjecture 1.28. (i) Any fusion ring has at most finitely many realizations over k, up to equivalence (possibly none). (ii) The number of tensor functors between two fixed fusion categories, up to a natural tensor isomorphism, is finite. Thus, the conjecture suggests that fusion categories and functors between them are discrete (“rigid”) objects and cannot be deformed. It was first proved in the case of unitary categories by Ocneanu; thus we call it “Ocneanu rigidity”. The conjecture is open in general but holds for categories over C (and hence for all fields of characteristic zero). Proving this will be the main goal of the next section.

2 Ocneanu rigidity 2.1 Main results 2.1.1 Müger’s squared norms. Let C be a fusion category. For every simple object V , we are going to define a number jV j2 2 k , the squared norm of V . We have already seen that V Š V  , so let us fix an isomorphism gV W V ! V  and consider its quantum trace tr.gV / ´ eV  B .gV ˝ id/ B iV 2 End.1/ Š k. Clearly, this is not an invariant of V , since gV is well defined only up to scaling. However, the product tr.gV / tr.gV1 / is already independent on the choice of gV and is an invariant of V . Definition 2.1 (Müger, [Mu1]). jV j2 D tr.gV / tr.gV1 /, and the global dimension of C is1 X dim C D jV j2 : V simple

If dim C ¤ 0, we say that C is nondegenerate. Definition 2.2. An isomorphism of tensor functors g W id !  is called a pivotal structure on C . A category equipped with a pivotal structure is said to be a pivotal category. In a pivotal tensor category, we can define dimensions of objects by dim V D tr.gV /. The following obvious properties hold: dim.V ˝ W / D dim V dim W and jV j2 D dim V dim V  . 1 To avoid confusion, we will use the notation dim for global dimensions, and italic dim for vector space dimensions.

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13

Definition 2.3. We say that a pivotal structure g is spherical if dim V D dim V  for all simple objects V . Remarks. 1. It is not known if every fusion category admits a pivotal or spherical structure. 2. For a simple object V one has tr.gV / ¤ 0. Indeed, otherwise 1 ,! V ˝ V   1, and then the multiplicity ŒV ˝ V  W 1  2, which is impossible in a semisimple category. Example 2.4. Let H be a finite dimensional semisimple Hopf algebra over k. Since k is algebraically closed, it is equivalent to saying that H has a decomposition: M End.V /: H D V simple

It is well known that the squared antipode S 2 is an inner automorphism (there exists g 2 H  with S 2 .x/ D gxg 1 ); this is nothing but the statement (proved above) that V is isomorphic to V  for simple H -modules V . Thus jV j2 D tr V .Sj2End.V / / and dim.Rep H / D trH .S 2 /. It is conjectured (by Kaplansky, [K]) that S 2 D 1; this would imply that Rep H admits a spherical structure, such that jV j2 D dim.V /2 and dim.Rep H / D dim.H /. For k D C, this is the well-known Larson–Radford theorem [LR]. 2.1.2 Main theorems Theorem 2.5 (Ocneanu, Blanchard–Wassermann; see [BW], [ENO]). If C is nondegenerate, then 1) it has no nontrivial first order deformations of its associativity constraints, and 2) any tensor functor from C has no nontrivial first order deformations of its tensor structure. Theorem 2.6 ([ENO]). Any fusion category over C is nondegenerate. The first theorem implies Ocneanu rigidity for nondegenerate fusion categories (see [ENO], 7.3, for the precise argument), and the second one proves the rigidity conjecture for fusion categories over C. In order to prove these theorems, we have to introduce and discuss the notions of module categories and weak Hopf algebras.

2.2 Module categories We have seen that the notion of a tensor category is the categorification of the notion of a ring. Similarly, the notion of a module category which we are about to define is the categorification of the notion of a module over a ring. Let C be a tensor category.

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Definition 2.7. A left module category over C is an abelian category M with an exact bifunctor ˝ W C M ! M and functorial isomorphisms ˛ W .˝/˝• ! ˝.˝•/ and  W 1 ˝ • ! • (where • 2 M) such that for any two functors obtained from  ˝     ˝• by inserting 1’s and parenthesis, all functorial isomorphisms between them composed of ˆ˙1 ’s, ˙1 ’s, ˛ ˙1 ’s and ˙1 ’s are equal. The definition of a right module category over C is analogous. We also leave it to the reader to define equivalence of module categories. There is an analog of the MacLane coherence theorem for module categories which claims that it is sufficient for ˆ, , ˛ and  to make the following diagrams commute: .. ˝ / ˝ / ˝ • ˛ 12;3;4

ˆ1;2;3 ˝ id

/ . ˝ . ˝ // ˝ •

˛ 1;23;4



. ˝ / ˝ . ˝ •/

/  ˝ .. ˝ / ˝ •/ 

˛ 1;2;34

id˝ ˛ 2;3;4

/  ˝ . ˝ . ˝ •// ,

˛;1;•

/  ˝ .1 ˝ •/ . ˝ 1/ ˝R• RRR RRR RR id˝  ˝ id RRRR  ( ˝•.

Examples 2.8. (i) C is a left module category over itself. (ii) Define the tensor category C op , which coincides with C as an abstract category, and has reversed tensor product ˝op , which is defined by X ˝op Y D Y ˝ X . The associativity and unit morphisms are defined in an obvious manner. Then C is a left module category over C op (as it is a right module category over itself). (iii) We deduce from (i) and (ii) that C is a left module category over C  C op . (iv) If C D Vect k and M D Rep A for a given algebra A over k, then M is a left module category over C . Note that if M is a left (right) module category over C , then its Grothendieck group Gr.M/ is a left (respectively, right) Gr.C /-module, with P a distinguished basis Mj and positive structure constants Nijr such that Xi  Mj D r Nijr Mr . In this way, we can associate to any object X 2 Obj.C / its left (right) multiplication matrix NX , which has positive entries, and in the semisimple case NX  D NXT . If C is a fusion category, we will be interested in semisimple finite module categories over C . Such a module category is called indecomposable if M is not module equivalent to M1 ˚ M2 for nonzero module categories Mi , i D 1; 2. As was mentioned above, the theory of module categories should be viewed as a categorical analog of the theory of modules (representation theory). Thus the main problem in the theory of module categories is

Lectures on tensor categories

15

Problem 2.9. Given a fusion category C , classify all indecomposable module categories over C which are finite and semisimple. The answer is known only for a few particular cases. For example, one has the following result (see [KO], [O1] for proof and references): c2 at level l, then Theorem 2.10. If C is the category of integrable modules over sl semisimple finite indecomposable module categories over C are in one-to-one correspondence with simply laced Dynkin diagrams of ADE type and with Coxeter number h D l C 2. 2.2.1 The category of bimodules. Let C be a tensor category. A structure of a left module category over C on an abelian category M is the same thing as a tensor functor C ! Fun.M; M/ (Fun.M; M/ is the monoidal category whose objects are exact functors from M to itself, morphisms are natural transformations, and the tensor product is just the composition of functors). This is just the categorical analog of the tautological statement that a module M over a ring A is the same thing as a representation  W A ! End.M /. If M is semisimple and finite, then M Š Rep A as an abelian category for a (nonunique) finite dimensional semisimple algebra A. Therefore, structures of a left module category over C on M are in one-to-one correspondence with tensor functors C ! Fun.M; M/ D A-bimod. Remark. In particular, if M has only one simple object (i.e., M Š Vectk as an abelian category), then C -module category structures on M correspond to fiber functors on C . Let us consider more closely the structure of the category A-bimod. Its tensor Q is the tensor product over A. The simple objects in this category are product ˝ Mij D Homk .Mi ; Mj /, where Mi 2 Obj.M/ are simple A-modules; and we have Q i 0j 0 D ıi 0j Mij 0 . Thus A-bimod is finite semisimple and satisfies all the axioms Mij ˝M L of a tensor category except one: 1 D i Mi i is not simple, but semisimple. Definition 2.11. A multitensor category is a category which satisfies all axioms of a tensor category except that the neutral object is only semisimple. A multifusion category is a finite semisimple multitensor category. Thus, A-bimod is a multifusion category. 2.2.2 Construction of module categories over fusion categories. Let B be an algebra in a fusion category C . The category M of right B-modules in C is a left module category over C: let X 2 Obj.C / and M be a right B-module (M ˝ B ! M ), then the composition .X ˝ M / ˝ B  ! X ˝ .M ˝ B/ ! X ˝ M gives us the structure of a right B-module on X ˝ M (and so it defines a structure of left C -module category

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on M). We will consider the situation when M is semisimple; in this case the algebra B is said to be semisimple. Theorem 2.12 ([O1]). Any semisimple finite indecomposable module category over a fusion category can be constructed in this way (but nonuniquely). Example 2.13. Let us consider the category C .G; !/, with G a finite group and ! 2 Z 3 .G; k / a 3-cocycle. Let H  G be a subgroup such that !jH D d for a cochain 2 C 2 .H; k /. Define the twisted group algebra B D k ŒH : L B D h2H Vh as an object of C (where Vh is the 1-dimensional module corresponding to h 2 H ), and the multiplication map B ˝ B ! B is given by .g; h/id W Vg ˝ Vh ! Vgh D Vg ˝ Vh . The condition !jH D d , which can be rewritten as .h; k/ .g; hk/!.g; h; k/ D .gh; k/ .g; h/ for all g; h; k 2 H , assures the associativity of the product for B (i.e., B is an algebra in C .G; !/). We call M.H; / the category of right B-modules in C .G; !/. Theorem 2.14 ([O3]). Assume char.k/ does not divide jGj. All semisimple finite indecomposable module categories over C .G; !/ have this form. Moreover, two module categories M.H1 ; 1 / and M.H2 ; 2 / are equivalent if and only if the pairs .H1 ; 1 / and .H2 ; 2 / are conjugate under the adjoint action of G. Proof. Let M be an indecomposable module category over C .G; !/. Since for every simple object we have X D Vg , X ˝ X  D Vg ˝ Vg 1 D 1, the multiplication matrix NX of X satisfies the equation NX NXT D id and thus NX is a permutation matrix. So we have a group homomorphism G ! Perm.simple.M//. But M is indecomposable, therefore G acts transitively on Y ´ simple.M/ and so Y D G=H . Thus M is the category of right B-modules in C .G; !/, where B D k ŒH  for a 2-cochain 2 C 2 .H; k /. The associativity condition for the product in B, as we saw above, is equivalent to .h; k/ .g; hk/!.g; h; k/ D .gh; k/ .g; h/ (i.e., !jH D d ). We are done.

2.3 Weak Hopf algebras Tensor functors C ! A-bimod are a generalization of fiber functors (which are obtained when A D k). So it makes sense to generalize reconstruction theory for them. This leads to Hopf algebroids, or, in the semisimple case, to weak Hopf algebras. 2.3.1 Definition and properties of weak Hopf algebras Definition 2.15 ([BNS]). A weak Hopf algebra is an associative unital algebra .H; m; 1/ together with a coproduct , a counit , and an antipode S such that: (1) .H; ; / is a coassociative counital coalgebra; (2)  is a morphism of associative algebras (not necessary unital);

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17

(3) . ˝ id/ B .1/ D ..1/ ˝ 1/  .1 ˝ .1// D .1 ˝ .1//  ..1/ ˝ 1/; (4) .fgh/ D .fg1 / .g2 h/ D .fg2 / .g1 h/; (5) m B .id ˝ S/ B .h/ D . ˝ id/ B ..1/  .h ˝ 1//; (6) m B .S ˝ id/ B .h/ D .id ˝ / B ..1 ˝ h/  .1//; (7) S.h/ D S.h1 /h2 S.h3 /. Here we used Sweedler’s notation: k .x/ D x1 ˝ x2 ˝    ˝ xk (k is the k-fold coproduct and summation is implicitly assumed). Remarks. 1. The notion of finite dimensional weak Hopf algebra is self-dual, i.e., if .H; m; 1; ; ; S/ is a finite dimensional weak Hopf algebra then .H  ;  ;  ; m ; 1 ; S  / is also a finite dimensional weak Hopf algebra. 2. Let H be a weak Hopf algebra. H is a Hopf algebra if and only if .1/ D 1 ˝ 1 (this is equivalent to the requirement that is an associative algebra morphism). The linear maps t W h 7! .11 h/12 and s W h 7! 11 .h12 / defined by (5) and (6) in the definition are called the target and source counital maps, respectively. The images A t D t .H / and As D s .H / are the target and source bases of H . Proposition 2.16 ([NV], Section 2). A t and As are semisimple algebras that commute with each other, and SjA t W A t ! As is an algebra antihomomorphism. An especially important and tractable class of weak Hopf algebras is that of regular weak Hopf algebras, defined as follows. Definition 2.17. A weak Hopf algebra H is regular if S 2 D id on A t and As . From now on all weak Hopf algebras we consider will be assumed regular. Let H be a finite dimensional weak Hopf algebra and consider the category C D Rep H . One can define the tensor product V ˝ W of two representations: V ˝ W ´ .1/.V ˝k W / as a vector space, and the action of any x 2 H on V ˝ W is given by .x/. As in the case of a Hopf algebra, the associativity morphism is the identity, t gives A t the structure of an H -module which is the neutral object in C , and the antipode S allows us to define duality. This endows C with the structure of a finite tensor category [NTV], Section 4. In the case when H is regular, each H -module M is also an A t ˝ As -module (by op Proposition 2.16), and hence it is an A t -bimodule (since As D A t ). Moreover, the forgetful functor C D H -mod ! A t -bimod is tensor. 2.3.2 Reconstruction theory. Let C be a finite tensor category, A a finite dimensional semisimple algebra and F W C ! A-bimod a tensor functor. Assume that the sizes of the matrix blocks of A are not divisible by char.k/ (for example, A is commutative or char.k/ D 0).

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Consider H D Endk .F / D End.Fx/, where Fx is the composition of F with the forgetful functor Forget to vector spaces; it is a unital associative algebra. Since any F .X/ is an A-bimodule, there exists an algebra antihomomorphism s W A ! H and an algebra morphism t W A ! H such that Œs.a/; t .a0 / D 0 for all a; a0 2 A. Moreover, x W Endk .F / ! Endk .F  F / in the same way as we can define a kind of coproduct  x / can be interpreted as an x for Tannakian formalism: .T / D J B T B J 1 . Thus .T element x / 2 H ˝A H D H ˝ H=ht .a/x ˝ y  x ˝ s.a/yi; .T x /.t.a/ ˝ 1 C 1 ˝ s.a// D 0 for all a 2 H . Now, since A is semisimple, such that .T there is a canonical map X mei ˝ e i n;  W H ˝A H ! H ˝k H; m ˝ n 7! i i

for dual bases .ei / and .e / of A relatively to the pairing .a; b/ D trA .La Lb /, where La is the operator of left multiplication by a (note that because of our assumption on the block sizes this pairing is nondegenerate). We can thus define the “true” coproduct x W H ! H ˝ H which turns out to be coassociative. DB One can also define a counit W H ! k by .T / D trA .T jFx .1/ / and an antipode S W H ! H by S.T /jFx .X / D .T jFx .X  / / . Theorem 2.18. The associative unital algebra H equipped with , and S as above is a regular weak Hopf algebra. Moreover, C Š Rep H as a tensor category. Thus, given a tensor category C over k and a finite dimensional semisimple algebra A with block sizes not divisible by char.k/, we have bijections (modulo appropriate equivalences): Finite dimensional regular weak Hopf algebras op H with bases A t DO A and AsV k DA



V V V V V V V V+

Finite tensor categories with tensor o functor F W C ! A-bimod

Finite semisimple indecomposable

/ module categories over C, equivalent to A-mod as abelian categories

If C is a fusion category, then C is a semisimple module category over itself. So C ŠL Rep H as a tensor category for a semisimple weak Hopf algebra H with base A D i2I kxi (with xi xj D ıij xi ). Corollary 2.19 (Hayashi). Any fusion category is the representation category of a finite dimensional semisimple weak Hopf algebra with a commutative base. Remark. It is not known to us if there exists a (nonsemisimple) finite tensor category which is not the category of representations of a weak Hopf algebra (i.e., does not

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admit a semisimple module category). Finding such a category is an interesting open problem.

2.4 Proofs 2.4.1 Nondegeneracy of fusion categories over C Proposition 2.20 ([N], [ENO]). In any fusion category, there exists an isomorphism of tensor functors ı W id ! ****. Proof. Recall that C Š Rep H for a finite dimensional semisimple regular weak Hopf algebra H . In the semisimple case, the generalization of Radford’s S 4 formula by Nikshych [N], Section 5, tells us that there exists a 2 G.H / such that S 4 .x/ D a1 xa for all x 2 H; where a 2 G.H / means a is invertible and .a/ D .1/.a ˝ a/ D .a ˝ a/.1/ (i.e., a is a group-like element). Thus we can define ı by ıV D a1 jV . Then for every H -modules V and W , the fact that ıV ˝W D ıV ˝ ıW follows from the group-like property of a. Theorem 2.21 ([ENO]). For fusion categories over C, for any simple object V one has jV j2 > 0. In particular, this implies that for any fusion category C over C one has dim C  1 and so is nondegenerate. Question. Does there exist > 0 such that for every fusion category C over C which is not Vectk , dim C > 1 C ? Proof of the theorem. We first do the pivotal case. P In this case dim.V ˝ W / D dim V dim W for all objects V , W , thus di dj D k Nijk dk , where di D dim.Xi / are the dimensions of the simple objects. In a shorter way we can rewrite these equalities as Ni dE D di dE, where dE D .d0 ; : : : ; dn1 /. For all i; j; k 2 I , Nik j D dim.Hom.Xi ˝ Xj ; Xk // D dim.Hom.Xj ; Xi ˝ Xk // D dim.Hom.Xi ˝ Xk ; Xj // D

(by rigidity)

(by semisimplicity)

j Nik :

Therefore NiT Ni dE D Ni  Ni dE D di  di dE D jXi j2 dE, so jXi j2 is an eigenvalue of NiT Ni associated to dE ¤ 0E and consequently jXi j2 > 0. Now we extend the argument to the non-pivotal case. Let us define the pivotal x of C, which is the fusion category whose simple objects are pairs .X; f /: extension C

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X is simple in C and f W X  ! X  satisfies f  f D ıX for the isomorphism of x has a canonical tensor functors ı W id ! **** constructed above. The category C   pivotal structure .X; f / ! .X ; f / (which is given by f itself), thus j.X; f /j2 > x ! C , .X; f / 7! X, preserves squared norms and 0. Finally the forgetful functor C so jXj2 > 0. 2.4.2 Proof of Ocneanu rigidity: the Davydov–Yetter cohomology. Let D be a tensor category. Define the following cochain complex attached to D: • C n .D/ D End.Tn /, where Tn is the n-functor D n ! D, .X1 ; : : : ; Xn / 7! X1 ˝    ˝ Xn (T0 D 1 and T1 D id). • The differential d W C n .D/ ! C nC1 .D/ is given by df D id ˝ f2;:::;nC1  f12;3;:::;nC1 C       C .1/n f1;:::;n1;nnC1 C .1/nC1 f1;:::;n ˝ id H n .D/ is the n-th space of the Davydov–Yetter cohomology ([Dav], [Y]). Example 2.22. Assume D D Rep H for a Hopf algebra H . Then C n .D/ D .H ˝n /Had . .C n ; d/ is a subcomplex of the co-Hochschild complex for H with trivial coefficients. Proposition 2.23 (see [Y]). H 3 .D/ and H 4 .D/ respectively classify first order deformations of associativity constraints in D and obstructions to these deformations. Examples 2.24. (i) Let G be a finite group and D D C .G; 1/. Then H i .D/ D H i .G; k/, and thus H i .D/ D 0 for i > 0 if k D C or jGj and char.k/ are coprime. (ii) Let G be a semisimple complex Lie group with Lie algebra g and consider V C D Rep G. Then H i .C / D . i g/G D H i .G; C/. In particular, H 3 .C / D C and H 4 .C / D 0. So there exists a unique one-parameter deformation of C D Rep G which is realized by Rep U„ .g/. The next result implies in particular the first part of Theorem 2.5. Theorem 2.25 ([ENO]). Let D be a nondegenerate fusion category over k. Then H i .D/ D 0 for all i > 0. Proof. The proof is based on the notion of categorical integral. Suppose that f R 2 C n .D/ (for X1 ; : : : ; Xn , fX1 ;:::;Xn : X1 ˝    ˝ Xn ! X1 ˝    ˝ Xn ). Define f 2 C n1 in the following way: for X1 ; : : : ; Xn1 2 Obj.D/, Z  X D tr V ..id ˝ gV / B fX1 ;:::;Xn1 ;V / tr.gV1 / f X1 ;:::;Xn1

V simple

where tr V ..id ˝ gV / B fX1 ;:::;Xn1 ;V / is equal to .id˝n ˝ eV  / B .id˝.n1/ ˝ gV ˝ id/ B .fX1 ;:::;Xn1 ;V ˝ id/ B .id˝n ˝ iV /:

21

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Remark. By definition,

R

id D dim D.

Assume now that f 2 Z n .D/ is a cocycle. Then if we put ' D Z 0 D df Z Z D id ˝ f2;:::;nC1  f12;3;:::;nC1 C    Z Z n nC1 f1:::;n1;nnC1 C .1/ f1;:::;n ˝ id    C .1/ D id ˝ '2;:::;n  '12;3;:::;n C       C .1/

n1

'1;:::;n1n C .1/

Lemma 2.26 ([ENO]).

R

n

Z

R

f , we have

f1:::;n1;nnC1 C .1/nC1 dim D  f1;:::;n :

f1;:::;n1;nnC1 D '1;:::;n1 ˝ id.

The proof of the lemma is based the theory of weak Hopf algebras, and we will omit it. See [ENO], Section 6. Thus when dim D ¤ 0, f D dim1 D .1/n1 d'. Remark. In the same way, for any tensor functor F W C ! D, one can define a cochain complex CFn .C / D End.Tn B F ˝n / and a differential d W CFn .C / ! CFnC1 .C / which is given by df D id ˝f2;:::;nC1 f12;3;:::;nC1 C  C.1/n f1;:::;n1;nnC1 C.1/nC1 f1;:::;n ˝id where f1;:::;i iC1;:::;nC1 acts on F .X1 /˝  ˝F .XnC1 / as f on F .X1 /˝  ˝F .Xi ˝ XiC1 /˝  ˝F .XnC1 / (we have used the tensor structure to identify F .Xi /˝F .XiC1 / and F .Xi ˝ XiC1 /). Then one can show (see [ENO]) that the corresponding cohomology spaces HFi .C / are trivial for nondegenerate categories, and that HF2 .C / (resp. HF3 .C /) classifies first order deformations of the tensor structure of F (resp. obstructions to these deformations). Thus the second part of Theorem 2.5 is proved.

3 Morita theory, modular categories, and lifting theory 3.1 Morita theory in the categorical context 3.1.1 Dual category with respect to a module category Problem 3.1. Let H be a finite dimensional (weak) Hopf algebra. C D Rep.H / is a finite tensor category. How to describe the category Rep.H  / in terms of C ?

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The answer is given by the next definitions. Definition 3.2. A module functor between module categories M1 , M2 over C is an additive functor F W M1 ! M2 together with a functorial isomorphism J W F . ˝1 •/ !  ˝2 F .•/ such that the following diagrams commute: F .. ˝C / ˝1 •/ J

F .˛/

/ F . ˝1 . ˝1 •//

J



. ˝C / ˝2 F .•/

/  ˝2 F . ˝1 •/ 

˛

id˝J

/  ˝2 . ˝2 F .•//

and J1;•

/ 1 ˝2 F .•/ F .1 ˝1 •/ NNN NNN 2 NN F .1 / NNN &  F .•/ .

Let C be a tensor category (not necessarily semisimple) and M a left module category over C.  D Definition 3.3. The dual category of C with respect to M is the category CM FunC .M; M/, the category of module functors from M to itself with tensor product being the composition of functors.

Thus the notion of the dual category is the categorification of the notion of the centralizer of an algebra in a module.  is a monoidal category and M is a left module category over it. Observe that CM  However, CM is not always rigid. For example, if C D Vectk and M D A-mod for  a finite dimensional associative algebra A over k, then CM D FunVectk .M; M/ D Fun.M; M/. This category contains the category A-bimod with tensor product ˝A which is not exact if A is not semisimple (while it must be exact in the rigid case). Thus, to insure rigidity of the dual category, we should perhaps restrict ourselves to a subclass of module categories. A subclass that turns out to produce a good theory is that of exact module categories. Namely (see [EO1]), a left module category is called exact if for any projective object P in C , and any X 2 Obj.M/, P ˝ X is also projective. Such a category is finite if and only if it has finitely many simple objects. In the particular case of a fusion category C , exactness for module categories coincides with semisimplicity. Theorem 3.4 ([EO1]). If C is a finite tensor category and M is a finite indecom is a finite tensor category. posable exact left module category over it, then CM Examples 3.5. (i) If C D Rep H and M D Rep A for a finite dimensional regular  D Rep.H op /. weak Hopf algebra with bases A, Aop , then CM

Lectures on tensor categories

23

(ii) Let C D C .G; !/ and M D M.H; / be as in example 2.13. Then one can  , where B D k ŒH  is consider the category of B-bimodules C .G; !; H; / ´ CM the twisted group algebra of H in C . Such categories are called group theoretical. Let C be a finite tensor category and M a finite indecomposable exact left module category over C . Then one can show ([ENO], [EO1], [O1]) that the following properties hold:   /M D C. (1) .CM

  /M D Z.C /. (2) .C  CM

(3) CC D C op (and then .C  C op /C D Z.C / by the previous one).

 (4) If M D B-mod for a semisimple algebra B in a fusion category C , then CM D B-bimod.  is also fusion. Moreover, we (5) If C is a nondegenerate fusion category, then CM  have dim CM D dim C and so dim Z.C / D .dim C /2 .

Remark. Note that property (1) is the categorical version of the double centralizer theorem for semisimple algebras (saying that the centralizer of the centralizer of A in a module M is A if A is a finite dimensional semisimple algebra). Property (2) is the categorical analog of the statement that if A0 is the centralizer of A in M then the centralizer of A ˝ A0 in M is the center of A. Finally, property (3) is the categorical version of the fact that the centralizer of A in A is Aop . 3.1.2 Morita equivalence of finite tensor categories. By now, all module categories are supposed to be finite and exact. Definition 3.6. Two finite tensor categories C and D are Morita equivalent if there  D D op . In exists an indecomposable (left) module category M over C such that CM this case we write C M D. Obviously, this notion is the categorical analog of Morita equivalence of associative algebras. Proposition 3.7 (Müger, [Mu1], [Mu2]). Morita equivalence of finite tensor categories is an equivalence relation. Proof. This relation is reflexive since CC D C op .  To prove the symmetry, assume that CM D D op , and define M _ ´ Fun.M; Vectk /.  op This is a left (indecomposable) module category over D and DM . Now let _ D C us prove transitivity. Suppose C M D and D N E. Take P D FunD .M _ ; N / (By analogy with ring theory, we could denote this category by M ˝D N .) Then CP D E op . Thus the transitivity condition is verified.

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Theorem 3.8 ([Mu1], [Mu2]; see also [O3]). Let C M D be a Morita equivalence of finite tensor categories. Then there is a bijection between indecomposable left module categories over C and D. It maps N over C to FunC .M; N / over D. This, obviously, is the categorical version of the well known characterization of Morita equivalent algebras: their categories of modules are equivalent. Corollary 3.9 ([O3]). Indecomposable left module categories over C .G; !; H; / are M.H; ; H 0 ; 0 / ´ FunC.G;!/ .M.H; /; M.H 0 ; 0 //: 3.1.3 Application to representation theory of groups. Let G be a finite group and consider the category D D Rep G. In fact, D D C .G; 1/M with M D M.G; 1/ D Vect k . Hence, indecomposable D-module categories are of the form M.G; 1; H; / D Rep C ŒH . Now recall that fiber functors are classified by module categories with only one simple object. In our case it corresponds to the case when C ŒH  is simple, which is equivalent to the requirement that is a nondegenerate 2-cocycle, in the sense of the following definition. Definition 3.10. A 2-cocycle on H is nondegenerate ifp H admits a unique projective irreducible representation with cocycle of dimension jH j. A group H which admits a nondegenerate cocycle is said to be of central type. Remarks. 1. It is obvious that a group of central type has order N 2 , where N is an integer. 2. Howlett and Isaacs [HI] showed that any group of central type is solvable. This is a deep result based on the classification of finite simple groups. Theorem 3.11 ([EG], [Mo]). Fiber functors on Rep G .i.e., Hopf twists on CŒG up to a gauge/ are in one-to-one correspondence with pairs .H; /, where H is a subgroup of G and a nondegenerate 2-cocycle on H modulo coboundaries and inner automorphisms. Proof. This follows from Theorem 3.8 and Corollary 3.9. We leave the proof to the reader. Corollary 3.12 ([TY]). Let D8 be the group of symmetries of the square and Q8 the quaternion group. Then Rep D8 and Rep Q8 are not equivalent (although they have the same Grothendieck ring). Proof. In Q8 , all subgroups of order 4 are cyclic and hence do not admit any nondegenerate 2-cocycle.

Lectures on tensor categories

25

On the other hand, D8 has two subgroups isomorphic to Z2  Z2 (not conjugate) and each has one nondegenerate 2-cocycle. Thus Q8 has fewer fiber functors (in fact only one) than D8 (which has three such). So we see that one can sometimes establish that two fusion categories are not equivalent (as tensor categories) by counting fiber functors. Similarly, one can sometimes show that two fusion categories are not Morita equivalent by counting all indecomposable module categories over them (since we have seen that Morita equivalent fusion categories have the same number of indecomposable module categories). Let us illustrate it with the following example. Example 3.13. We want to show that Rep.Zp  Zp / and Rep Zp2 are not Morita equivalent. First remember that Rep G D C .G; 1; G; 1/ and module categories over it are parametrized by .H; /, where H is a subgroup of G and 2 H 2 .H; C /. On the one hand, Zp2 has three subgroups (Zp2 itself, Zp , and 1), all with a trivial second cohomology. Thus Rep Zp2 has 3 indecomposable module categories. On the other hand, Zp  Zp has p C 3 subgroups: Zp  Zp , p C 1 copies of Zp , and 1. Moreover, Zp  Zp has p 2-cocycles up to coboundaries. Thus Rep.Zp  Zp / has 2p C 2 > 3 module categories.

3.2 Modular categories and the Verlinde formula Let C be a braided tensor category. Then we have a canonical (non-tensor) functorial isomorphism u W id !  given by the composition V ! V ˝ V  ˝ V  ! V  ˝ V ˝ V  ! V  (the maps are the coevaluation, the braiding, and the evaluation). This isomorphism is called the Drinfeld isomorphism. Using the Drinfeld isomorphism, we can define a tensor isomorphism ı W id !  by the formula ıV D .uV  /1 uV . Definition 3.14. A ribbon category is a braided tensor category together with a pivotal structure g W id !  such that g  g D ı. We refer the reader who wants to learn more about ribbon categories (especially the graphical calculus for morphisms, using tangles) to [K], [BaKi] or [Tu]. Assume now that C is a ribbon category. Recall for any simple object V 2 C one can define the dimension dim V . It is known (see, e.g., [K]) that dim V  D dim V . For any two objects V , W one can define the number SV W 2 End.1/ Š k to be .eV  ˝ eW  / B .gV ˝ idV  ˝ gW ˝ idW  / B .idV ˝ W V  ˝ idW  / B .idV ˝ V  W ˝ idW  / B .iV ˝ iW /: Now assume that C is fusion, with simple objects Xi ’s. We can define a matrix S with entries Sij D SXi Xj . Then S has the following properties:

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(1) Sij D Sj i ; (2) Sij D Si  j  ; (3) Si0 D dim Xi ¤ 0. Definition 3.15. A ribbon fusion category is called modular if S is nondegenerate. Proposition 3.16 ([Mu2], [Tu]). If C is a nondegenerate fusion category with a spherical structure, then Z.C / is a modular category. Proposition 3.17 ([BaKi], Theorem 3.1.7). In a modular category C , X Sik Skj D .dimC /ıij  : k

Thus if p C is a modular category, then dim C ¤ 0 and we can define new numbers sij D Sij = dim C (here we must make a choice of the square root). Theorem 3.18 (Verlinde formula, [BaKi]). X sir sjr Nij˛ s˛r D : s0r ˛ So sir =s0r are eigenvalues of the multiplication matrix Ni . In particular, they are algebraic integers (i.e., roots of a monic polynomial with integer coefficients - the characteristic polynomial of Ni ). Hence: Proposition 3.19. For every r,

dim C .dim Xr /2

D

si r si  r 2 s0r

is an algebraic integer.

This result will be very useful to prove classification theorems in Section 4. 3.2.1 Galois property of the S-matrix. A remarkable result due to J. de Boere, J. Goeree, A. Coste and T. Gannon states that the entries of the S-matrix of a modular category lie in a cyclotomic field, see [dBG], [CG]. Namely, one has the following theorem. Theorem 3.20. Let S D .sij /i;j 2I be the S-matrix of a modular category C . There exists a root of unity such that sij 2 Q. /. Proof. Let fXi gi2I be the representatives of isomorphism classes of simple objects of C ; let 0 2 I be such that X0 is the unit object of C and the involution i 7! i  of I be defined by Xi Š Xi  . By the definition of modularity, any homomorphism f W K.C/ ! C is of the form f .ŒXi / D sij =s0j for some well-defined j 2 I . Hence x one has g.sij =s0j / D sig.j / =s0g.j / for a well defined for any automorphism g of Q action of g on I .

Lectures on tensor categories

27

P Now remember from the previous subsection that one has the following properties: D ıij  , sij D sj i , and s0i  D s0i ¤ 0. P k sik skjP Thus, j sij sj i  D 1 and hence .1=s0i /2 D j .sj i =s0i /.sj i  =s0i  /. Applying the automorphism g to this equation we get   X s s   X s ıg.i/g.i  / 1 ji ji jg.i / sjg.i  / D D : g 2 Dg  s0i s0i s0g.i / s0g.i  / s0g.i / s0g.i  / s0i j

j

It follows that g.i  / D g.i / and g..s0i /2 / D .s0g.i / /2 . Hence g..sij /2 / D g..sij =s0j /2  .s0j /2 / D .sig.j / /2 : Thus g.sij / D ˙sig.j / . Moreover the sign g .i / D ˙1 such that g.s0i / D g .i /s0g.i / is well defined since s0i ¤ 0, and g.sij / D g..sij =s0j /s0j / D g .j /sig.j / D g .i/sg.i/j . In particular, the extension L of Q generated by all entries sij is fix nite and normal, that is Galois extension. Now let h be another automorphism of Q. We have gh.sij / D g. h .j /sih.j / / D g .i / h .j /sg.i /h.j / and hg.sij / D h. g .i /sg.i/j / D h .j / g .i /sg.i /h.j / D gh.sij /; and the Galois group of L over Q is abelian. Now the Kronecker–Weber theorem (see, e.g., [Ca]) implies the result.

3.3 Lifting theory First recall that a fusion category over an algebraically closed field k can be regarded as a collection of finite dimensional vector spaces Hijk (D Hom.Xi ˝ Xj ; Xk /), together with linear maps between direct sums of tensor products of these spaces which satisfy some equations (given by axioms of tensor categories). Thus one can define a fusion category over any commutative ring with R to be a collection of free finite rank Rmodules Hijk together with module homomorphisms between direct sums of tensor products of them which satisfy the same equations. By a realization of a fusion ring A over R we will mean a fusion category over R such that Nijk ´ dim.Hijk / are the structure constants of A. If I is an ideal in R and C a fusion category over R then it is clear how to define the reduced (=quotient) fusion category C =I over R=I with the same Grothendieck ring. Tensor functors between fusion categories over k can be defined in similar terms, as collections of linear maps satisfying algebraic equations; this allows one to define tensor functors between fusion categories over R (and their reduction modulo ideals) in an obvious way.

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Now let k be any algebraically closed field of characteristic p > 0, W .k/ the ring of Witt vectors of k, I the maximal ideal of W .k/ generated by p, and K the algebraic closure of the fraction field of W .k/ (char.K/ D 0). z of C to W .k/ is Definition 3.21. Let C be a fusion category over k. A lifting C a realization of Gr.C/ over the ring W .k/ together with an equivalence of tensor z =I  categories C ! C. In a similar way, one defines a lifting of a tensor functor F W C ! D: it is a z !D z over W .k/ together with an equivalence of tensor functors tensor functor Fz W C  Fz =I ! F . Theorem 3.22 ([ENO]). Let C be a nondegenerate fusion category over k. Then there exists a unique lifting of C to W .k/. Proof. This follows from the fact that liftings are classified by H 3 .C / and obstructions by H 4 .C /. And we know from Section 2 that the Davydov–Yetter cohomology vanishes for nondegenerate categories. Theorem 3.23 ([ENO]). Let F W C ! D be a tensor functor between nondegenerate fusion categories over k. Then there exists a unique lifting of F to W .k/. Proof. Again, liftings of F are parametrized by HF2 .C / and obstructions by HF3 .C /, which are trivial in the nondegenerate case. Corollary 3.24 ([EG2]). Any semisimple Hopf algebra H over k with tr.S 2 / ¤ 0 z over W .k/. (i.e., also cosemisimple) lifts to H y DH z ˝W .k/ K, which is a Hopf algebra over a field of Hence one can define H characteristic zero. This allows one to extend results from the characteristic zero case to positive characteristic. For example, applying the Larson–Radford theorem [LR] y , one can find: (see Corollary 4.26 below) to H Corollary 3.25 (Kaplansky’s 7th conjecture, [EG2]). If H is a semisimple and cosemisimple Hopf algebra over any algebraically closed field, then S 2 D 1. Corollary 3.26 ([ENO]). A nondegenerate braided (resp. symmetric) fusion category over k is uniquely liftable to a braided (resp. symmetric) fusion category over W .k/. Proof. A braiding on C is the same as a splitting C ! Z.C / of the natural (forgetful) tensor functor Z.C / ! C . Theorem 3.23 implies that such a splitting is uniquely liftable. Thus a braiding is uniquely liftable. Now prove the result in the symmetric case. A braiding gives rise to a categorical equivalence B W C ! C op , and it is symmetric if and only if the composition of B and B 21 is the identity. Hence the corollary follows from Theorem 3.23.

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We conclude the section with mentioning a remarkable theorem of Deligne on the classification of symmetric fusion categories over C. Theorem 3.27 ([De]). Any symmetric fusion category over C is Rep G for a finite group G. With some work, one can extend this result using Corollary 3.26: Corollary 3.28 ([EG3]). Any symmetric nondegenerate fusion category over k (of characteristic p) is Rep G for a finite group G of order not divisible by p.

4 Frobenius–Perron dimension 4.1 Definition and properties Let C be a finite tensor category with simple objects X0 ; : : : ; Xn1 . Then for every object X 2 Obj.C /, we have a matrix NX of left multiplication by X : ŒX ˝ Xi W This matrix has nonnegative entries, and in the Grothendieck ring we Xj  D .NX /ij .P have: XXi D j .NX /ij Xj . Let us now recall the classical Theorem 4.1 (Frobenius–Perron). Let A be a square matrix with nonnegative entries. Then (1) A has a nonnegative real eigenvalue. The largest such eigenvalue .A/ dominates in absolute value all other eigenvalues of A. Thus the largest nonnegative eigenvalue of A coincides with the spectral radius of A. (2) If A has strictly positive entries, then .A/ is a simple eigenvalue, which is strictly positive, and its eigenvector can be normalized to have strictly positive entries. Moreover, if v is an eigenvector with strictly positive entries, then the corresponding eigenvalue is .A/. Thus to all X 2 Obj.C/ one can associate a nonnegative number dC .X / D .NX /, its Frobenius–Perron dimension. Examples 4.2. (i) TheYang–Lee category: X 2 D 1CX , so NX D p 1C 5 . 2

01 11

and dC .X / D

(ii) Let C D Rep H for a finite dimensional quasi-Hopf algebra H , then dC .X / D dim.X/ for all H -modules X . The following proposition follows from the interpretation of dC .X / as the spectral radius of NX .

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Proposition 4.3. For all objects X of C , infinity.

log.length.X ˝n // log n

! dC .X / when n goes to

Theorem 4.4 ([ENO], [E]). The assignment X 7! dC .X / extends to a ring homomorphism Gr.C / ! R. Moreover, dC .Xi / > 0 for i D 0; : : : ; n  1. P Proof. Consider X D i Xi 2 Gr.C / and denote by MX the matrix of right multiplication by X. For i; j 2 I , .MX /ij D ŒXi ˝ X W Xj   dim.Hom.Xi ˝ X; Xj // X dim.Hom.Xk ; Xi ˝ Xj // > 0: D k

Hence by the Frobenius–Perron theorem, there exists aP unique eigenvector of MX (up to scaling) with strictly positive entries, say R D i ˛i Xi : RX D R with  D .MX /. Now for all Y 2 Gr.C /, .YR/X D YR and then by the uniqueness of R there is ˇY 2 R such that YR D ˇY R. Since R has positive coefficients, applying again the Frobenius–Perron theorem, we obtain ˇY D .NY / D dC .Y /. Consequently, dC .Y C Z/R D .Y C Z/R D YR C ZR D .dC .Y / C dC .Z//R and dC .Y Z/R D Y ZR D YdC .Z/R D dC .Y /dC .Z/R. So Y 7! dC .Y / extends to a ring homomorphism Gr.C / ! R. Suppose that dC .Xi / D 0; then Xi R D 0 and so Xi Xj D 0 for all j 2 I , which is not possible. Thus dC .Xi / > 0. Remark. It is clear that the Frobenius–Perron dimension can be defined for any finite dimensional ring with distinguished basis and nonnegative structure constants (even if it has no realization) and does not depend on the corresponding category. Proposition 4.5. dC is the unique character of Gr.C / that maps elements of the basis to strictly positive numbers. P Proof. Let  be another such character. Then .Xi /.Xj / D Nijk .Xk /. Thus the vector with positive entries .Xk / is an eigenvector of the matrix Ni with eigenvalue .Xi /. So by the Frobenius–Perron theorem, .Xi / D dC .Xi /. Corollary 4.6. Quasi-tensor functors between finite tensor categories preserve Frobenius-Perron dimension. Corollary 4.7. dC .X/ D dC .X  /. Properties of the Frobenius–Perron dimension (1) ˛ D dC .X/ is an algebraic integer (it is a root of the characteristic polynomial of NX ).

Lectures on tensor categories

31

(2) For all g 2 Gal.Q=Q/, jg˛j ˛ (use part two of the Frobenius–Perron theorem). In particular, ˛  1. (3) ˛ D 1 () X ˝ X  D 1 (in this case X is called invertible). Proof. If X ˝ X  D 1, then 1 D dC .1/ D dC .X /dC .X  /. Since dC .X /  1 and dC .X  /  1, we find that dC .X  / D 1. Conversely, consider iX W 1 ,! X ˝ X  and compute dC .X ˝ X  / D dC .1/ C dC .Coker iX / D 1 C dC .Coker iX /: Now if dC .X/ D 1, then dC .X ˝ X  / D 1. So dC .Coker iX / D 0 and hence Coker iX Š 0. Consequently, iX is an isomorphism and thus 1 Š X ˝ X  . (4) ([GHJ]) If ˛ < 2, then ˛ D 2 cos n for n  3. Proof. Since dC is a character, ˛ is the largest characteristic value of NX . But the largest the spectral radius p characteristic value of a positive integer matrix A (i.e.,   of AAT ) is, by Kronecker’s theorem, of the form 2 cos n , or is  2. Theorem 4.8 ([EO1]). Let C be a finite tensor category. Then C Š Rep H as a tensor category for a finite dimensional quasi-Hopf algebra H if and only if every object X of C has an integer Frobenius–Perron dimension. Proof. First suppose that every P object X is such that dC .X / 2 N. Then one can consider the object P D i dC .Xi /Pi , where Pi are projective covers of Xi , and define a functor F W C ! Vect k , X 7! Hom.P; X /, which is exact. Since F ./ ˝ F ./ and F . ˝ / extend to exact functors C  C ! Vectk that map simple objects Xi  Xj to the same images, they are isomorphic. Thus F is quasi-tensor and C Š Rep H . If C Š Rep H , then reconstruction theory says there exists a quasi-fiber functor on C. We know that such a functor preserves Frobenius–Perron dimensions, so they are integers. Corollary 4.9. If H1 , H2 are finite dimensional quasi-Hopf algebras such that Rep H1 Š Rep H2 as tensor categories, then H1 and H2 are equivalent by a twist. Proof. In the proof of Theorem 4.8 there is no choice in the definition of the quasi-fiber functor F . Thus (by reconstruction theory) H is unique up to a twist. Remark. This is not true in the infinite dimensional case. For example, consider the category C D Rep.SLq .2// of representations of the quantum group SLq .2/ with q not equal to a nontrivial root of unity. Then there are many fiber functors on C which are not isomorphic (even as usual functors). More precisely, for every m  2 one can find a tensor functor F W C ! Vect k such that dim.F .V1 // D m (where V1 is the standard 2-dimensional representation of SLq .2/). Such F can be classified and yield quantum groups of a non-degenerate bilinear form [B], [EO2].

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Finally, let us give a number-theoretic property of the Frobenius–Perron dimension in a fusion category, which allows one to dismiss many fusion rings as non-realizable. Theorem 4.10 ([ENO]). If C is a fusion category over C, then there exists a root of unity such that for every object X of C dC .X / 2 ZŒ . Example 4.11. Consider the fusion ring A with basis 1, X , Y and fusion rules X Y D 2X C Y , X 2 D 1 C 2Y and Y 2 D 1 C X C 2Y . The computation of dC .X / reduces to a cubic equation whose Galois group is S3 . So we cannot find any root of unity such that dC .X/ 2 ZŒ , and consequently A is not realizable.

4.2 FP-dimension of the category Let C be a finite tensor category with simple objects X0 ; : : : ; Xn1 . We denote by Pi the projective cover of Xi (i D 0; : : : ; n  1). Definition 4.12. The Frobenius–Perron dimension of the category C is dC .C / D P i dC .Xi /dC .Pi /. ExamplesP4.13. (i) If C is semisimple (and hence fusion), then we have dC .C / D i dC .Xi /2 . (ii) If C D Rep H for a finite dimensional quasi-Hopf algebra H , then dC .C / D dim.H /. The usefulness of this notion is demonstrated, for example, by the following result. Proposition 4.14 ([EO1]). The Frobenius–Perron dimension of the category is invariant under Morita equivalence. Remember that Z.C/ is Morita equivalent to C  C op . Thus we have Corollary 4.15. Let C be a finite tensor category. Then dC .Z.C // D dC .C /2 . We note that for spherical categories these results appear in [Mu1], [Mu2]. The following theorem plays a crucial role in classification of tensor categories, and in particular allows one to show that many fusion rings are non-realizable. Theorem 4.16 ([EO1]). If C is a full tensor subcategory of a finite tensor category D, d .D/ then dC is an algebraic integer. C .C / Examples 4.17. (i) Let D D C .G; 1/ and C D C .H; 1/ for a finite group G and its subgroup H . Then Theorem 4.16 says that jH j divides jGj (because an algebraic integer which is also a rational number is an integer). Thus Theorem 4.16 is a categorical generalization of Lagrange’s theorem for finite groups.

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(ii) Let D D Rep A and C D Rep B for a finite dimensional Hopf algebra A and a quotient B D A=I of A by a Hopf ideal I . Theorem 4.16 says d i m.B/ divides d im.A/ (this is the famous Nichols–Zoeller theorem [NZ]). The same applies to quasi-Hopf algebras (in which case the result is due to Schauenburg, [S]). Theorem 4.18 ([ENO]). If C is a fusion category with integer dC .C /, then we have dC .Xi /2 2 N for all i 2 I . of C generated by direct summands Proof. Let Cad be the full tensor subcategory L  of Xi ˝ Xi (i 2 I ), and define B D .X i ˝ Xi /. This object has an integer i FP-dimension: dC .B/ D dC .C / 2 N. Then consider M D NB ˝m , the left multiplication matrix by B ˝m in Cad . This matrix has positive entries for large enough m (since any simple object of Cad is contained in B ˝m ). Let Y0 ; : : : ; Yp be the simple objects of Cad . The vector .dC .Y0 /; : : : ; dC .Yp // is an eigenvector of M with integer eigenvalue dC .B/n . By the Frobenius–Perron theorem, this eigenvalue is simple. Thus the entries of the eigenvector are rational (as dC .Y0 / D 1) and hence integer (as they are algebraic integers). Consequently, dC .Xi ˝ Xi / D dC .Xi2 / 2 N. Example 4.19. Let C be a Tambara–Yamagami 1.26). p P (TY) category (see Example Then dC .g/ D 1 for g 2 G. Also, X 2 D g, so d .X / D jGj. Thus C g2G dC .C / D 2jGj. In the particular case of the Ising model (G D Z2 ), dC .1/ D dC .g/ D 1 and p dC .X/ D 2, and dC .C / D 4.

4.3 Global and FP-dimensions Until the end of the paper, and without precision, we will assume that our categories are over C. 4.3.1 Comparison of global and FP-dimension. Let C be a fusion category. Theorem 4.20 ([ENO]). For every simple object V in C , one has jV j2 dC .V /2 , and hence dimC dC .C /. Moreover, if dim C D dC .C /, then jV j2 D dC .V /2 for any simple V . Proof. It is sufficient to consider the pivotal case (otherwise one can take the pivotal x and recall that the forgetful functor F W C x ! C preserves squared norms extension C and FP-dimension, because it is tensor). In this case Ni dE D di dE (where di D dim Xi and dE DP.d0 ; : : : ; dn1 P /), thus by the FP theorem jdi j dC .Xi /, and this is an equality if i jdi j2 D i dC .Xi /2 .

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Remark. In general, the FP-dimension of a fusion category and its global dimension are not equal, or even Galois-conjugate (and the same is true for dC .V /2 and .dim V /2 , for any simple object V ). Now denote respectively by D and  the global and FP-dimensions of C . We already know D= 1 (previous theorem). Moreover we have Theorem 4.21 ([ENO]). D= is an algebraic integer. Proof. We can assume C is spherical. Otherwise one may consider its pivotal extension, which can be shown to be spherical (see [ENO]), and whose global and FP-dimensions are respectively 2D and 2). In this case Z.C/ is modular, of global and FP-dimensions D 2 and 2 (respectively). Let s D .sij /ij be its S-matrix. It follows from the Verlinde formula that the matrices Ni have common eigenvalues sij =s0j , and the corresponding eigenvectors are the columns of s. Since s is nondegenerate, there exists a unique label r such that the simple objects of Z.C /). sir =s0r D dC .YP i /, where Yi areP 2 ri si r D ır  r =s0r , where we used the symmetry Then 2 D i dC .Yi /2 D i ss0r s0r 2 2 D of s and the fact that s D .ıi  j /ij . So we find that r D r  and 2 D 1=s0r 2 2 2 2 2 D =.dim Xr / . Consequently D = D .dim Xr / , hence D= is an algebraic integer. Corollary 4.22 ([ENO]). Let C be a nondegenerate fusion category over a field k of characteristic p. Then its FP-dimension  is not divisible by p. z be the lifting of C , and C bDC z ˝W .k/ K Proof. Assume that  is divisible by p. Let C where K is the algebraic closure of the fraction field of W .k/. Then Theorem 4.21 b is divisible by , hence by p. So the global says that the global dimension D of C dimension of C is zero. Contradiction (C is nondegenerate). 4.3.2 Pseudo-unitary fusion categories Definition 4.23. A fusion category C (over C) is called pseudo-unitary if dim C D dC .C /. Remark. Unitary categories (those arising from subfactor inclusions, see [GHJ]) all satisfy this condition (so the terminology is coherent). Proposition 4.24 ([ENO]). Any pseudo-unitary fusion category C admits a unique spherical structure, in which dimV D dC .V /. Proof. Let b W id ! **** be an isomorphism of tensor functors, and g W id !  an isomorphism of additive functors such that g 2 D b. Let fi D dC .Xi /. Define di D tr.gXi / and dE D .d0 ; : : : ; dn1 /; then fi D jdi j by pseudounitarity. Further,

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we can define the action of g on Hom.Xi ˝ Xj ; Xk /; let .Ti /j k denote the trace of this operator. Then Ti dE D di dE, and j.Ti /j k j .Ni /j k . Thus, ˇX ˇ X .Ti /j k dk ˇˇ .Ni /j k fk D fi fj : fi fj D jdi dj j D ˇˇ This means that the inequality in this chain is an equality. In particular .Ti /j k D ˙.Ni /j k , and the argument of di dj equals the argument of .Ti /j k dk whenever .Ni /j k > 0. This implies that whenever Xk occurs in the tensor product Xi ˝ Xj , the ratio di2 dj2 =dk2 is positive. Thus, the automorphism of the identity functor defined by jXi D di2 =jdi j2 is a tensor automorphism. Let us twist b by this automorphism, i.e., replace b by b 1 . After this twisting, the new dimensions di will be real. Thus, we can assume without loss of generality that di were real from the beginning. It remains to twist the square root g of b by the automorphism of the identity functor given by jXi D di =jdi j (i.e., replace g by g ). After this twisting, the new Ti is Ni and the new dk is fk . This means that g is a pivotal structure with positive dimensions. It is obvious that such a structure is unique. We are done. Theorem 4.25 ([ENO]). Any fusion category of integer FP-dimension  is pseudounitary. In particular it is canonically spherical. Proof. Let D D D1 ; : : : ; Dm be the algebraic conjugates of D D dim C . Then x such that gi .D/ D Di , and the corresponding categories consider gi 2 Gal.Q=Q/ Ci D gi .C /. We know that Q dim Ci D Di and dC .Ci / D , so Di = 1 is an algebraic integer. Hence Q i .Di =/ is an algebraic integer 1. But it is also a rational number (because i Di ;  2 N), so it is an integer which is necessarily 1, and therefore Di D  for all i . In particular D D . Corollary 4.26 (The Larson–Radford theorem, [LR]). If H is a finite dimensional semisimple Hopf algebra over C with antipode S , then S 2 D 1. Proof. Let C D Rep H . On the one hand we know that dC .C / D dim.H / 2 N, hence C is pseudo-unitary. By example 2.4, it means dim.H / D dim C D tr.S 2 /. On the other hand, S is of finite order, so S 2 is semisimple and its eigenvalues are roots of unity. Consequently S 2 D 1.

4.4 Classification A natural classification problem for fusion categories is the following one. Problem 4.27. Classify fusion categories over C of given Frobenius–Perron dimension. The next theorem solves this problem in the case of the Frobenius–Perron dimension being a prime number p. Namely, it generalizes to the quasi-Hopf algebra case a result of Kac and Zhu on semisimple Hopf algebras of prime dimension p.

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Let C be a fusion category over C. Theorem 4.28 ([ENO]). If dC .C / D p is a prime, then C D C .Zp ; !/. In particular, any semisimple quasi-Hopf algebra H of prime dimension p is of the form H D Fun.Zp / with associator defined by ! 2 H 3 .Zp ; C / D Zp . Proof. dC .C / D p is a prime, then dC .Z.C // D p 2 2 N. Hence Z.C / has a canonical spherical structure in which di W D dim Xi D dC .Xi / for any simple object Xi . Moreover, since C is itself spherical (because it is of integer FP-dimension), Z.C / p is modular and hence p 2 =di2 is an algebraic integer. Thus di D 1 or p (as di2 2 N). p If there exists i such that di D p, then using the forgetful functor F W Z.C / ! C p we find a simple object F .Xi / in C with FP-dimension p (it is simple because the dimensions of its simple constituents must be square roots of integers). Since dC .C / D p, it is the only simple object in C . This is a contradiction (there must be a neutral object). Consequently, all simple objects in Z.C /, and hence in C also (using F ), have FP-dimension 1, i.e., are invertible. But fusion categories all whose simple objects are invertible are all of the type C .G; !/. In our case the group G must have order p, so the result is proved. With quite a bit more work, this theorem can be extended to the case of products of two primes. Theorem 4.29. If dC .C / D pq for two prime numbers p q, then either p D 2 and C is a Tambara–Yamagami category attached to the group Zq , or C is Morita equivalent to C .G; !/ with jGj D pq. Proof. The case p D q is done in [ENO, Proposition 8.32] and the case p < q is treated in [EGO].

Open problems In conclusion we formulate two interesting open problems. (1) Let us fix N 2 N (and still work over C). E. Landau’s theorem (1903) says that the number of finite groups which have N irreducible representations is finite. In the same way, the number of semisimple finite dimensional quasi-Hopf algebras which have N irreducible representations is finite (see [ENO]). It is natural to ask if the number of fusion categories over C with N simple objects is finite. In the case N D 2 this is shown in [O2], but the case N D 3 is already open. (2) Does there exists a semisimple Hopf algebra H over C whose representation category Rep H is not group-theoretical?

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For quasi-Hopf algebras, it exists (consider, e.g., a TY category related to G D Zp  Zp with the isomorphism G _ ! G corresponding to an elliptic quadratic form, see [ENO]).

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M. Müger, From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Algebra 180 (2003), 81–157.

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The Drinfeld associator of gl.1j1/ Jens Lieberum Mathematisches Institut, Rheinsprung 21, 4051 Basel, Switzerland e-mail: [email protected]

x in a completion of the Abstract. We determine explicitly a rational even Drinfeld associator ˆ universal enveloping algebra of the Lie superalgebra gl.1j1/˚3 . More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal group-like Drinfeld associator ˆ. x by a weight system. As a related result of independent interest, The associator ˆ is mapped to ˆ we show how O. Viro’s generalization 1 of the multi-variable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine ˆ by using the / of a planar tetrahedron . invariant 1 .

Contents Introduction . . . . . . . . . . . . . . . . . . . . . 1 Drinfeld associators in Ayk .3/ . . . . . . . . . 2 The ƒ.n/-algebras C.n/ and D.n/ . . . . . . y m .3/ . . . . . . . . . 3 Drinfeld associators in C k 4 Tensor products and duality of gl.1j1/-modules 5 The tensor functor W0 . . . . . . . . . . . . . 6 ƒ./-linear weight systems . . . . . . . . . . 7 The Kontsevich integral of unitrivalent graphs . 8 The Alexander series of a tetrahedron . . . . . 9 Associativity and gl.1j1/-modules . . . . . . . 10 Proof of Theorem 1 . . . . . . . . . . . . . . . y and Viro’s Alexander invariant . . . . . . . . 11 r References . . . . . . . . . . . . . . . . . . . . . .

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Introduction Building on concepts of Mac Lane and Kohno, Drinfeld introduced associators in 1989–90 in [Dr1] in context with a weakened version of the coassociativity axiom of Hopf algebras and quasitriangular Hopf algebras. Around the same time, the development of a new approach to knot theory started with the concept of Vassiliev

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invariants ([Vas]). The relation between Vassiliev invariants and Drinfeld’s work was established by Kontsevich’s analytic construction of a universal Vassiliev invariant Z and its algebraic description that requires the existence of a Drinfeld associator. The most useful and most convenient associators for topological applications are horizontal even group-like Drinfeld associators (see [BN2], [LM2], [LMO], [Lie]). In the main result of this paper we determine explicitly an even horizontal groupx in a completion of U.gl.1j1//˝3 . More generally, we define an like associator ˆ algebra C .n/ over ƒ.n/ D QŒd1˙1 ; : : : ; dn˙1  that is generated by trivalent diagrams on n strings modulo some graphical relations. The elements di correspond to certain elements in the center of U.gl.1j1//˝n , hence it is natural to treat them as coefficients of trivalent diagrams. The topological significance of the elements di comes from the construction of the Alexander polynomial using gl.1j1/, where exp.di =2/ arises as the indeterminate corresponding to a meridian of the i -th component of a link (see Corollary 32). Let C 0 .3/ be the Q-subalgebra of C .3/ generated by trivalent diagrams. In a completion Cc 0 .3/ of C 0 .3/ there exists a unique Drinfeld associator of the form y 0 .3/; ˆ D exp.F  Œt 12 ; t 23 / 2 C where F 2 QŒŒC; D; E  QŒŒd1 ; d2 ; d3  is a formal power series that starts with F D

C C 4D 4C 2 C 36CD C 48D 2  31E 1  C 24 5760 2903040 6C 3 C 96C 2 D C 240CD 2 C 192D 3 C 13CE  184DE  C  464486400

and C D d1 d3  d2 .d1 C d2 C d3 /;

D D .d1 C d2 /2 C .d2 C d3 /2 ;

E D .d1 C d2 /2 .d2 C d3 /2 : The main result of this paper is the following theorem. Theorem 1. The series F is the unique solution of the equation s sinh.F k/ '.d2 /'.d1 C d2 C d3 / D cosh.F k/ C d1 d3 k '.d1 C d2 /'.d2 C d3 / where '.x/ D 2 sinh.x=2/=x and k 2 D d1 d2 d3 .d1 C d2 C d3 /. Due to a computation of P. Vogel the solution F is given explicitly by where

F D X ‰.d1 d3 X; d2 .d1 C d2 C d3 /X /;

(1)

  1  X 1=2 1=2 up v q .u C 2/pCqC1 ‰.u; v/ D 2.p C q C 1/ p q p;qD0

(2)

The Drinfeld associator of gl.1j1/

41

and s X D d11 d31

! '.d2 /'.d1 C d2 C d3 / 1 : '.d1 C d2 /'.d2 C d3 /

(3)

y 0 .3/ can be characterized as the unique even horizontal The associator ˆ 2 C group-like Drinfeld associator inside of completions of k ˝ C 0 .3/, where k is any y 1 .3/ is the unique even commutative Q-algebra. The image of ˆ in a quotient C group-like Drinfeld associator inside of this quotient (see Theorem 13). By mapping x mentioned before we see that the gl.1j1/-weight system maps ˆ to the associator ˆ x every even horizontal group-like Drinfeld associator to ˆ. The quantum supergroup Uq .gl.1j1// has been used by O. Viro to extend the multi-variable Alexander polynomial of links to an invariant 1 of colored embedded trivalent graphs ([Vir]). In general, the invariant of a trivially embedded tetrahedron colored by representations of a quantum group provides the relation between R-matrices and 6j -symbols ([Tu2]). Although this construction neither extends in its full generality to quantum supergroups nor to versions of Vassiliev invariants for 3-manifolds the value 1 . / and its relation to the Kontsevich integral Z turned out x explicitly. The translation between 1 . / and ˆ x is in to be useful to determine ˆ the spirit of a general connection between well-behaved invariants of trivalent graphs and associators that is investigated by D. Bar-Natan and D. Thurston. We extend the relation between 1 and the Kontsevich integral Z to arbitrary trivalent graphs (Theorem 31). This is a result of independent interest. It generalizes an unpublished proof of A. Vaintrob who related the multi-variable Alexander polynomial of links to the Kontsevich integral. With the standard definition the Kontsevich integral of trivalent graphs ([MuO]) Theorem 31 would only hold up to a factor that depends on the colored graph but not on its embedding. In order to avoid this factor, to simplify computations, and to emphasize the roles played by cyclic orientations of vertices and half-framings we introduce a different normalization of the Kontsevich integral of oriented trivalent graphs (Theorem 20). The paper is organized as follows. In Section 1 we recall definitions and properties of a module A./ of trivalent diagrams on an oriented unitrivalent graph  and of y of A.3 / where n consists of n intervals. Drinfeld associators in a completion A.3/ In Section 2 we introduce the module C ./ and investigate its structure for  D n (Theorem 8 and Corollary 9). Section 3 contains results about Drinfeld associators y 0 .3/ and in a quotient C y 1 .3/ of C y 0 .3/ (Theorem 13). In particular, we establish in C the existence and uniqueness of the series F and deduce equations (1) to (3) from Theorem 1. In Sections 4 to 9 we prepare the proof of Theorem 1 that will be given in Section 10. We start by recalling some representation theory of gl.1j1/ in Section 4 and use it in Sections 5 and 6 to associate linear maps W ı W C ./ ! Q to colored trivalent graphs  . In Section 7 we introduce the non-standard normalization of the Kontsevich integral Z mentioned above.

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y by composing Z and W ı and compute In Section 8 we define theAlexander series r y . Remarkably, this computation is r for trivially embedded colored tetrahedra possible without knowing an explicit formula for an associator. Section 9 contains computations involving morphisms between tensor products of three simple gl.1j1/modules, among them a formula for a change of bases induced by the associativity of the tensor product. In Section 10 we apply these formulas. We compute the Alexander series of for a different diagram and obtain an expression that depends non-trivially on an associator. Then we prove Theorem 1 by comparing this expression with the value from Section 8. y Sections 4 to 8 are also used in Section 11 where we relate the Alexander series r 1 to Viro’s Alexander invariant  (Theorem 31). Acknowledgements. I would like to thank D. Bar-Natan, C. Kassel, G. Masbaum, D. Thurston, and P. Vogel for helpful discussions, and O. Viro for writing [Vir] and for sending me a preliminary version of that paper.

y k .3/ 1 Drinfeld associators in A A graph is called unitrivalent if all of its vertices have valency one or three. Let  be a unitrivalent graph with oriented edges and cyclically oriented vertices. As an exception, we allow circles as connected components of  that we consider as a single oriented edge without vertex. The graph  may have multiple edges between vertices. When  has no univalent vertex, we call it a trivalent graph. Let V .G/ (resp. E.G/) be the set of vertices (resp. edges) of a unitrivalent graph G. A trivalent diagram D with skeleton  is a unitrivalent graph G whose univalent vertices are glued to  n V ./ by an injective gluing map. The unitrivalent graph G has the following properties: trivalent vertices of G are cyclically oriented, but in contrast to , edges of G are not oriented. In addition, we require that each connected component of G has at least one univalent vertex. We represent a trivalent diagram graphically by a generic picture of  [ G in the plane. We use thicker lines to draw  than we use for G. We assume that cyclic orientations of oriented vertices are always counterclockwise. When it is of importance we indicate orientations of edges of  by arrows, and we include the names i; j; : : : of edges (resp. vertices) in our graphical representation of trivalent diagrams by writing them close to the corresponding edges (resp. vertices). Homeomorphisms h W  [ G !  [ G 0 between trivalent diagrams on  have to respect orientations of edges and vertices and induce a homeomorphism of  that is homotopic to the identity. By abuse of language we call the homeomorphism class of a trivalent diagram simply trivalent diagram. For a vertex v 2 V ./ and an edge i 2 E./ that is incident to v, we define sv;i 2 f˙1g by sv;i D 1 if the edge i is oriented towards v and sv;i D 1 otherwise. We specify relations between trivalent diagrams by using graphical representations of the part where these diagrams differ.

43

The Drinfeld associator of gl.1j1/

Definition 2. Let A./ be the Q-vector space generated by trivalent diagrams on  modulo the relations (STU) and (InvV).

D

.STU/ .InvV/



v

sv;i

; v

C sv;j

j

i

v

C sv;k

D 0: k

The signs on the right side of relation (STU) depend on the cyclic order of the trivalent vertex and on the orientation of the shown part of  in this relation. The degree of a trivalent diagram D D  [ G is defined by deg.D/ D .1=2/ #V .G/. This definition induces a grading on A./. It is well known (see [BN1]) that the relations (IHX) and (AS) below are consequences of the (STU) relation. .IHX/

D



;

.AS/

D

:

` A basis of A./ is not known explicitly. Let n D niD1 Ii be the disjoint union of oriented intervals Ii . The bijection between f1; : : : ; ng and the intervals of n is a part of the definition of n . The vector spaces A.n/ D A.n / are particularly interesting for several reasons. One reason is that for connected trivalent graphs  there exist isomorphisms A./ Š A.b1 .// where b1 is the first Betti number. A second reason is that A.n/ is an algebra in the following way. We represent trivalent diagrams on n graphically in the strip R  Œ0; 1 such that @Ii D fi g  f0; 1g and these intervals direct from .i; 1/ to .i; 0/. We say that f1g  f1; : : : ; ng (resp. f0g  f1; : : : ; ng) is the upper (resp. lower) boundary of a trivalent diagram on n . The product ab of trivalent diagrams a, b is induced by gluing the lower boundary points of a to the corresponding upper boundary points of b. The 1-element of A.n/ is the diagram n . For a commutative Q-algebra k we define the N-graded k-module Ak ./ ´ k ˝Q A./ and denote its homogeneous components k ˝Q A./i by Ak ./i . The case k 6D Q will only be important in this section and in Section 3. When k D Q we omit the symbol k. We define the completion Ayk ./ by Ayk ./ D

1 Y

Ayk ./i :

iD0

WeP represent elements of completions of graded vector spaces by formal power series 1 iDk ai with homogeneous elements ai of degree i . We consider completions as metric spaces (and in particular as topological spaces) by d

1 X iDk

ai ;

1 X iDk



bj D

1 X iDk

ıai ;bi 2i

44

Jens Lieberum

where ıai ;bi 2 f0; 1g is 0 iff ai D bi . The algebra structure on Ak .n/ ´ Ak .n / extends in a unique way to a topological algebra structure on Ayk .n/ ´ Ayk .n /. Define continuous k-linear maps i W Ayk .n/ ! Ayk .n C 1/ (i D 0; : : : ; n C 1), where i .D/ (i D 1; : : : ; n) is obtained from the trivalent diagram D by replacing the i-th interval of D by two copies and by summing over all ways of lifting the univalent vertices that are glued to the i -th interval to the copies of that interval. This sum has 2` terms when ` univalent vertices are glued to the i -th interval of D. We label the new intervals in i .D/ by i and i C 1 and replace labels j with j > i by j C 1. Define nC1 .D/ as the union of D with a new skeleton component labeled n C 1, and define 0 .D/ by first adding to D a new skeleton component labeled 0 followed by replacing all labels i of the skeleton components by i C 1. Define i W Ayk .n/ ! Ayk .n  1/ by i .D/ D 0 if the i -th interval of D contains a univalent vertex of D n , and by deleting the i -th interval and by replacing labels j > i by j  1 otherwise. y Let f W f1; : : : ; mg ! f1; : : : ng be an injective map. For a 2 A.m/ we denote f .1/:::f .m/ y by a 2 A.n/ the element obtained by applying f to the labels of the skeleton components of trivalent diagrams in a and by adding intervals with labels i 2 f1; : : : ; ng n f .f1; : : : ; mg/ to the skeleton of these diagrams. Let t ij be the unique trivalent diagram on n of degree 1 where the i -th edge of n is connected to the j -th edge of n by a single edge. Define R 2 Ayk .2/ by R D exp.t 12 =2/. Now we are ready to define a Drinfeld associator. Definition 3. A Drinfeld associator ˆ in Ayk .3/ is a solution of equations (DA1)– (DA4) in Ayk .n/ (n D 4; 3; 3; 2): (DA1)

0 .ˆ/ ı 2 .ˆ/ ı 4 .ˆ/ D 3 .ˆ/ ı 1 .ˆ/;

(DA2)

1 .R12 / D ˆ312 ı R13 ı .ˆ132 /1 ı R23 ı ˆ;

(DA3) (DA4)

ˆ ı ˆ321 D 1; 1 .ˆ/ D 2 .ˆ/ D 3 .ˆ/ D 1:

Equation (DA4) implies that there exists a unique P 2 Ayk .3/ such that ˆ D exp.P /. Let P .Ayk .n// be the closed k-submodule of Ayk .n/ consisting of series that involve only trivalent diagrams D with the property that D n n is non-empty and connected. When P 2 P .Ayk .n// we say that exp.P / is group-like. A Drinfeld associator is called horizontal, if it lies in the closed subalgebra of Ayk .3/ generated by t 12 and t 23 . A horizontal Drinfeld associator uniquely determines a formal series S 2 khhA; Bii in non-commutative, associative indeterminates A and B such that ˆ D S.t 12 ; t 23 / (see [BN3], Fact 9 and Corollary 4.4). When ˆ is group-like the series S satisfies S D exp.p/ for a Lie series p in A and B over k. P We say that a Drinfeld associator ˆ is even if ˆ D 1 iD0 a2i with a2i 2 Ak .3/2i . For the following fact see [Dr2], Theorem A00 and [BN4], Corollary 4.2. (DA5)

45

The Drinfeld associator of gl.1j1/

y Fact 4. There exists an even horizontal group-like Drinfeld associator in A.3/. Associators in Ayk .3/ are not unique, but for two associators ˆ1 , ˆ2 there exists an element T 2 Ayk .2/ satisfying 1 .T / D 2 .T / D 1 and T 21 D T such that ˆ1 D 0 .T /2 .T /ˆ2 1 .T 1 /3 .T 1 / (see Theorem 8 of [LM1]). We say that ˆ1 and ˆ2 are related by a twist T . Grouplike (resp. even) associators are related by group-like (resp. even) twists. The relation between horizontal associators is more involved: the proofs of Fact 4 rely on an action of the formal Grothendieck–Teichmüller group on horizontal group-like Drinfeld associators and on the existence of a horizontal group-like Drinfeld associator ˆKZ 2 AyC .3/.

2 The ƒ.n/-algebras C .n/ and D.n/ A well-known construction induces a map A.n/ ! U.gl.1j1//˝n (see [FKV]). Instead of studying this map and its image Un we will introduce relations between trivalent diagrams in this section that belong to the kernel of this map. This will lead us to a graphical definition of an algebra C .n/ which is small enough such that we can explicitly describe a basis and that has a more convenient algebraic structure than Un (e.g. C.n/ will not contain nilpotent elements but Un does). Since gl.1j1/ will be important to analyze the structure of C .n/ and motivates its definition, we will first recall the definition gl.1j1/. Let V D V0 ˚V1 be a Z=.2/-graded vector space. The algebra End.V / has a natural Z=.2/-grading with homogeneous components End.V /0 D End.V0 / ˚ End.V1 / and End.V /1 D Hom.V0 ; V1 / ˚ Hom.V1 ; V0 /. The map str W End.V / ! Q defined by str.'/ D tr.'j End.V0 / /  tr.'j End.V1 / / is called supertrace. The number sdim.V / D dim.V0 /  dim.V1 / D str.idV / is called the superdimension of V . We consider End.V / as a Lie superalgebra gl.V / with bracket Œ  ;   induced by ŒA; B D AB  .1/deg.A/ deg.B/ BA for homogeneous elements A; B 2 End.V / D gl.V /. For dim V0 D dim V1 D 1 the Lie superalgebra gl.V / is isomorphic to gl.1j1/ which has a homogeneous basis given by the following four matrices:         1 0 1 0 0 1 0 0 H D ; DD ; ED ; F D : (4) 0 1 0 1 0 0 1 0 The non-vanishing brackets of basis elements are given by ŒH; E D ŒE; H  D 2E;

ŒF; H  D ŒH; F  D 2F;

ŒE; F  D ŒF; E D D:

46

Jens Lieberum

Since all brackets involving D vanish, we see that D is in the center of gl.1j1/, hence Di D 1˝i1 ˝ D ˝ 1˝ni is in the center of U.gl.1j1//˝n . We regard U.gl.1j1//˝n as a ƒC .n/ ´ QŒd1 ; : : : ; dn -algebra, where di acts by multiplication with Di . Let ƒ.n/ ´ QŒd1˙1 ; : : : ; dn˙1 . Notice that by the Poincaré–Birkhoff–Witt theorem (see [Kac]) U.gl.1j1//˝n is a free ƒC .n/-module, hence the canonical map U.gl.1j1//˝n ! ƒ.n/ ˝ƒC .n/ U.gl.1j1//˝n μ U.n/ is injective. Instead of describing elements in the kernel of A.n/ ! U.gl.1j1//˝n directly it will be more convenient to describe elements in the kernel of ƒ.n/ ˝Q A.n/ ! U.n/ in Definition 5 below. More generally, we first adapt the definition of the ring ƒ.n/ to unitrivalent graphs  such that the ring ƒ.n/ belongs to the graph n . For this purpose, recall some facts about H ´ H 1 .; @; Z/. For an edge e 2 E./ let ˛e 2 H be given by the 1-cocycle that evaluates on a 1-chain c to the coefficient of e in c. Then H is the abelian group generated by ˛e .e 2 E.// and the relations sv;i ˛i C sv;j ˛j C sv;k ˛k D 0 for each trivalent vertex v of  that is incident to the three edges i; j; k 2 E./. A simple computation or a graphical argument using H Š Hom.H1 .; @/; Z/ imply that ˛e D 0 iff the number of connected components of  increases when we cut  at a point p in the interior of e, and at least one of the new connected components contains no univalent vertex besides p. Let ƒ./ be the commutative Q-algebra generated by elements di ; di1 (i 2 E.// modulo relations di di1 D 1 for each i 2 E./ and relations sv;i di C sv;j dj C sv;k dk D 0 for each vertex as above. If ˛e D 0 for some e 2 E./, then de D 0 and the invertibility of de implies ƒ./ D f0g1 . Otherwise, ƒ./ is isomorphic to a localization of a Q-algebra of polynomials in rank.H / indeterminates. Definition 5. Let C ./ be the quotient of ƒ./ ˝Q A./ by the relations (CL1A) and (CL2A).

i

(CL2A)

D 2di2



(CL1A)

i

i

j

D di dj

i

j

; i

:

Both relations in the definition of C ./ relate ‘coefficients’ to ‘legs’ of trivalent diagrams what explains the letters C and L in the names of the relations. The following relations are consequences of the definition of C ./. (LS)

i

i

D

i

i

;

(IntV)

D

D 0:

1 This case is not important for our purpose so we did not need to work with a more complicated definition of ƒ./ in this case.

47

The Drinfeld associator of gl.1j1/

Relation (LS) (‘leg slide’) follows by two-fold application of relation (CL2A): first add a pair of legs and then remove a pair of legs on the edge i . Adding legs by using (CL2A) is possible due to the invertibility of di2 . In context with relation (STU), relation .LS/ is equivalent to relations (IntV) (‘internal vertex’). The definition of deg.D/ for a trivalent diagram D and deg.de / D 1 (e 2 E./) induce a Z-grading of C ./. Consider the N-graded Q-subalgebra ƒC ./ of ƒ./ generated by elements de (e 2 E./). Define C C ./ D .ƒC ./ ˝ A.//

C 0 ./ D  .A.//

and

where  W ƒ./˝A./ ! C./ denotes the canonical projection. Let C .n/ D C .n / and ƒ.n/ D ƒ.n /. There exists a unique structure of a ƒ.n/-algebra on C .n/ such that the map  is a homomorphism of rings. The space C C .n/ ´ C C .n / (resp. C 0 .n/ ´ C 0 .n /) is a ƒC .n/ ´ ƒC .n /-subalgebra (resp. Q-subalgebra). In the rest of this section we will establish a basis of C .n/ (see Corollary 9). This will mainly be important for proving uniqueness results about associators. The urgent reader can skip the rest of this section and come back to it when needed. A unitrivalent diagram on a set M is a unitrivalent graph with oriented trivalent vertices and at least one univalent vertex on each connected component together with an assignment of a label in M to each univalent vertex. Examples of unitrivalent diagrams can be obtained from trivalent diagrams D on n by labeling the univalent vertices of D n n according to the connected components of n . Definition 6. Let D.n/ be the ƒ.n/-module generated by unitrivalent diagrams on the set f1; : : : ; ng modulo the relations (IHX), (AS), (IntV), (CL1B), (CL2B). i i i

(CL1B) j

(CL2B)

i

D di2

i

k

j

D di dj

k j

; :

Relation (CL1B) concerns connected components of unitrivalent diagrams because all univalent vertices of the diagrams in the relation have labels. Relation (CL2B) can be applied to parts of connected components of unitrivalent diagrams. Let D be a unitrivalent diagram on f1; : : : ; ng. Let Wi be the set of all linear orders on the set of univalent vertices of D labeled i with the following property: for all connected components C of D and all univalent vertices l1 < l2 of C labeled i a univalent vertex l3 labeled i with l1 < l3 < l2 also belongs to C . We define W D W1      Wn and 1 X .D/ D Dw ; #W w2W

where Dw is the trivalent diagram on n obtained by gluing the univalent vertices labeled i of D to the i-th interval of n according to the order w.

48

Jens Lieberum

Proposition 7. The definition of  induces an isomorphism of Z-graded ƒ.n/-modules D.n/ ! C .n/. Proof. First verify that  is well-defined: it is clear that  is compatible with relations (AS), (IHX), (IntV) and it is easy to see that  is compatible with relation (CL2B). By relation (LS) the second vertex2 on the interval i in relation (CL1A) can be moved freely on that interval. We use this to see that  is compatible with relation (CL1B) in the case j 6D i 6D k. The compatibility of  with relation (AS) implies compatibility with relation (CL1B) in the cases i D j and i D k. When D and D 0 are trivalent diagrams on n that differ only by the order of their univalent vertices on n , then by relation (STU) the element D  D 0 2 C .n/ can be expressed in terms of diagrams D 00 such that D 00 n n has more trivalent vertices than D n n . This implies that  is surjective. The proof of the injectivity of  is more difficult and similar to the proof of Theorem 8 in [BN1]. Let a ;

G.n/ D b

1

d c

1

;

1

1

; e

f

 ˇ ˇ ˇ 1  a  b  n; 2  c < d  n; 2  e  f  n :

1

The disjoint union of diagrams turns D.n/ into a commutative ƒ.n/-algebra whose unit element is the empty diagram. Theorem 8. The commutative ƒ.n/-algebra D.n/ is freely generated by G.n/. The proof of Theorem 8 will occupy the rest of this section. We obtain the following corollary of Theorem 8 by using Proposition 7 and the ascending filtration of C .n/ defined for trivalent diagrams D by the number of connected components of D n n . Corollary 9. The ƒ.n/-module C .n/ is free. For any order on G.n/ a basis of C .n/ is given by ordered monomials in .G.n//. By Corollary 9 the ƒC .n/-module C C .n/ is torsion-free. This can be seen directly by using that the Q-vector space C C .n/ admits an Nn -grading where the elements di act by isomorphisms of degree .0; : : : ; 0; 1; 0; : : : ; 0/ where the 1 is at the i -th place. Using the same grading and Corollary 9 one sees that for n > 1 the ƒC .n/-module C C .n/ is not free. We call a unitrivalent diagram D a comb if it is a tree whose unique spanning tree for the set of trivalent vertices is empty or a point or homeomorphic to an interval. We say that a unitrivalent diagram D is a wheel when it contains a circle whose complement is a disjoint union of intervals and each of these intervals has exactly one labeled univalent vertex. The first step in the proof of Theorem 8 is the following lemma. 2 Notice that our definition of  is slightly different from the standard definition of  (see [BGRT], Def. 2.7) because the standard definition would not be compatible with relation (CL1B).

49

The Drinfeld associator of gl.1j1/

Lemma 10. The ƒ.n/-algebra D.n/ is generated by G.n/. Proof. By relation (IntV) the algebra D.n/ is generated by trees and wheels. By relation (AS) wheels of odd degree whose univalent vertices have the same label are trivial in D.n/. Relation (CL2B) then implies that wheels of odd degree are trivial 1by a monomial in di˙1 in D.n/ and wheels of even degree are related to 1 (i D 1; : : : ; n). Now we consider trees. By relation (IHX) it is sufficient to consider combs. By relation (CL2B) we only have to consider combs of degrees 1, 2, 3, and 4. All combs of degree 1 appear in our list of generators, so there is nothing to do. Now we treat combs of degree 3. In the computation below we use relations (CL2B), (IHX) and (AS), and (CL2B) and (AS) to write such a comb as a linear combination of combs that have two univalent vertices labeled by 1. d12

i

j

i 1 1 j

D k

`

i 1 1 j

i 1

D k

`

k

1

`

1

D dk d`

k 1

j

i

1 j

 k 1 1

`

`

j

k 1 1 `

1

 di d` i

i

C

1 `

1

 dj dk i

j



1

C di dj k

(5)

1

:

j

k

`

By relation (AS), the symmetry of the comb of degree 3, and by invertibility of d1 we see that we can write a comb of degree 3 as a linear combination of elements of G.n/. We continue with combs of degree 4. The proof of the first equality below is similar to the computation in equation (5), and the second equality follows from relations (CL1B) and (AS). 1 d13 dm

i mj

1 1 1

D dk d` k

`



1 1 1

 dj dk i

D d12 dk d`

j

i j

1

i

1 1 1

 di d` `

C dj dk

k 1

i `

1 1 1

C di dj j

C di d`

k 1

k j

`

C di dj

 `

1

k

(6) :

By equation (6) and relation (AS) we can write combs of degree 4 as linear combinations of elements of G.n/. We apply equation (6) to combs of degree 2 as follows: d1

j

` k

D d1 dj2

j j j

D dk k

`

1

j `

C d`

1

k j

C dj

1

` k

:

This completes the proof. In the proof of the following lemma we use the Lie superalgebra gl.1j1/. Lemma 11. The elements of G.n/ are ƒ.n/-linearly independent in D.n/.

50

Jens Lieberum

Proof. The algebra Sym.gl.1j1/˚n / is generated by Hi ; Di ; Ei ; Fi (i D 1; : : : ; n), where Xi denotes the element X of the i -th copy of gl.1j1/ (see equation (4)). We consider Sym.gl.1j1/˚n / as a module over QŒd1 ; : : : ; dn  where di acts by multiplication with Di . A well-known construction using the element ! (see equation (32)) and equation (9) shows that there exist morphisms of ƒ.n/-algebras un W D.n/ ! ƒ.n/ ˝QŒd1 ;:::;dn  Sym.gl.1j1/˚n /gl.1j1/ satisfying un . i  un 1

j/

 j i

1

1

i

j

!

D d1 .Ej Fi C Fj Ei / C di .E1 Fj C F1 Ej / C dj .Ei F1 C Fi E1 /; (7) D d12 .Fi Ej C Fj Ei / C d1 di .E1 Fj C Ej F1 /

un

un .1

D .1=2/.dj Hi C di Hj / C Fi Ej  Ei Fj ;

(8)

C d1 dj .Ei F1 C E1 Fi / C 2di dj F1 E1 ; 1/

D

2d12 :

The ƒ.n/-algebra D.n/ has an N3 -grading given for unitrivalent diagrams D by @.D/ D .@1 .D/; @2 .D/; @3 .D//, where @1 .D/ is the number of connected components of D of degree 1, @2 .D/ is the number of connected components in D of even degree, and @3 .D/ is the number of connected components in D of odd degree  3. The elements of G.n/ are homogeneous with respect to @. The formulas for un .D/ (D 2 G.n/) from above imply that elements of G.n/ of the same degree are ƒ.n/-linearly independent. It follows from computations of [FKV] that the maps un from the proof of Lemma 11 are compatible with relations (IntV) and satisfy u n .i

j/ D 2di dj ;   (9) 1  un C   D un : 2 Equations (9) or direct computations imply that un is compatible with relations (CL1B) and (CL2B). The maps un are not injective because the elements in equations (7) and (8) are nilpotent. Therefore, we cannot use the maps un to complete the proof of Theorem 8.



Proof of Theorem 8. Let Qn be the quotient field of ƒ.n/. Consider the commutative Qn -algebra Dn D Qn ˝ƒ.n/ D.n/. Let the degree of a unitrivalent diagram be the number of its components. Then Dn admits the structure of a connected primitively generated graded Hopf algebra of finite type over Qn whose space of primitive elements Pn is the homogeneous part of degree 1 of Dn and whose counit is the augmentation map. By [MiM] the algebra Qn is freely generated by Pn . By Lemmas 10 and 11 the map D.n/ ! Dn induced by ƒ.n/  Qn is injective and maps G.n/ to a basis of Pn . This completes the proof.

51

The Drinfeld associator of gl.1j1/

y m .3/ 3 Drinfeld associators in C k Let C 1 .n/ be the quotient of C 0 .n/ by the ideal I1 generated by trivalent diagrams D y 0 ./, ƒ y C .n/, C y 0 .n/, C y 1 .n/) O C ./ (resp. C O C .n/, C with b1 .D n n / > 0. Define ƒ k k k k k k by extending coefficients of ƒC ./ (resp. C 0 ./, ƒC .n/, C C .n/, C 0 .n/, C 1 .n/) from Q to the commutative Q-algebra k followed by completion. Define continuous morphisms of k-algebras QiW ƒ O C .n/ ! ƒ O C .n C 1/  k k

.i D 0; : : : ; n C 1/;

Q i .dj / D dj for j < i,  Q i .dj / D dj C1 for j > i ,  Q i .di / D di C diC1 , and  O C .n/ ! ƒ O C .n  1/ Qi W ƒ k k

.i D 1; : : : ; n/;

Qi .dj / D dj for j < i, Qi .dj / D dj 1 for j > i, Qi .di / D 0. y Q i ˝ i W ƒ O C .n/ ˝ A.n/ O C .n C 1/ ˝ Ayk .n C 1/ and Then the maps  ! ƒ k k C C y y  1/ induce continuous linear maps O .n/ ˝ A.n/ O .n  1/ ˝ A.n Qi ˝ i W ƒ ! ƒ k k y m .n/ ! C y m .nC1/ i W C k k

and

y m .n/ ! C y m .n1/ i W C k k

.m D0 C0 ; 0; 1/: (10)

The proof that i in equation (10) is well defined uses an Nn -grading of C m .n/. A direct proof that i is well defined uses that C C .n/ is a torsion-free ƒC .n/-module and requires a computation to ensure compatibility with relation (CL1A). y C .3/ (resp. C y m .3/ with m D 0; 1) is a Definition 12. A Drinfeld associator ˆ in C k k y C .n/ (resp. C y m .n/). solution of equations (DA1)–(DA4) in C k k y m .n/ commutes with i and i there Since the canonical map m from Ayk .n/ to C k y m .3/ by Fact 4. We use the exist even horizontal group-like Drinfeld associators in C k y m .2/ (m D0 C0 ; 0; 1) as defined in Section 1. For example, notion of a twist T 2 C k   exp.d12 d22 / and exp d1 d2 .d1 C d2 / y C .2/. With the definition of even, horizontal, and groupare non-trivial twists in C k y m .3/ (m D 0; 1) as in Section 1, we have the following like Drinfeld associators in C k theorem. Theorem 13. (1a) There exists exactly one even horizontal group-like Drinfeld assoy 0 .3/. ciator in C k y 1 .3/. (1b) There exists exactly one even group-like Drinfeld associator in C k (2) The unique Drinfeld associator in (1a) and (1b) is equal to y m .3/  C y m .3/ .m D 0; 1/ exp.F  Œt 12 ; t 23 / 2 C k O for a unique F 2 QŒŒC; D; E  ƒ.3/, where the inclusion is given by C D 2 2 .d1 C d2 / C .d2 C d3 / , D D .d1 C d2 /2 .d2 C d3 /2 , E D d1 d3  d2 .d1 C d2 C d3 /.

52

Jens Lieberum

Proof. Existence of F: Let ˆ be an even horizontal group-like Drinfeld associator in y A.3/. By .DA5/ we have ˆ D exp.p.t 12 ; t 23 // for a Lie series p in A and B that involves only terms of even degrees. The free Lie algebra with two generators A and B is spanned linearly by a set K that is defined recursively by A; B 2 K and by adA .c/; adB .c/ 2 K for c 2 K. Therefore Lemma 14 below implies that ˆm D m .ˆ/ D exp.F  Œt 12 ; t 23 / for some F 2 QŒŒ.d1 C d2 /2 ; .d2 C d3 /2 ; E. Equation (DA3) implies that p saty .3/. As a consequence, F is invariant under the isfies p.t 23 ; t 12 / D p.t 12 ; t 23 / 2 C 2 permutation of .d1 C d2 / and .d2 C d3 /2 . This implies F 2 QŒŒC; D; E. Uniqueness of ˆ1 : The proofs of Theorems 8 and 9 of [LM1] can be adapted to y 1 .3/ is related to ˆ1 by an even group-like see that any even group-like associator in C k y 1 .2/ (see also Lemma 4.17 of [BN2]). Corollary 9 implies that T D 1. twist T 2 C k Therefore ˆ1 is unique. Uniqueness of F : Corollary 9 implies that the ƒC .n/-module M D C C .n/=ƒC .n/I1  C 1 .n/ is torsion-free and 0 6D Œt 12 ; t 23  2 M . This implies that F 2 ƒC .3/ is uniquely determined by ˆ1 . It is easy to see that C , D, E are algebraically independent by using that d1 C d2 , d2 C d3 , E are algebraically independent. Therefore F is uniquely determined as a formal series in C , D, E. Uniqueness of ˆ0 : Except for the uniqueness of the Lie series p equation (DA5) y 0 .3/. As in the first holds also for horizontal group-like Drinfeld associators in C k part of the proof we see that an even horizontal group-like Drinfeld associator ‰ in y 0 .3/ can be expressed as ‰ D exp.F 0  Œt 12 ; t 23 / for some F 0 2 kŒŒC; D; E. Let C k y 0 .n/ ! C y 1 .n/ be the canonical projection. Then W C k k .‰/ D exp.F 0  Œt 12 ; t 23 / D ˆ1 D exp.F  Œt 12 ; t 23 / by the uniqueness of ˆ1 which implies F D F 0 by the uniqueness of F . This completes the proof. It follows from Theorem 13 that any even group-like associator in Ayk .3/ is mapped y 1 .3/ by the canonical projection. The to an associator with rational coefficients in C k next lemma was used in the proof of Theorem 13. Lemma 14. The following identities hold in C 0 .3/: Œt 12 ; Œt 12 ; Œt 12 ; t 23  D .d1 C d2 /2 Œt 12 ; t 23 ; 23

23

12

23

2

12

23

Œt ; Œt ; Œt ; t  D .d2 C d3 / Œt ; t ; Œt 12 ; Œt 23 ; Œt 12 ; t 23  D .d1 d3  d2 .d1 C d2 C d3 //Œt 12 ; t 23  D Œt 23 ; Œt 12 ; Œt 12 ; t 23 :

(11) (12) (13)

53

The Drinfeld associator of gl.1j1/

Proof. In the computation in C 0 ./ below, the first equality follows from relations (LS) and (STU). For the second equality we apply relations (LS), (AS), and (CL2A). The third equality follows from relation (CL1A). i

k j

i

1 D 2

k

i

k

C

C

j i

1 D 2

j

k

i

j i

j

k

 j

di2 2

j

k



D di2

k!

i

i

 di2

i

k!

(14) j

k

: j

Now we prove equation (15) by the computation in C 0 .3/ below. The three equalities in this computation follow by applying relations (CL2A) and (AS), equation (14), relations (STU), (AS), and (CL2A), respectively: D d11 d3

 D d11 d3 d12



 .1=2/d12

D .1=2/.d1 C d2 /d3

(15)

:

In the computation below, we apply relations (STU) and (AS) to obtain the first three equalities. The fourth equality follows from equations (14) and (15) and from relation (CL2A) and (AS). The fifth equality is implied by equation (14). Œt 12 ; Œt 12 ; Œt 12 ; t 23  h h D t 12 ; t 12 ; h D t 12 ; 

ii

C

D D d22

C2  .1=2/d2 d3

C .d1 C d2 /d3  .1=2/d1 d3

i

C

C d11 d2

C

C

C d1 d21 C d12

54

Jens Lieberum

D .d12 C d22 /Œt 12 ; t 23  C .1=2/.d1 C d2 /d3  C d1 d21 d22  .1=2/d2 d3   .1=2/d1 d3 C d11 d2 d12

 

D .d1 C d2 /2 Œt 12 ; t 23 : This proves equation (11). Equation (12) follows by applying the Q-algebra automorphism of C 0 .3/ induced by interchanging the intervals 1 and 3 of the skeleton. The proof of equation (13) is similar to the proof of equations (11) and (12). Lemma 14 and Theorem 13 imply that the denominator of the homogeneous part of degree 2n in exp.F  x/ 2 QŒŒC; D; E; x (deg.x/ D 2) is a lower bound for this y denominator in any even group-like associator in A.3/. Now we come to P. Vogel’s proof of the formula for F using Theorem 1. Proof of equations (1) to (3). There exists a unique solution ‰ 2 QŒŒu; v of p p p 1 C u D cosh.‰ uv/ C u=v sinh.‰ uv/ D 1 C u‰ C .1=2/uv‰ 2 C    : (16) By Theorem 1 the series ‰ is related to F by equations (1) and (3). Equation (16) implies the following equation in R D QŒŒu; vŒ.uv/1=2 : p p p p .1 C u=v/e 2‰ uv  2.1 C u/e ‰ uv C 1  u=v D 0: p p p The solutions in RŒ v; . u C v/1  of this quadratic equation are p p p p p v.1 C u/ ˙ u 1 C v.u C 2/ 1 C u ˙ .1 C u/2  1 C u=v ‰ uv e D D p p p uC v 1 C u=v where we can determine the sign ˙ because ‰ 2 QŒŒu; v. Let H 2 RŒŒw be the unique solution of p p p p p v 1 C uw C u 1 C vw H uv D 2 RŒŒw: e p p uC v Then ‰ D H.u C 2/. Let H 0 be the partial derivative of H with respect to w. Then p   p p p 1 vu uv 0 H uv D p Cp H uve p p 2. u C v/ 1 C uw 1 C vw p p p p p uv v 1 C uw C u 1 C vw D p : p p 2. u C v/ .1 C uw/.1 C vw/ Therefore 0

1

X

D H D p 2 .1 C uw/.1 C vw/ p;q0

1=2 p

!

! 1=2 up v q w pCq ; 2 q

55

The Drinfeld associator of gl.1j1/

which implies H D

X p;q0

1=2 p

!

! 1=2 up v q w pCqC1 q 2.p C q C 1/

because H is 0 for w D 0. This implies equation (2) by substituting w D u C 2. By solving equation (16) iteratively one sees that the series ‰ in equation (2) satisfies ‰ 2 QŒuŒŒv  QŒŒu; v and that the coefficient of v q of ‰ is a polynomial in u of degree q. It would be interesting if the associator of Theorem 13 could be used to investigate y m .3/ (m D 0; 1) by the canonical the coefficients of the image of ˆKZ 2 AyC .3/ in C C projection. The simplest relation holds for m D 1: y 1 .3/ there exists a unique Remark 15. For every group-like Drinfeld associator ˆ 2 C k .2/ such that ˆ is related to the associator series G.ˆ/ 2 kŒŒd1 C d2 ; d1 d2   ƒC k   1 12 23 y .  exp.F  Œt ; t / 2 C .3/ of Theorem 13 by the twist exp G.ˆ/  k

The existence of the twist in Remark 15 follows as in the proof of Theorem 13 by using the structure of Ck1 .2/. The uniqueness of the twist uses the structure of Ck1 .1/ and the triviality of H 2 .K • / where the cochain complex K • consists of the T P i spaces K n D niD1 Ker.i /  Ck1 .n/ and the coboundary maps ı n D nC1 iD0 .1/ i (compare Proposition 4.4 and Remark 4.9 of [BN2]).

4 Tensor products and duality of gl.1j1/-modules For t D . ; ; / 2 Q  Q  Z=.2/ there exists a unique 2-dimensional gl.1j1/module V t with sdim.V t / D 0 such that for all v 2 V t with deg.v/ D  we have H  v D . C 1/v, D  v D v and E  v D 0. For each triple t D . ; ; / 2 Q  Q  Z=.2/ we fix a choice of a vector 0 6D v t 2 V t with deg.v t / D  . Denote F  v t by w t . Then H  w t D .  1/w t , D  w t D w t , F  w t D 0 and E  w t D v t . In particular, the vectors v t and w t form a basis of V t . It is easy to see that the modules V t are simple. For triples ti D . i ; i ; i / 2 Q  Q  Z=.2/ (i D 1; 2) with 1 C 2 6D 0 and for e2 D .0; 1; 0/; e3 D .0; 0; 1/ 2 Q  Q  Z=.2/ we define gl.1j1/-linear maps t1 ;t2 W

V t1 Ct2 Ce2 ! V t1 ˝ V t2 by

t1 ;t2 .v t1 Ct2 Ce2 /

t1 ;t2 W

V t1 Ct2 e2 Ce3 ! V t1 ˝ V t2 by

D v t1 ˝ v t2 ; and

t1 ;t2 .w t1 Ct2 e2 Ce3 /

D w t1 ˝ w t2 :

Since t1 ;t2 and t1 ;t2 are non-trivial, well defined, and have non-isomorphic simple images, equation (17) holds for reasons of dimension: V t1 ˝ V t2 Š V t1 Ct2 Ce2 ˚ V t1 Ct2 e2 Ce3

if 1 C 2 6D 0:

(17)

56

Jens Lieberum

The Z=.2/-graded space .V t / D HomQ .V t ; Q/ becomes a gl.1j1/-module by .a  ˇ/.v/ D .1/deg.a/ deg.ˇ / ˇ.a  v/ for all v 2 V t and homogeneous elements a 2 gl.1j1/; ˇ 2 .V t / . Let ˛ 2 .V t / be given by ˛.w t / D 1, ˛.v t / D 0. We have H  ˛ D . C 1/˛, D  ˛ D  ˛, E  ˛ D 0, and deg.˛/ D  C 1. This implies .V t / Š V t 

where t  D . ;  ;  C 1/:

(18)

For . ; / 2 QZ=.2/ there exists a unique 1-dimensional representation I D Q of gl.1j1/ with sdim.I / D .1/ and H  v D v for all v 2 I . The formulas Dv DE v DF v D0 I .

The modules I and V t form a complete set of isomorphism types hold for all v 2 of simple gl.1j1/-modules up to isomorphisms of degree 0. Define a gl.1j1/-linear map t W V t  ˝ V t ! I00 D Q by t .v t



˝ wt / D 1

t .w t

and



˝ v t / D .1/ :

The map t can be obtained as the composition of ˝ id with the evaluation map, where is the isomorphism that we used to prove equation (18). We define 0     t W I0 ! V t ˝ V t by t .1/ D w t ˝ v t  .1/ v t ˝ w t . Then we have . where

t

t

˝

t/

ı.

t

˝

t/

D

t

D.

t

˝

t/

ı.

D idV t . There exist unique gl.1j1/-linear maps t1 ;t2

ı

t1 ;t2

D

and

t1 Ct2 Ce2

t1 ;t2

ı

t

˝

t1 ;t2 ,

t1 ;t2

D

t /; t1 ;t2

(19) satisfying

t1 Ct2 e2 Ce3 :

We have t1 ;t2 ı t1 ;t2 D 0 and t1 ;t2 ı t1 ;t2 D 0 because these maps are homomorphisms between non-isomorphic simple modules. By equation (17) and Schur’s lemma (see [Kac]) the elements t1 ;t2

ı

t1 ;t2 ;

t1 ;t2

ı

(20)

t1 ;t2

are a basis of the vector space Endgl.1j1/ .V t1 ˝ V t2 / of gl.1j1/-linear endomorphisms of V t1 ˝ V t2 of degree 0. In the following lemma we present some relations between the morphisms introduced in this section. Lemma 16. For ti D . i ; i ; i / 2 Q  Q  Z=.2/ (i D 1; : : : ; 4) with t3 D t1 C t2 C e2 and t4 D t1 C t2  e2 C e3 we have t1 ;t2 t1 ;t2

D.

t4

˝

t2 /

1 . 3 2 D .1/1 . 3 D .1/2

ı.

t3

t4 ;t2

˝ t1

˝

˝

t2 /

ı.

t3 /

ı.

t2 /

D.

t3 ;t2 t1

˝

t1

˝

˝

t2 /

t1 ;t3 /:

t4 /

ı.

t1

˝

t1 ;t4 /;

57

The Drinfeld associator of gl.1j1/

Proof. We compare the three elements of Homgl.1j1/ .V t1 ˝ V t2 ; V t4 / from the first equation by comparing the images of v t1 ˝ v t2 and w t1 ˝ w t2 which determine the morphisms. We obtain .

˝

t4

t2 /

ı.

t4 ;t2

˝

t2 /.v t1

˝ v t2 / D .

t1 ;t2 .v t1

.

t1

˝

t4 /

ı.

˝

t1

t1 ;t4 /.v t1

t4

˝

t2 /.v t4

˝ v t2 ˝ v t2 / D 0;

˝ v t2 / D 0;

˝ v t2 / D .

t1

˝

t4 /.v t1

˝ v t1 ˝ v t4 / D 0

as the value of v t1 ˝ v t2 , and .

˝

t4

t2 /

ı.

t4 ;t2

˝

D.

t4

˝

t2 /.F

D.

t4

˝

t2 /.w t4

t2 /.w t1

˝ w t2 /

 .v t4 ˝ v t2 / ˝ w t2 / ˝ v t2 ˝ w t2 C .1/4 v t4 ˝ w t2 ˝ w t2 / D w t4 ; t1 ;t2 .w t1

.

t1

˝

t4 /

ı.

t1

˝

t1 ;t4 /.w t1

˝ w t2 / D w t4 ;

˝ w t2 /

D.

t1

˝

t4 /.w t1

˝ F  .v t1 ˝ v t4 //

D.

t1

˝

t4 /.w t1

˝ w t1 ˝ v t4  .1/1 w t1 ˝ v t1 ˝ w t4 / D w t4 :

We illustrate a small difference in the proof of the second equation by a sample computation. .

t3

˝ D

t2 /

1 . 1

D.

t3

ı. t3

˝

t3 ;t2

˝

˝

t2 /.v t1

t2 /.E



˝ v t2 /

 .w t3 ˝ w t2 / ˝ v t2 /

3 2 v t3 ˝ w t2 ˝ v t2 C .1/3 w t3 ˝ v t2 ˝ v t2 t2 / 1 1



3 vt : 1 3 The rest of the proof is straightforward. D .1/2

5 The tensor functor W0 Let  be a unitrivalent graph with oriented edges and vertices whose edges i 2 E./ are colored by pairs . i ; i / 2 Q  Q. We say that the coloring of  is admissible if the following two conditions hold at every trivalent vertex v of  that is incident to i; j; k 2 E./: sv;i i C sv;j j C sv;k k D 0; sv;i i C sv;j j C sv;k k D sv;i sv;j sv;k :

(21)

58

Jens Lieberum

When  is admissibly colored then ƒ./ 6D 0 by equation (21) because i 6D 0 for all i 2 E./. Let us define a category C0 . Objects of C0 are finite sequences .c1 ; : : : ; ck / of triples ci D . i ; i ; i / 2 Q  Q  Z=.2/: Morphisms from c 0 D .c10 ; : : : ; ck00 / to c 1 D .c11 ; : : : ; ck11 / with cji D . ji ; ji ; ji / consist of unitrivalent graphs  with admissible coloring, where the univalent vertices of  are related to c 0 and c 1 as follows: the boundary of  is decomposed into two disjoint subsets @ D @0  [ @1 , @i  is in bijection with f1; : : : ; ki g, the j -th vertex of @i  is incident to an edge of  that is colored by . ji ; ji /, and that edge is directed towards the boundary iff ji C i 0 mod 2. We represent morphisms of C0 graphically by generic pictures of  in R  Œ0; 1 where the i -th boundary point of @j  has coordinates .i; j /. The composition b ı a 2 HomC0 .c 0 ; c 2 / of morphisms a 2 HomC0 .c 0 ; c 1 /, b 2 HomC0 .c 1 ; c 2 / is defined graphically by placing b onto the top of a and by shrinking the result to R  Œ0; 1. We define a tensor product of objects of C0 by concatenation of sequences. For a 2 HomC0 .c 0 ; c 1 /; b 2 HomC0 .c 2 ; c 3 / we define a ˝ b 2 HomC0 .c 0 ˝ c 2 ; c 1 ˝ c 3 / graphically by placing a to the left of b. To an object c D .c1 ; : : : ; ck / of C0 with ci D . i ; i ; i / we assign the gl.1j1/module W0 .c/ D V t1 ˝    ˝ V tk

with ti D ..1/i i ; .1/i i ; i /:

(22)

Let t D . ; ; 0/. We do not specify colors or orientations of edges in pictures, when they are uniquely determined by the context or when the formulas hold for arbitrary orientations. Assume that the graphs in equations (23) to (25) are colored by . ; /. Then we define     W0 (23) D t; D t ; W0     W0 (24) D t ; W0 D t ;     W0 D t ; W0 (25) D t : Let ci D . i ; i ; i /. Consider orientations and colors of the graphs such that 2 HomC0 .c1 ; .c2 ; c3 //

and

and

2 HomC0 ..c1 ; c2 /; c3 /:

Then we define ti D ..1/i i ; .1/i i ; i / and (   if 1 C 2 C 3 0 mod 2; t2 ;t3 D W0 1 t2 ;t3 if 1 C 2 C 3 1 mod 2; (   3 t1 ;t2 if 1 C 2 C 3 0 mod 2; W0 D if 1 C 2 C 3 1 mod 2: t1 ;t2

(26) (27)

59

The Drinfeld associator of gl.1j1/

Let t1 ;t2 2 Homgl.1j1/ .W0 ..c1 ; c2 //; W0 ..c2 ; c1 /// be the superpermutation of tensor factors induced by t1 ;t2 .v ˝ w/ D .1/deg.v/ deg.w/ w ˝ v. We define   W0 2 HomC0 ..c1 ; c2 /; .c2 ; c1 //: (28) D t1 ;t2 for We have the following lemma. Lemma 17. The map W0 induces a tensor functor from C0 to gl.1j1/-modules. Proof. The tensor category C0 has a presentation by admissibly colored generators3 ;

;

;

;

;

;

;

;

(where for the last three graphs all orientations are possible) modulo the relations ;

D D D D

;

D

D ;

;

D

;

(29)

;

D

(30) ;

D

D

;

where the pictures in equations (29), (30) represent words in the generators and ˝; ı such that all compositions are defined. The value of W0 on objects and generators of C0 has been defined in equations (22) to (28). It remains to verify the compatibility of W0 with the defining relations of C0 . The relation on the left side of equation (29) follows from equation (19). Assume that the graphs on the right side of equation (29) are colored and oriented such that they are mapped by W0 to elements of Homgl.1j1/ .V t1 ˝ V t2 ; V t3 / for certain parameters ti D . i ; i ; i / 2 Q  Q  Z=.2/. We will distinguish two cases. For 1 C 2 C 3 1 mod 2 we compute     D t1 ;t2 D . t4 ˝ t2 / ı . t4 ;t  ˝ t2 / D W0 ; W0 2

where we used the first equation of Lemma 16 and the equations (22) to (27). For 1 C 2 C 3 0 mod 2 we compute   D .1/3 3 t1 ;t2 D .1/2 C3 1 . t3 ˝ t2 / ı . t3 ;t  ˝ t2 / W0 2     1 ; D . t3 ˝ t2 / ı .1/ 1 t3 ;t  ˝ t2 D W0 2

3 Some

of these generators are superfluous, but facilitate the statement of the defining relations of C0 .

60

Jens Lieberum

where we used the second equation of Lemma 16 this time. The equation     W0 D W0 is verified similarly. For the verification of the compatibility of W0 with the relations in equation (30) we use that W0 maps graphs to morphisms of degree 0 together with simple properties of the superpermutation t1 ;t2 and the map t . This completes the proof. The morphisms t1 ;t2 verify t1 ;t2 D .1/1 2 t2 ;t1 t2 ;t1 whereas for t1 ;t2 we have t1 ;t2 D .1/.1 C1/.2 C1/ t2 ;t1 ı t2 ;t1 . Using equation (26) this translates as follows to trivalent vertices v of colored trivalent graphs  that are incident to i; j; k 2 E./. We define sv 2 f˙1g by sv D 1 iff jsv;i C sv;j C sv;k j D 1. Then we have     : (31) D sv W0 W0 When sv D 1 we distinguish the cases sv;i C sv;j C sv;k =-3 where v is called a source, and sv;i C sv;j C sv;k D 3 where v is called a sink.

6 ƒ./-linear weight systems Maps from trivalent diagrams to modules are called weight systems. In this section we will combine a well-known construction of weight systems related to gl.1j1/ with the results of the previous section. Define a category C that has the same objects as C0 . The set HomC .c; c 0 / is a direct sum of modules C ./ where  2 HomC0 .c; c 0 /. The graphical definition of the composition of morphisms in C0 extends to trivalent diagrams and induces Q-linear maps ˇ W ƒ.1 / ˝Q ƒ.2 / ! ƒ.1 ı 2 /

and

˛ W C .1 / ˝Q C .2 / ! C .1 ı 2 /

with the property that for all pi 2 ƒ.i /, Di 2 C .i / (i D 1; 2) we have ˛..p1 D1 / ˝ .p2 D2 // D ˇ.p1 ˝ p2 /˛.D1 ˝ D2 /: We consider C0 as a subcategory of C in the obvious way. For objects c of C we define a functor W by W .c/ D W0 .c/. Now we extend W0 to morphisms of C . Define ! 2 gl.1j1/˝2 by ! D .1=2/.H ˝ D C D ˝ H / C F ˝ E  E ˝ F: (32) P Sometimes we use the notation ! D  a ˝b for !. The elements ! and D ˝D are a basis of the space of invariants in gl.1j1/ ˝ gl.1j1/ by the adjoint representation. Consider an object c D .c1 ; : : : ; ck / of C with ci D . i ; i ; i /. Let  D idc 2 C0 . Define Tc 2 EndC .c/ as the trivalent diagram of degree 1 on  that connects the

The Drinfeld associator of gl.1j1/

61

first interval of  with the second interval. We define W .Tc / 2 Endgl.1j1/ .W .c// by W .Tc /.v1 ˝    ˝ vk / X D .1/deg.v1 / deg.b /C1 C2 a  v1 ˝ b  v2 ˝ v3 ˝    ˝ vk ;

(33)



where vi 2 W .ci /.4 For morphisms D of C0 we define W .D/ D W0 .D/. For any unitrivalent graph  with admissible coloring, we consider gl.1j1/-modules as a ƒ./-module, where di acts by multiplication with i when . i ; i / is the color of the edge i . Equation (21) implies that this definition is compatible with the defining relations of ƒ./. We have the following lemma. Lemma 18. The definition of W extends uniquely to a functor from C to gl.1j1/modules that is ƒ./-linear on C ./ and Q-linear on general morphisms. Proof. Using relation (STU) we see that the modules C ./ are generated by diagrams D such that D n  contains no trivalent vertex. For the diagrams D one can choose pictures such that horizontal stripes around intervals in D n  are equal to Tc12 for various objects c of C . Linearity of W now implies that the definition of W determines the value of all morphisms of C . We have to verify that W is independent of the choices from above and compatible with the defining relations of C ./. It follows from Lemma 17 and general properties of ! (! is invariant, supersymmetric, and of degree 0) that the definition of W induces well-defined Q-linear maps W0 from A./ to Q for all admissibly colored unitrivalent graphs . It remains to show that the ƒ./-linear maps W W C ./ ! Q defined by W ı  D W0 are well y ./. As in the proof of defined, where  denotes the canonical map from A./ to C Lemma 11 this follows from particular properties of gl.1j1/ and !: equations (34) and (35) (see [FKV]) imply the compatibility of W with relations (CL1A) and (CL2A).         W D 2di dj W ; W D 0; W D 0; (34) i j i j     1 W D W C   : (35) 2 Now consider a trivalent diagram D on , where  is an admissibly colored trivalent graph. Let p 2  n V .D/ be a point on an edge with color . p ; p /. By cutting D at p, we obtain a diagram Dp 2 EndC .. p ; p ; 0//. Lemma 19. There exists a unique linear map W ı W EndC .;/ ! Q such that for a trivalent diagram D on an admissibly colored trivalent graph  and any p 2  nV .D/ on an edge with color . p ; p / we have W .Dp / D p W ı .D/

tp ;

where tp D . p ; p ; 0/:

4 The explicit appearance of the factor .1/1 C2 in equation (33) is sometimes avoided by introducing an antisymmetry relation for univalent vertices that are glued to .

62

Jens Lieberum

The map W ı is ƒ./-linear on C ./  EndC .;/. Proof. We only have to show that the definition of W ı .D/ does not depend on the choice of p, because then Lemma 18 implies the compatibility of W ı with the defining relations of C ./  EndC .;/ and the ƒ./-linearity. Consider two points p1 ; p2 2  n V .D/ on edges with colors . i ; i / (i D 1; 2). We treat the case 1 6D 2 in detail. Let ti D . i ; i ; 0/. Pictorial representations of the diagrams D, Dp1 , and Dp2 are shown in equation (36), where the box labeled x represents a morphism x 2 EndC ..t1 ; t2 //. D D p1

x

p2 ,

Dp1 D x

p2 ,

Dp2 D p1

x

:

(36)

It follows from equations (20), (26), (27) and definitions that there exist a; b 2 Q such that W .x/ D a

t1 ;t2

ı

a D W 1  2

t1 ;t2

Cb !

t1 ;t2

ı

t1 ;t2

b W C 1  2

! ;

which implies W .Dpi / D

    a aCb b W W C D i 1  2 1  2 1  2

ti :

(37)

in equation (37) does not depend on pi (i 2 f1; 2g). It follows that The value aCb 1 2 W .D/ is well-defined for diagrams D whose colored skeleton has two edges colored by . 0i ; 0i / (i D 1; 2) with 01 6D 02 because then we can treat the case 1 D 2 by a two-fold application of the argument above. The remaining case is well known from computations concerning the 1-variable Alexander polynomial of links. Alternatively, there is a proof for 1 D 2 (or, more generally, 1 6D  2 ) similar to the proof above, where a picture of D is chosen as in equation (36), but with different orientations and with x 2 EndC ..t1 ; t2 //. ı

7 The Kontsevich integral of unitrivalent graphs Recall from [LM1] the definition of the Kontsevich integral Z of framed q-tangles. We assume that Z is defined using an even group-like horizontal Drinfeld associator y y in A.3/. For a q-tangle T we have Z.T / 2 A..T // where .T / is the underlying 1-manifold of T . In this section we will study an extension of Z to graphs (compare [MuO]). We consider a category G na whose morphisms are isotopy classes of oriented (half-)framed unitrivalent graphs G with cyclically oriented vertices. By definition,

The Drinfeld associator of gl.1j1/

63

G is properly embedded into R  Œ0; 1  R and we have G \ R  fi g  R D f1; : : : ; ni g  fi g  f0g for i D 0; 1 and for certain n0 ; n1  0. We represent a strand of G with a righthanded half twist of the framing graphically by . The objects of G na are nonassociative words in the symbols C and . For example, ..C// is an object of G na . Unitrivalent graphs G 2 HomG na .w0 ; w1 / are related to wi as follows: the i-th symbol a 2 fC; g of w0 (resp. w1 ) corresponds to the i -th lower (resp. upper) boundary point p D .i; 0; 0/ (resp. p D .i; 1; 0/) of G where a D C (resp. a D ) means that the graph G must be oriented downwards (resp. upwards) at p. na The category Ay has the same objects as G na . The set HomAy na .w0 ; w1 / is the y direct sum of all modules A./ where  is a unitrivalent graph whose boundary is partitioned into two ordered sets called lower and upper boundary as in Section 5, and a 2 fC; g in w0 and w1 is related to the boundary points in a graphical representation of  in the same way as above. We consider the invariant Z of q-tangles from [LM1] na as a tensor functor from the subcategory T na  G na of q-tangles to Ay . Let us recall how Z depends on the orientations of edges of q-tangles. Let  be an oriented 1-dimensional manifold with boundary and let  0   be a set of connected components of . Let D be a trivalent diagram on . Define S 0 .D/ by inverting the orientation of all components of  0 and by multiplying the result by .1/m where m is the number of univalent vertices of D n  that are glued to  0 . Let T be a q-tangle and T 0  T be a set of connected components of T . Define ST 0 .T / by inverting the orientation of all components of T 0  T . With this notation we have Z.ST 0 .T // D S.T 0 / .Z.T //:

(38)

We omit the index of S when all components of a unitrivalent graph (resp. of the skeleton of a trivalent diagram) are concerned. For a 2 EndAy.C/ we define    S  .a/ D ˝ ı ˝a˝ ı ˝ : y  End y.C/ with S.a/ 6D S  .a/. It is unknown if there exist elements a 2 A.1/ A na In diagrams of morphisms G of G we use projections of generic representatives of G to the first two coordinates such that the cyclic order at trivalent vertices is counterclockwise in the projection. For a unitrivalent graph  we define  op by inverting the cyclic order of all trivalent vertices of . This induces maps y y op /; a 7! aop ; and Homna .w0 ; w1 / ! Homna .w0 ; w1 /; G 7! G op : A./ ! A. G G We denote the Kontsevich integral of the trivial knot by and regard it as an element y of A.1/. Since is equal to 1 in degree 0 the element is invertible and there exist unique roots of n (n 2 Z) that are equal to 1 in degree 0. The elements k (k 2 Q) satisfy S. k / D S  . k /. In the formulas below the box labeled k represents the element k or S. k / according to the orientation of the interval with the box.

64

Jens Lieberum

Theorem 20. (1) For any choice of a; b 2 EndAy na .C/;

c 2 HomAy na .; .CC//;

and d 2 HomAy na .C; .//

there exists a unique extension of Z to a tensor functor from G na to Ay 

a



Z

D

a

 ;

S .a/

D

b

 1=2





Z

 D c;

;

 1=2



Z

D d;

ı c D c op and (2) If all morphisms G of G na .

satisfying

S .b/ S .b/



Z

na

and Z

 

D exp



 =4 :

ı d D d op holds then we have Z.G op / D Z.G/op for

Sketch of proof. First we consider oriented graphs whose vertices are oriented boxes with a distinguished lower boundary (called coupons). A category G 0na is defined in the same way as G na except that morphisms are embedded framed graphs with coupons Ca D



a



2 HomG 0na .s; t /

that are colored by elements a 2 HomAy na .s; t /. It is easy to see that there exists a na unique extension of Z to a tensor functor from G 0na to Ay that verifies Z.Ca / D a for all coupons Ca as above. This general construction implies that part (1) of the theorem holds iff Z









DZ

and

Z







DZ



:

These two equations follow from a well-known identity in HomAy...CC//; C/: 1

0

C AD

B Z@

 1=2

:

1. 1=2 /

(2) The isotopy invariance of Z implies Z





DZ



op



:

65

The Drinfeld associator of gl.1j1/

When the upper two strands at a trivalent vertex are both oriented downwards or both oriented upwards then the equation ! 1 D   2 and the symmetry properties of c and d imply   op Z DZ



:

Part (2) of the theorem now follows from part (1) and the equations above. From now on we will fix the choices below in the definition of the extension of Z to unitrivalent graphs.     0 0 0 0 Z ; Z ; (39) D D 0  1 Z





D

 1=2

 1=2

 1

;

Z





D

 1=2

 1=2 0

:

y With this definition we have Z.G/ 2 A..G// where .G/ is the underlying graph of G. For explicit computations in the following sections we list more values of Z.      1=2 0 0  1=2 Z D D ; Z ; 1=2 1=2   Z Z Z Z Z

    

    

D D D D D

 1=2

0

 1=2

0 0

0

 1  1=2

 1=2

 1=2  1=2

0

 1=2 0

 1=2

;

Z

;

Z

;

Z

;

Z

;

Z

    

    

D D D D D

 1=2

0

 1=2

 1 0

0

 1=2

 1=2 0

 1=2

 1=2  1=2

; (40)

0  1=2

;

0

; ; :

The computation of these values can be simplified by first generalizing symmetry properties of Z from q-tangles (Proposition 3.1 of [LM2]) to trivalent graphs.

66

Jens Lieberum

8 The Alexander series of a tetrahedron For a trivalent diagram D on an admissibly colored trivalent graph  we define

b

W ı .D/ D W ı .D/ h.1=2/ #V .D/1 2 h1 QŒŒh  QŒŒhŒh1 :

b

(41)

y 0 ./ ! h1 QŒŒh. Notice This definition induces a continuous linear map W ı W C that .1=2/ #V .D/ > deg.D/ when   D has trivalent vertices. For an admissibly x D  .Z.G// where colored framed unitrivalent graph G  R2  I we define Z.G/ 0 y y the continuous linear map  W A..G// ! C ..G// is defined on trivalent diagrams D by .D/ D D and the skeleton .G/ of D is colored according to G.

b

 y x Definition 21. The invariant r.G/ D W ı Z.G/ of an admissibly colored framed trivalent graph G is called the Alexander series of G. In the following three sections we will compute the Alexander series of a trivially embedded colored tetrahedron T in two different ways. With the first computation we y /. This will be possible without will determine the value of the Alexander series r.T knowing an associator, because we can choose a diagram of T where the associator y /. Then we will use the isotopy invariance of r. y We gives a trivial contribution to r.T y / will choose a diagram of T where the associator enters into the computation of r.T by a formal power series of 2  2-matrices. This way we will obtain information about the associator by this second computation that will suffice to prove Theorem 1. Let c be the trivial knot with color c D . ; / and with 0-framing. It follows y from Section 2.2 of [BNG] that for c D .1; 0/ we have r. / D 1=.e h=2  e h=2 /. c For c D . ; / we obtain from this value the more general formula y r.

c

/D

1 : e h=2  e h=2

(42)

b

x // because by using that the left side of equation (34) suffices to compute W ı .Z. c is 0-framed and we know the structure of C .1/ (see [Thu] for a more general c result). For a trivalent diagram D on  2 HomC0 .c; c 0 / the definition

.D/ D W .D/hdeg D 2 Homgl.1j1/ .W .c/; W .c 0 //ŒŒh W y 0 ./ ! Homgl.1j1/ .W .c/; W .c 0 //ŒŒh. Recall

W C induces a continuous linear map W the definition '.x/ D .e x=2  e x=2 /=x D 2 sinh.x=2/=x from the introduction. Equation (42) and Lemma 19 imply that for the trivalent diagram on a skeleton y .t/ with t D . ; ; / we have t 2 EndC 0

. / D .1='. h// W

t:

(43)

ı .id ˝G/ ı 2 EndG na .;/ is called Let G 2 EndG na .C/. Then H D the closure (or trace) of G. When H is admissibly colored and the upper (or lower) edge of G has color . ; / then the definition of Z, Lemma 19, and equations (41)

67

The Drinfeld associator of gl.1j1/

and (43) imply that for t D . ; ; 0/ we have y / r.H

D

t

h.1=2/ #V ..H // x W .Z.G//: h'. h/

(44)

For i D 1; 2; 3 let ti D . i ; i ; i / 2 Q  Q  Z=.2/ with 1 C 2 ; 2 C 3 ; 1 C 2 C 3 2 Q :

(45)

We will use in the following sections the triples ti D . i ; i ; i / 2 Q QZ=.2/ (i D 4; : : : ; 11) defined by t4 D t1 C t2 C e2 ; t5 D t4 C t3 C e2 ; t6 D t2 C t3 C e2 ; ti D ti3  2e2 C e3 .i  7/

(46)

and the colors ci D ..1/i i ; .1/i i / (i D 1; : : : ; 11). Equation (17) implies that the tensor product of three modules decomposes as V t1 ˝ V t2 ˝ V t3 Š V t5 ˚ V t˚2 ˚ V t11 : 8

(47)

The following lemma concerns simple modules of multiplicity 1 in equation (47). Lemma 22. With ti as in equation (46) the following formulas hold. .

t1

˝

t2 ;t3 /

ı

t1 ;t6

D.

t1 ;t2

˝

t3 /

ı

t4 ;t3 ;

.

t1

˝

t2 ;t3 /

ı

t1 ;t9

D.

t1 ;t2

˝

t3 /

ı

t7 ;t3 ;

t1 ;t6

ı.

t1

˝

t2 ;t3 /

D

t4 ;t3

ı.

t1 ;t9

ı.

t1

˝

t2 ;t3 /

D

t7 ;t3

ı.

˝

t1 ;t2 t1 ;t2

˝

t3 /; t3 /:

Proof. We compute .

t1

˝

t2 ;t3 /

ı

t1 ;t6 .v t5 /

D v t1 ˝ v t2 ˝ v t3 D .

t1 ;t2

˝

t3 /

ı

t4 ;t3 .t t5 /:

This implies the first equation. The second equation is proved similarly. The remaining two equations follow from equation (47) and similar computations. Consider the three diagrams Tk (k D 1; 2; 3) shown in Figure 1.

c5 T1 D c1

c2 c4

c11

c6

c5

c3 ; T2 D c1

c2 c7

c8

c9

c11

c3 ; T3 D c1

c6

c2 c7

c8

Figure 1. Three colored diagrams whose closures are planar tetrahedra.

c3 :

68

Jens Lieberum

Fix a choice of k 2 f1; 2; 3g. Define i D i .k/ by i D 1 iff the edge of Tk labeled by ci points downwards. Then equations (45) and (46) ensure that the coloring of Tk is admissible. Let bi D ..1/i i ; .1/i i ; i /. Define b D b5 if k D 1, b D b11 if k D 2, and b D b8 if k D 3. Then Tk D Uk ı Ak ı Lk where Uk 2 HomG ..b1 .b2 b3 //; b/ consists of the upper half of Tk , Lk 2 HomG .b; ..b1 b2 /b3 // consists of the lower half of Tk , and Ak 2 HomG ...b1 b2 /b3 /; .b1 .b2 b3 /// consists of three vertical colored strands. It follows from equations (39), (26), and Lemma 22)

ıZ x to the following morphism from V t5 ŒŒh to that the graph L1 is mapped by W .V t1 ˝ V t2 ˝ V t3 /ŒŒh:

.Z.L x 1 // D W ..L1 // D . W

t1 ;t2

˝

t3 / ı

t4 ;t3

D.

t1

˝

t2 ;t3 / ı

t1 ;t6 :

(48)

ıZ x to the following morphism from .V t1 ˝ V t2 ˝ The graph U1 is mapped by W V t3 /ŒŒh to V t5 ŒŒh (see equations (40), (43), (27)):

.Z.U x 1 // D '. 5 h/'. 6 h/W ..U1 // W D 5 6 '. 5 h/'. 6 h/ t1 ;t6 ı .

t1

˝

(49)

t2 ;t3 /:

The associator ˆ is a series with constant term 1, and all higher order terms of ˆ involve a commutator (see (DA5)). Therefore, by equation (47) and Schur’s lemma, the action of ˆ on .V t1 ˝ V t2 ˝ V t3 /ŒŒh restricts to t5

2 Endgl.1j1/ .V t5 /  Endgl.1j1/ .V t1 ˝ V t2 ˝ V t3 /ŒŒh:

This implies t1 ;t6

ı.

t1

˝

t2 ;t3 /

.Z.A x 1 // ı . ıW

t1

ı

t2 ;t3 /

ı

t1 ;t6

D

t5 :

(50)

Let S1 be the closure of T1 . Equations (44), (48), (49), and (50) allow to compute y r.S1 / without knowing ˆ: y 1/ r.S

t5

D

h

.Z.T x 1 // D 6 h'. 6 h/ W 5 '. 5 h/

t5

D .e 6 h=2 e 6 h=2 /

t5 :

(51)

In general, the Alexander series of a planar tetrahedron is given by the following lemma. Lemma 23. Let T be a planar tetrahedron with admissible coloring and blackboard framing. There exists a unique edge e of T such that by reversing the orientation of e we obtain a tetrahedron with one source and without a sink. Let . ; / be the color of e. Then we have y / D e h=2  e h=2 : r.T Proof. By Lemma 7.2.A of [Vir] there are four isotopy classes of planar oriented tetrahedra with blackboard framing. By Lemma 7.2.B of [Vir] two of these tetrahedra do not have an admissible coloring. By Lemma 7.2.C of [Vir] the remaining two oriented tetrahedra have a unique edge e as in the lemma. For one admissibly colored

69

The Drinfeld associator of gl.1j1/

y 1 / in equation (51). The second tetrahedron is tetrahedron S1 we have computed r.S y 2 / proceeds along the the closure S2 of T2 (see Figure 1). The computation of r.S y 1 /. same lines as the computation of r.S

9 Associativity and gl.1j1/-modules For the diagram T3 in Figure 1 we will see in Section 10 that the contribution of the

.Z.T x 3 // is non-trivial. We will use Lemma 24 associator ˆ in the computation of W and Corollary 25 below for a similar purpose in this computation as we used Lemma 22 to deduce equation (48). 1 C2 C3 / Lemma 24. For ti D . i ; i ; i / as in equations (45), (46) and  D .21 . C2 /.2 C3 / we have two bases of Homgl.1j1/ .V t8 ; V t1 ˝ V t2 ˝ V t3 / that are related by !    .1/2 3  t1 ˝ 2 3 t1 ;t2 ˝ t3 / ı t2 ;t3 / ı t1 ;t9 t7 ;t3 D : .1/2 1 . t1 ˝ t2 ;t3 / ı t1 ;t6 . t1 ;t2 ˝ t3 / ı t4 ;t3 1 1 C2 (52)

Proof. We compute .

˝

t1

t2 ;t3 /.

D.

t1

˝

D w t1 ˝

t1 ;t9 .w t8 // t2 ;t3 /.F

 .v t1 ˝ v t9 //

t2 ;t3 .v t9 /

C .1/1 v t1 ˝

t2 ;t3 .w t9 /

1 w t ˝ E  .w t2 ˝ w t3 / C .1/1 v t1 ˝ w t2 ˝ w t3 2 C 3 1 2 .1/2 3 D w t1 ˝ v t2 ˝ w t3  w t ˝ w t2 ˝ v t3 2 C 3 2 C 3 1 C .1/1 v t1 ˝ w t2 ˝ w t3 ;

D

and similarly (or simpler) .

t1

˝

t2 ;t3 /.

t1 ;t6 .w t8 //

D .1/2 w t1 ˝ v t2 ˝ w t3 C w t1 ˝ w t2 ˝ v t3 ; .1/2 2 w t ˝ v t2 ˝ w t3 C w t1 ˝ w t2 ˝ v t3 1 C 2 1 .1/1 C2 1  v t1 ˝ w t2 ˝ w t3 ; 1 C 2

.

t1 ;t2

˝

t3 /.

t7 ;t3 .w t8 //

D

.

t1 ;t2

˝

t3 /.

t4 ;t3 .w t8 //

D w t1 ˝ v t2 ˝ w t3 C .1/1 v t1 ˝ w t2 ˝ w t3 :

We see that the vectors . as well as . t1 ;t2 ˝ t3 /.

t1 ˝

t2 ;t3 /.

t7 ;t3 .w t8 //

t1 ;t9 .w t8 // and . t1 ˝

and .

t1 ;t2

˝

t3 /.

t2 ;t3 /.

t4 ;t3 .w t8 //

t1 ;t6 .w t8 //

are linearly

70

Jens Lieberum

independent. Equation (47) implies that Homgl.1j1/ .V t8 ; V t1 ˝ V t2 ˝ V t3 / is twodimensional, so we have found two bases of that space. Verify that equation (52) holds when we evaluate the morphisms in this equation on w t8 . This implies the lemma because w t8 generates V t8 . Dually to Lemma 24 we have the following corollary. Corollary 25. With ti and  as in Lemma 24 two bases of Homgl.1j1/ .V t1 ˝ V t2 ˝ V t3 ; V t8 / are related by  t1 ;t9 ı . t1 ˝ t1 ;t6 ı . t1 ˝

 t2 ;t3 / D t2 ;t3 /

.1/2 1 1 2

!

1

t7 ;t3

.1/2 3 2 C3



t4 ;t3

ı. ı.

t1 ;t2 t1 ;t2

˝ ˝



t3 / t3 /

:

Proof. By Lemma 24 linear maps fi , gi are related by .f1 ; f2 /T D A.g1 ; g2 /T for a certain matrix A. Therefore, maps fi0 , gi0 satisfying fi0 ı fj D ıij t8 and gi0 ı gj D ıij t8 are related by .f10 ; f20 /T D .AT /1 .g10 ; g20 /T . This implies the corollary. In the rest of this section we prepare the analogue of equation (50) for the computation in Section 10. By the definition of the gl.1j1/-module structure of V t1 ˝ V t2 we have F  .v t1 ˝ v t2 / D w t1 ˝ v t2 C .1/1 v t1 ˝ w t2 ; E  .w t1 ˝ w t2 / D .1/1 2 w t1 ˝ v t2 C 1 v t1 ˝ w t2 :

(53) (54)

For 1 6D  2 equations (53) and (54) are formulas for a change of bases in the two dimensional eigenspace of H on V t1 ˝ V t2 and imply . 1 C 2 /w t1 ˝ v t2 D 1 F  .v t1 ˝ v t2 /  .1/1 E  .w t1 ˝ w t2 /; . 1 C 2 /v t1 ˝ w t2 D .1/1 2 F  .v t1 ˝ v t2 / C E  .w t1 ˝ w t2 /:

(55)

Lemma 26. With ti and  as in Lemma 24 the following formulas hold: .

t1 ;t2

˝

t3 /

ı.

t1

˝

t2 ;t3 / D

t4 ;t3

ı

.

t1 ;t2

˝

t3 /

ı.

t1

˝

t2 ;t3 /

D

t7 ;t3

ı

.

t1 ;t2

˝

t3 /

ı.

t1

˝

t2 ;t3 /

D

t7 ;t3

ı

.

t1 ;t2

˝

t3 /

ı.

t1

˝

t2 ;t3 /

D

t4 ;t3

.1/2 1 t1 ;t6 1 C 2 .1/2 3 t1 ;t9  2 C 3 ; t1 ;t6 C

ı

t1 ;t9 :

t4 ;t3

ı

t1 ;t6 ;

t7 ;t3

ı

t1 ;t9 ;

71

The Drinfeld associator of gl.1j1/

Proof. We have .

t1 ;t2

˝

t3 /

D.  D

ı.

t1

t1 ;t2

˝

t4 ;t3

ı

˝

t2 ;t3 /.v t1

t3 /.v t1 t1 ;t6

˝ v t6 /

˝ v t2 ˝ v t3 / D v t4 ˝ v t3

C

.1/2 1 1 C 2

t4 ;t3

ı



.v t1 ˝ v t6 /:

t1 ;t6

Using equation (55) we see that .

t1 ;t2

˝

t3 /

ı.

t1

˝

t2 ;t3 /.w t1

˝ w t6 /

D.

t1 ;t2

˝

t3 /.w t1

˝ F  .v t2 ˝ v t3 //

D.

t1 ;t2

˝

t3 /.w t1

˝ w t2 ˝ v t3 C .1/2 w t1 ˝ v t2 ˝ w t3 /

t1 ;t2

˝

t3 /

D.



.1/2 1 F  .v t1 ˝ v t2 / ˝ w t3 1 C 2  .1/1 C2  E  .w t1 ˝ w t2 / ˝ w t3 1 C 2

.1/2 1 w t ˝ w t3 1 C 2 4  .1/2 1 D ı C t4 ;t3 t1 ;t6 1 C 2 D

 t4 ;t3

ı

t1 ;t6

.w t1 ˝ w t6 /:

Since an element of Homgl.1j1/ .V t1 ˝ V t6 ; V t4 ˝ V t3 / is determined by the images of v t1 ˝ v t6 and w t1 ˝ w t6 this implies the first equation of the lemma. The remaining three equations are proved similarly. The following corollary holds for reasons of symmetry. Corollary 27. With ti and  as in Lemma 24 the following formulas hold: .

t1

˝

t2 ;t3 /

ı.

t1 ;t2

˝

t3 /

D

t1 ;t6

ı

.

t1

˝

t2 ;t3 /

ı.

t1 ;t2

˝

t3 /

D

t1 ;t9

ı

.

t1

˝

t2 ;t3 /

ı.

t1 ;t2

˝

t3 /

D

t1 ;t9

ı

.

t1

˝

t2 ;t3 /

ı.

t1 ;t2

˝

t3 /

D

t1 ;t6

.1/2 3 2 C 3 .1/2 1 t7 ;t3  1 C 2 t4 ;t3 ; t4 ;t3

ı

C

t1 ;t6

ı

t4 ;t3 ;

t1 ;t9

ı

t7 ;t3 ;

t7 ;t3 :

Proof. Let V;W 2 Hom.V ˝W; W ˝V / be the linear map induced by the permutation of tensor factors. When we interchange the labels t3n2 and t3n (n D 1; 2; 3) in the equations of Lemma 26, replace the equations X D Y 2 Hom.V ˝ W; V 0 ˝ W 0 / by V 0 ;W 0 XV;W D V 0 ;W 0 Y V;W , and apply the properties preceding equation (31) and and , then we obtain the equations of the corollary. similar equations for

72

Jens Lieberum

By equation (47) commutators of elements of Endgl.1j1/ .V t1 ˝ V t2 ˝ V t3 / lie in the subspace Endgl.1j1/ .V t8 / of Endgl.1j1/ .V t1 ˝ V t2 ˝ V t3 /. We make some explicit computations. Lemma 28. With ti and  as in Lemma 24 we have Œ.

t1 ;t2

ı

t1 ;t2 /

D Œ.

˝

ı

t1 ;t2

t3 ; t1 t1 ;t2 /

˝.

˝

t2 ;t3

t3 ; t1

ı

t2 ;t3 /

˝.

t2 ;t3

ı

t2 ;t3 /

DŒ

t1

˝.

t2 ;t3

ı

t2 ;t3 /; .

t1 ;t2

ı

t1 ;t2

DŒ

t1

˝.

t2 ;t3

ı

t2 ;t3 /; .

t1 ;t2

ı

t1 ;t2 /

t1 ;t2

˝

D

2

.1/ 1  . 1 C 2 .1/2 3  . 2 C 3

t1 ;t2

t3 /

˝

ı

t3 /

ı

ı

t4 ;t3

t7 ;t3

˝ ˝

t3 

ı.

t7 ;t3

ı

t3 /

t4 ;t3

t1 ;t2

ı.

˝

t1 ;t2

˝

t3 / t3 /:

Proof. For the first commutator we compute Œ.

t1 ;t2

ı

D.

t1 ;t2 / t1 ;t2

˝

˝ 2

t3 ; t1

t3 /

ı

˝.

t4 ;t3

ı

t2 ;t3

t1 ;t6

.1/ 1 . t1 ;t2 ˝ t3 / ı 1 C 2  . t1 ˝ t2 ;t3 / ı t1 ;t6 ı C

2

ı

t2 ;t3 /

ı.

t1

t4 ;t3

ı

t4 ;t3

ı.

˝

t2 ;t3 / t1 ;t6

t1 ;t2

ı. ˝

t1

˝

t2 ;t3 /

t3 /

.1/ 3 . ˝ t2 ;t3 / ı t1 ;t6 ı t4 ;t3 ı . t1 ;t2 ˝ t3 / 2 C 3 t1 .1/2 1 D . t1 ;t2 ˝ t3 / ı t4 ;t3 ı t1 ;t6 ı . t1 ˝ t2 ;t3 / 1 C 2 .1/2 3  . ˝ t2 ;t3 / ı t1 ;t6 ı t4 ;t3 ı . t1 ;t2 ˝ t3 / 2 C 3 t1 .1/2 1 D .. t1 ;t2 ˝ t3 / ı t4 ;t3 ı t7 ;t3 ı . t1 ;t2 ˝ t3 / 1 C 2 .1/2 3 C . t1 ;t2 ˝ t3 / ı t4 ;t3 ı t4 ;t3 ı . t1 ;t2 ˝ t3 // 2 C 3 .1/2 3  .. t1 ;t2 ˝ t3 / ı t7 ;t3 ı t4 ;t3 ı . t1 ;t2 ˝ t3 / 2 C 3 .1/2 1 C . t1 ;t2 ˝ t3 / ı t4 ;t3 ı t4 ;t3 ı . t1 ;t2 ˝ t3 // 1 C 2 .1/2 1  D . t1 ;t2 ˝ t3 / ı t4 ;t3 ı t7 ;t3 ı . t1 ;t2 ˝ t3 / 1 C 2 .1/2 3  . t1 ;t2 ˝ t3 / ı t7 ;t3 ı t4 ;t3 ı . t1 ;t2 ˝ t3 /; 2 C 3 

73

The Drinfeld associator of gl.1j1/

where the first equality follows from Lemma 26 and Corollary 27, the second equality is a consequence of Lemma 22, and the third equality is implied by Corollary 25 and Lemma 24. The remaining equations can be proven similarly. The following lemma will be used to express the action of ˆ on .V t1 ˝V t2 ˝V t3 /ŒŒh in terms of the commutators of our basis elements from Lemma 28. Lemma 29. With ti D . i ; i ; i / as in equations (45), (46), and Y D .1/1 C2 C3

2 EndC ..b1 ; b2 ; b3 //

(bi D ..1/i i ; .1/i i ; i /) we have W .Y / D . 1 C 2 /. 2 C 3 /Œ. t1 ;t2 ı t1 ;t2 / ˝ t3 ; t1 ˝ . t2 ;t3 ı t2 ;t3 /: P Proof. With ! D  a ˝ b as in equation (32) we compute X .1/deg.v t1 / deg.b / a  v t1 ˝ b  v t2 D . 1 . 2 C 1/ C 2 . 1 C 1//v t1 ˝ v t2 ; 2 

2

X

.1/deg.w t1 / deg.b / a  w t1 ˝ b  w t2 D . 1 . 2  1/ C 2 . 1  1//w t1 ˝ w t2 :



Equation (33) implies  2W .1/1 C2 D .a C b/.  2W .1/2 C3 D .c C d /



t1 ;t2

t1

ı 

˝.

t1 ;t2 /

t2 ;t3

ı

˝

t3

t2 ;t3 /

C .a  b/.

C .c  d /

t1 ;t2

t1

˝.

ı

t1 ;t2 /

t1 ;t2

˝

ı

t3 ;

t1 ;t2 /;

where a D 1 2 C 2 1 , b D 1 C 2 , c D 2 3 C 3 2 , and d D 2 C 3 . By Lemma 28 we have h    i W .Y / D W .1/1 C2 ; W .1/2 C3 D .1=4/..a C b/.c C d / C .a  b/.c  d /  .a C b/.c  d /  .a  b/.c C d //Œ. t1 ;t2 ı t1 ;t2 / ˝ t3 ; t1 ˝ . t2 ;t3 ı D bd Œ.

t1 ;t2

ı

t1 ;t2 /

˝

t3 ; t1

˝.

t2 ;t3

ı

t2 ;t3 /

t2 ;t3 /:

This completes the proof.

10 Proof of Theorem 1 Let T3 D U3 ı A3 ı L3 be the colored graph in Figure 1. The upper half U3 of T3 is

ıZ x to the following morphism from .V t1 ˝ V t2 ˝ V t3 /ŒŒh to V t8 ŒŒh mapped by W

74

Jens Lieberum

(see equations (39), (40), (43), (27)):

.Z.U x 3 // D '. 8 h/1=2 '. 1 h/1=2 '. 6 h/1=2 '. 2 h/1=2 W ..U3 // W D  6 '. 8 h/1=2 '. 1 h/1=2 '. 6 h/1=2 '. 2 h/1=2

t1 ;t6

ı.

t1

ı

t2 ;t3 /:

(56)

ıZ x to the following For similar reasons the lower half L3 of T3 is mapped by W morphism from V t8 ŒŒh to .V t1 ˝ V t2 ˝ V t3 /ŒŒh:

.Z.L x 3 // D '. 1 h/1=2 '. 7 h/1=2 W ..L3 // W D 7 '. 1 h/1=2 '. 7 h/1=2 . Since Endgl.1j1/ .V t8 / D Q satisfying x

t8

D

t1 ;t6

ı.

t1

˝

t8

t1 ;t2 /

˝

t3 /

ı

t7 ;t3 :

(57)

there exists a formal power series x 2 QŒŒh

t2 ;t3 /

.Z.A x 3 // ı . ıW

t1 ;t2

˝

t3 /

ı

t7 ;t3 :

(58)

We will determine x in two different ways. Using equations (56), (57), and (58) we compute y 3/ r.S

t4

h

.Z.T x 3 // W 8 '. 8 h/ x 6 7 '. 6 h/1=2 '. 2 h/1=2 '. 7 h/1=2 h D 8 '. 8 h/1=2 D

(59) t4 :

y 3 / D e 2 h=2  e 2 h=2 D  2 h'. 2 h/ for the closure By Lemma 23 we have r.S S3 of T3 . Equation (59) implies s s '.v/'.u C v C w/ 2 8 '. 2 h/'. 8 h/ D ; (60) xD 6 7 '. 6 h/'. 7 h/ '.u C v/'.v C w/ with  D 2 8 =. 6 7 / as in Lemma 24 and u D 1 h; v D 2 h; w D 3 h. Now we use equation (58) directly to derive an equation for x that depends on a Drinfeld associator. We start with a general remark. Let R be a commutative ring with 1. Let M2 .RŒŒh/ be the algebra of 2  2-matrices over RŒŒh. Let a; b 2 hRŒŒh be elements of the augmentation ideal of RŒŒh. Then we have     0 a cosh.c/ a sinh.c/=c (61) exp D 2 M2 .RŒŒh/; b 0 b sinh.c/=c cosh.c/ where c 2 D ab (the result does not depend on the choice of a (formal) root c of c 2 because cosh.c/ and sinh.c/=c are even power series in c). Notice that the bases of Lemma 24 and Corollary 25 establish isomorphisms M2 .QŒŒh/ Š Endgl.1j1/ .V t˚2 /ŒŒh  End.V t1 ˝ V t2 ˝ V t3 /ŒŒh: 8

(62)

75

The Drinfeld associator of gl.1j1/

Let F 2 QŒŒd1 ; d2 ; d3  be the formal power series of Theorem 13. Let Y be as in Lemma 29 with 2 D 1. Lemmas 28 and 29 imply

.Y / D a. F .u; v; w/W

t1 ;t2

C b.

˝

t1 ;t2

t3 /

˝

ı t3 /

t7 ;t3

ı

ı

t4 ;t3

t4 ;t3

ı

ı.

t7 ;t3

t1 ;t2

ı.

˝

t1 ;t2

t3 /

˝

t3 /;

(63)

where a D .u C v/ w F .u; v; w/; and b D u.v C w/  F .u; v; w/: By equations (58), (38), Corollary 25, and equations (61) to (63) with c 2 D ab D uvw.u C v C w/F .u; v; w/2 we have x

t8

D

t1 ;t6

ı. D .

ı.

t1

t1 ;t2

˝

t7 ;t3

ı.

˝ t3 /

t2 ;t3 /

ı

t1 ;t2

.exp.F .d1 ; d2 ; d3 /  Y // ıW

t7 ;t3

˝

t3 /

 3 =. 2 C 3 /

.Y // ı . ı exp.F .u; v; w/W

t1 ;t2

˝

t3 /

ı

D . cosh.c/  . 3 b=. 2 C 3 // sinh.c/=c/ D .cosh.c/ C uwF .u; v; w/ sinh.c/=c/ t8 : Equations (60) and (64) imply cosh.c/ C uw sinh.c/=.c=F .u; v; w// D

t4 ;t3

ı.

t1 ;t2

˝

t3 //

(64)

t7 ;t3 t8

s

'.v/'.u C v C w/ : '.u C v/'.v C w/

Since this formula holds for arbitrary values of u; v; w 2 Q h with u C v 6D 0, v C w 6D 0, and u C v C w 6D 0 we have proven the equation between formal power series stated in Theorem 1.

y and Viro’s Alexander invariant 11 r y to Viro’s Alexander invariant 1 . In this section we relate the Alexander series r Proofs that are direct translations of proofs from [Vir] will only be sketched in what y on different trivalent follows. We start with deriving relations between values of r framed graphs with admissible coloring. When the colors and orientations of the lower three edges of are fixed, then by equation (21) there are two possible colors of the upper edge in an admissible coloring of a graph. In the first case the upper and lower

76

Jens Lieberum

edge point into the same direction, and we compute     1 y y c D r. y r r / c c

(65)

c

by using equations (39), (40), (41), (42), (43), and Lemma 19. In the second case b ı Z maps W to x where x 2 QŒŒh and is a morphism between nonb C isomorphic simple gl.1j1/-modules. Therefore, we obtain   y d D 0 if c 6D d: r (66) c

Consider two parallel strands a

b

colored by a D . ; /, b D . 0 ; 0 / and define

s 2 f˙1g (resp. s 0 2 f˙1g) iff the left (resp. right) strand points downwards. Then we have  X   a b y y y a b D c (67) r if s C s 0 0 6D 0; r r c a

c

b

where the sum runs over the two colors c such that the coloring of

a

b c

a

b

is admissible.

In proofs of equation (67) and equation (68) below, we use equation (17) to show that the left sides of these equations are equal to linear combinations   X y p ; xp r p

for certain xp 2 QŒŒh and we use equation (65) to determine the coefficients xp (see the proof of 9.2.A in [Vir] for more details). Now consider

a

c

d

b e

with colors a; b; c; d; e 2 Q  Q. Let a D . ; /, b D

. 0 ; 0 /, and define s; s 0 2 f˙1g as above. We assume that s C s 0 0 6D 0. Then ! a b X  a b a b c c y y y y f f r D r r r ; (68) c d

e

f

d

e

d

e

where the sum runs over the one or two colors f such that the coloring of admissible and

a d

c

b e



a

c

d

b f e

a

b

coincides with

a d

c

b e

a d

b f e

is

as oriented colored graph. Let i

be the edge colored c in c . When the restriction s C s 0 0 6D 0 is satisfied (resp. d e violated) we say that equation (68) can (resp. cannot) be applied to the edge i . For a strand colored by c D . ; / with a right-handed half-twist, it follows by a direct computation that     h=4 y y D e : (69) r r c c

The Drinfeld associator of gl.1j1/

77

At a trivalent vertex v of G we obtain from part (2) of Theorem 20 and equation (31) that     y y r D sv r : (70) y that are needed to state the following propoWe have collected all properties of r sition. y of embedded colored framed trivalent graphs G is Proposition 30. The invariant r uniquely determined by its value on the trivial knot in equation (42), its values on planar tetrahedra in Lemma 23, and by the skein relations in equations (65) to (70). Proof. Consider a diagram of G. We use equations (67) and (70) to replace each crossing in that diagram by a planar graph with two trivalent vertices (notice that despite the restriction in equation (67) this is always possible). By equation (69) we may assume that the resulting planar graph has blackboard framing. Let n be the number of connected components plus the number of trivalent vertices of a planar trivalent graph G. The connected components F of R2 n G are called the faces of the diagram, and the trivalent vertices (resp. the edges) in the closure of a face F are called the vertices (resp. edges) of that face. Let ` be the minimal number of vertices of a face of G. 5 We will prove the proposition by induction on the pairs .n; `/ 2 N  N0 with lexicographical order. For ` D 0 and n D 1 theP graph G is a trivial knot. For ` D 0 y / D 0 where the sum runs and n > 1, we use equation (67), the property c r. c over the same values of c as the sum in equation (67), and equation (65) to show that y r.G/ D 0 in this case. For ` D 1 the graph G cannot have an admissible coloring, so we do not need to consider that case. In the case ` D 2 we can apply equation (65) y which will reduce n by 2, or we can apply equation (66) to show that r.G/ D 0. Now let `  3. Case 1: Assume that equation (68) can be applied to an edge of F . Then we can reduce ` by one while preserving n. Case 2: When equation (68) cannot be applied to an edge of F 6 we choose a trivalent vertex v of G that is connected to a vertex of the face F by an edge e, and that is not itself a vertex of F . Such a vertex v exists because ` > 2 was minimal. Case 2a: We assume that equation (68) can be applied to e. Then we proceed as shown schematically in Figure 2. Equation (68) can be applied to the two edges i , j in the second step in Figure 2 because equation (68) could not be applied to an edge of F in the first picture. This way we again decrease ` by 1 while preserving n. Case 2b: When equation (68) cannot be applied to e, we proceed as follows. We first use equation (65) to add a bubble to one edge of F . This will increase the number of edges and vertices of F by two. Let e be the new edge of F that belongs to the bubble, let . ; 0/ be the color of e, and let v be one vertex of e. As shown in reasons of Euler characteristic we have `  5 (see [BeS]), but we do not need this in the proof. that case ` must be  4 and even.

5 For 6 In

78

Jens Lieberum

v

e F

:: :

Ý

i

F j

:: :

Ý

F

:: :

Figure 2. Reducing the number of vertices of a face.

Figure 3, our plan is to apply equation (68) ` times to push v around F , and then to use equation (65) again to remove the bubble. :: :Ý

:: :Ý

:: : Ý  Ý

:: :Ý

:: :Ý

:: :

Figure 3. Changing the colors of the edges of a face.

y y 0 / on diagrams This will express r.G/ as a linear combination of values r.G G of the same shape as G, but where orientations of the edges of the face F may have changed and where the colors of these edges have changed by additive constants .˙ ; ˙1/. There are infinitely many possible choices of 2 Q such that equation (68) can be applied ` times as needed above. For any such choice of , we can apply case 2a to all diagrams G 0 as above. This completes the proof. 0

Let B D QŒŒhŒh1  be the quotient field of QŒŒh, M D exp.Qh/  B  , and W D Q. Define ˇ W M  W ! M by ˇ.m; w/ D mw . Let G be a trivalent graph with admissible coloring. Define the colored graph q.G/ by replacing all colors . ; / by .exp. h=4/; / 2 M  W . Then the colors of q.G/ verify condition 2.8.A of [Vir] for the 1-palette P D .B; M; W; ˇ/. Viro’s Alexander invariant 1 (see Section 6.3 y in of [Vir]), considered as a map G 7! 1 .q.G//, verifies the same equations as r Proposition 30 (see 7.2.D and Section 9 of [Vir]). This implies the following theorem. Theorem 31. For an admissibly colored framed trivalent graph G  S 3 we have y r.G/ D 1 .q.G//: y More generally, the invariants r.G/ for all admissible colorings of G determine Viro’s Alexander invariant of graphs with ’universal’ colors as in Section 7.4 of [Vir]. y Theorem 31 and 7.7.G of [Vir] imply the following relation between r.L/ and the multi-variable Alexander polynomial rL . Corollary 32. For a link L with n components colored by . i ; 0/ we have y rL .e 1 h=2 ; : : : ; e n h=2 / D r.L/: Corollary 32 can also be proven directly by using the characterization of r by axioms from [Tu1]. Except for some technical details this direct proof is a translation

The Drinfeld associator of gl.1j1/

79

of the proof given in [Vir]. I learned about Corollary 32 from A. Vaintrob, but he never y was defined using the universal Vassiliev invariant Z, published his proof. Since r Corollary 32 implies that the coefficient of hk in rL .e 1 h=2 ; : : : ; e n h=2 / is a Vassiliev invariant of degree k C 1 (this can also be proven directly using [Har] or [Tu1]).

References [BN1]

D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423–472.

[BN2]

D. Bar-Natan, Non-associative tangles, in Geometric topology (Athens, GA, 1993), ed. by W. H. Kazez, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI, 1997, 139–183.

[BN3]

D. Bar-Natan, Vassiliev and quantum invariants of braids, in The interface of knots and physics (San Francisco, 1995), Proc. Symp. Appl. Math. 51, Amer. Math. Soc., Providence, RI, 1996, 129–144.

[BN4]

D. Bar-Natan, On associators and the Grothendieck-Teichmüller group I, Selecta. Math. (N.S.) 4 (1998), 183–212.

[BNG]

D. Bar-Natan, S. Garoufalidis, On the Melvin-Morton-Rozansky conjecture, Invent. Math. 125 (1996), 103–133.

[BGRT] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, The Århus integral of rational homology 3-spheres I: A highly non trivial flat connection on S 3 , Selecta Math. (N.S.) 8 (2002), 315–339. [BeS]

A.-B. Berger, I. Stassen, The skein relation for the .g2 ; V /-link invariant, Comment. Math. Helv. 75 (2000), 134–155.

[Dr1]

V. G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419–1457.

[Dr2]

V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal.Q=Q/, Leningrad Math. J. 2 (1991), 829–860.

[FKV]

J. M. Figueroa-O’Farrill, T. Kimura, A. Vaintrob, The universal Vassiliev invariant for the Lie superalgebra gl.1j1/, Commun. Math. Phys. 185 (1997), 93–127.

[Har]

R. Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), 365–378.

[Kac]

V. G. Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1977), 31–64.

[Kas]

C. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer-Verlag, New York 1995.

[Le]

T. Q. T. Le, On denominators of the Kontsevich integral and the universal perturbative invariant of 3-manifolds, Invent. Math. 135 (1999), 689–722.

[LM1]

T. Q. T. Le, J. Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), 41–64.

[LM2]

T. Q. T. Le, J. Murakami, Parallel version of the universal Vassiliev-Kontsevich invariant, J. Pure Appl. Algebra 121 (1997), 271–291.

[LMO]

T. Q. T. Le, J. Murakami, T. Ohtsuki, On a universal quantum invariant of 3-manifolds, Topology 37 (1998), 539–574.

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[Lie]

J. Lieberum, Universal Vassiliev invariants of links in coverings of 3-manifolds, J. Knot Theory Ramifications 13 (2004), 515–555.

[MiM]

J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264.

[Mur]

J. Murakami, A state model for the multi-variable Alexander polynomial, Pacific J. Math. 157 (1993), 109–135.

[MuO]

J. Murakami, T. Ohtsuki, Topological quantum field theory for the universal quantum invariant, Commun. Math. Phys. 188 (1997), 501–520.

[Thu]

D. P. Thurston, Wheeling: A diagrammatic analogue of the Duflo isomorphism, Ph.D. Thesis, University of Calfornia, Berkeley 2000. arXiv:math/0006083

[Tu1]

V. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1 (247), 97–147; English transl. Russian Math. Surveys 41 (1986), no. 1, 119–182.

[Tu2]

V. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter, Berlin 1994.

[Vas]

V. A. Vassiliev, Cohomology of knot spaces, in Theory of singularities and its applications (ed. by V. I. Arnold), Adv. Soviet Math. 1, Amer. Math. Soc., Providence, RI, 1990, 23–69.

[Vir]

O. Viro, Quantum relatives of the Alexander polynomial, Algebra Anal. 18 (2006), no. 3, 63–157; English transl. St. Petersbg. Math. J. 18 (2007), 391–457.

Integrable systems associated with elliptic algebras Alexander Odesskii and Vladimir Rubtsov School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom of Great Britain and Northern Ireland and Landau Institute of Theoretical Physics, 117234, Kosygina, 2 Moscow, Russia e-mail: [email protected] Département de Mathématiques, UMR 6093 du CNRS, Université d’Angers, 49045 Angers, France and ITEP, 25, Bol. Cheremushkinskaya, 117259, Moscow, Russia e-mail: [email protected]

Abstract. We construct new integrable systems (IS), both classical and quantum, associated with elliptic algebras. Our constructions are based both on a construction of commuting families in skew fields and on properties of the elliptic algebras and their representations. We give some examples showing how these IS are related to previously studied systems.

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Commuting families in some non-commutative algebras . . . . . . . . . . 1.1 Commuting families in skew fields . . . . . . . . . . . . . . . . . . 1.2 Poisson commuting families . . . . . . . . . . . . . . . . . . . . . . 2 Elliptic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition and main properties . . . . . . . . . . . . . . . . . . . . . 2.2 The algebra Qn .E; / . . . . . . . . . . . . . . . . . . . . . . . . . 3 Integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Commuting elements in the algebras Qn .EI / . . . . . . . . . . . . 3.2 Commuting elements in B2;:::;2 ./ . . . . . . . . . . . . . . . . . . 4 Some examples of elliptic integrable systems . . . . . . . . . . . . . . . . 4.1 Low-dimensional example: the algebra q2 .E; / . . . . . . . . . . . 4.2 SOS eight-vertex model of Date–Miwa–Jimbo–Okado . . . . . . . . 4.3 Elliptic analogs of the Beauville–Mukai systems and the Fay identity 5 Discussion and future problems . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. 82 . 84 . 84 . 85 . 88 . 88 . 88 . 91 . 91 . 96 . 97 . 97 . 98 . 100 . 102 . 103

82

Alexander Odesskii and Vladimir Rubtsov

Introduction This paper is an attempt to establish a direct connection between two close subjects of modern Mathematical Physics: integrable systems (IS) and elliptic algebras. More precisely, we construct IS on a large class of elliptic algebras. We will start with a short account of the subject of the story and then we will briefly describe the IS’s under consideration. In [11], B. Enriquez and the second author proposed a construction of commuting families of elements in skew fields. They used the Poisson version of this construction to give a new proof of the integrability of the Beauville–Mukai integrable systems associated with a K3 surface S ([2]). The Beauville–Mukai systems appear as Lagrangian fibrations of the form S Œg ! jLj D P.H 0 .S; L//, where S Œg is the Hilbert scheme of g points of S , equipped with the symplectic structure introduced in [34], and L is a line bundle on S. Later, the authors of [10] explained that these systems are natural deformations of the “separated” (in the sense of [25]) versions of Hitchin’s integrable systems, more precisely, of their description in terms of spectral curves (already present in [27]). Beauville–Mukai systems can be generalized to surfaces with a Poisson structure (see [4]). When S is the canonical cone Cone.C / of an algebraic curve C , these systems coincide with the separated version of Hitchin’s systems. A quantization of Hitchin’s system was proposed in [3]. The paper [11] shows that the construction of commuting families yields a quantization of the separated version of this system on the canonical cone. This construction depends on a choice of quantization of functions on Cone.C /: In [11], we also conjectured that one can determine such choices in the construction of quantized Beauville–Mukai systems that these systems become isomorphic to the Beilinson–Drinfeld systems at the birational level. A part of this program was realized in [12] for the case of S D T  .P 1 n P /; where P  P 1 is a finite subset. Another main theme of the paper is a certain family of elliptic algebras. These algebras (with 4 generators) appeared in the works of Sklyanin [42], [43] and were later generalized (for any number of generators) and intensively studied by Feigin and one of us ([15], [16], [17]). They can be considered as deformations of certain quadratic Poisson structures on symmetric algebras. The geometric meaning of these Poisson structures was explained in [18] (see also [39], [32]): these are natural Poisson structures of moduli spaces of holomorphic bundles on an elliptic curve. We will use the recent survey [36] as our main source of results and references in the theory of elliptic algebras. Here is an account of the relations between elliptic algebras and IS. Elliptic algebras appeared in Sklyanin’s approach to integrability of the Landau–Lifshitz model ([42], [43]) using the methods of the Faddeev school (quantum inverse scattering method and R-matrix approach). Later, Cherednik observed a relation between the elliptic algebras defined in [15] and the Baxter–Belavin R-matrix (see [7]). An interesting observation of Krichever and Zabrodin giving an interpretation of a generator in the

Integrable systems associated with elliptic algebras

83

Sklyanin elliptic algebra as a hamiltonian of the 2-point Ruijsenaars IS ([30]) was later generalized in [5] to the case of the double elliptic 2-point classical model. However, all applications of these algebras to the IS theory had a somewhat indirect character until the last two years. Another relation between elliptic algebras and IS was obtained in a recent paper of Sokolov–Tsyganov ([44]), who constructed (using Sklyanin’s definition of quadratic Poisson structures) some classical commuting families associated with these Poisson algebras. The integrability of these families is implied by Sklyanin’s method of separation of variables (SoV) and is technically based on a generalization of classical methods going back to Jacobi, Liouville and Stäckel (which are close to the classical part of theorems in [11] and [1]). However, all their results are stated in a “non-elliptic” language. Recently, an example of an integrable system associated with linear and quadratic Poisson brackets given by the elliptic Belavin–Drinfeld classical r-matrix was proposed in [29]. This system (an elliptic rotator) appears both in finite and infinitedimensional cases. The authors of [29] give an elliptic version of 2-dimensional ideal hydrodynamics on the symplectomorphism group of the 2-dimensional torus as well as on a non-commutative torus. The elliptic algebras also appear in the context of non-commutative geometry (see [8]). It would be interesting to relate them to the numerous modern attempts to define a non-commutative version of IS theory. On the other hand, the IS which we construct here appear directly in the frame of elliptic algebras. Let us describe the results of this paper. We construct two commuting families in elliptic algebras (Theorem 3.1 and Proposition 3.4). The first commuting family is related to a quantum version of a bi-hamiltonian system which was introduced in [37]. In this paper, a family of compatible elliptic Poisson structures was introduced. This family contains three quadratic Poisson brackets such that their generic linear combination is the quasi-classical limit qn .E/ of an elliptic algebra Qn .E; /. The famous Magri–Lenard scheme yields the existence of a classical integrable system associated with the elliptic curve. The corresponding integrable system has as its phase space a 2m-dimensional component of the moduli space of parabolic rank two bundles on the given elliptic curve E. More precisely, the coordinate ring of the open dense part of this component has the structure of a quadratic Poisson algebra isomorphic to q2m .E/. The quantization of this system was unknown because it is not known how to quantize the Magri–Lenard scheme. When n D 2m, we construct a quantum integrable system based on the approach of [11]; we conjecture that this is a quantization of the classical system from [37] and check this in the first nontrivial case (m D 3). Whereas Theorem 3.1 takes place for n D 2m, we think that there are some interesting integrable quantum systems in the case n D 2mC1. It would be interesting to study the bi-hamiltonian structures giving the algebra q2mC1 .E/ using the results of Gelfand–Zakharevich ([24]) on the geometry of bi-hamiltonian systems in the case of odd-dimensional Poisson manifolds. The precise quantum version of these systems

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in the context of the elliptic algebras Q2mC1 .E; / is still obscure and should be a subject of further studies. The second commuting family (Proposition 3.4) is associated with a special choice of the elliptic algebra. This family is obtained, on the one hand, by direct application of the construction from [11] to the elliptic algebras and, on the other hand, by using the properties of the “bosonization” homomorphism, constructed in earlier works ([16], [17]). Some of these families (under the appropriate choice of numerical parameters) may be interpreted as algebraic examples of completely integrable systems. We give a geometric interpretation to some of them describing a link with the Lagrangian fibrations on symmetric powers of the cone over an elliptic curve, giving a version of the Beauville–Mukai systems (see [2], [25], [11], [33]). The theorems in [11] may also be interpreted as an algebraic version of the SoV method (as it was argued in [1]). Hence, it is very plausible that some of our quantum commuting families arising from the generalization of the Jacobi–Liouville–Stäckel conditions are the quantum elliptic versions of the IS from [44]. We hope to return to this question in a future paper. We give also some low-dimensional examples of our construction.

1 Commuting families in some non-commutative algebras 1.1 Commuting families in skew fields Let A be an associative algebra with unit. We will assume later that A is contained in a skew field K. Fix a natural number n  2. We will assume that there are n subalgebras Bi  A, 1  i  n such that for any pair of indices i ¤ j , 1  i; j  n, any pair of elements b.i/ 2 Bi and b.j / 2 Bj commute with each other (while the algebras Bi are not assumed to be commutative). Let us consider a matrix M of size n  .n C 1/ 0 0 1 n1 n 1 b.1/ b.1/ : : : b.1/ b.1/ 1 n1 n C Bb 0 B .2/ b.2/ : : : b.2/ b.2/ C (1.1) B : :: :: :: :: C @ :: : : : : A 0 1 n1 n b.n/ : : : b.n/ b.n/ b.n/ such that all the elements of i -th row belong to the i -th subalgebra B.i / . We will denote by M i the n  n matrix obtained from the matrix M by removing i-th row. The corresponding Cartier–Foata determinant will be denoted by M i . Its definition repeats verbatim the standard one: in each matrix Mi the entries lying in different rows commute together, so that each summand in the standard definition of the determinant is (up to sign) the product of n elements of different rows, whose product is order-independent.

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The following theorem was proved in [11]. Theorem 1.1. Assume that the matrix M 0 is invertible. Then the elements Hi ´ .M 0 /1 M i , i D 1; : : : ; n, pairwise commute.1 The proof of theorem is achieved by some tedious but straightforward induction procedure. Similar results were obtained in the framework of multi-parametric spectral problems in Operator Analysis ([28]) and in the framework of Seiberg–Witten integrable systems associated with a hyperelliptic spectral curves in [1]. The important step in the proof of Theorem 1.1 is the following “triangle” relations which are similar to the usual Yang–Baxter relation: M i .M 0 /1 M j D M j .M 0 /1 M i ; ij

0 1

B .M /

B

kj

ik

0 1

D B .M /

ij

B ;

(1.2) (1.3)

where B ij is the co-factor of the matrix element bji , 0  i; j  n. Theorem 1.1 can be reformulated to give the following result: Corollary 1.2. Let A be an algebra, .fi;j /0in;1j n be elements of A such that fi;j fk;` D fk;` fi;j for any i, j , k, ` such that j ¤ `. P For any I  f0; Q: : : ; ng; J  f1; : : : ; ng of the same cardinality, we set I;J D 2Bij.I;J / . / i2I fi;.i/ . Here Bij.I; J / denotes the set of bijections between I and J . Assume that the I;J are all invertible. Set i ´ f1;:::;ng;f0;:::;i;:::;ng . Then the elements L Hi D .0 /1 i all commute together.

1.2 Poisson commuting families The following observation is straightforward. Lemma 1.3. If B is an integral Poisson algebra, then there is a unique Poisson structure on Frac.B/ extending the Poisson structure of B. This structure is uniquely defined by the relations f1=f; gg D ff; gg=f 2 ;

f1=f; 1=gg D ff; gg=.f 2 g 2 /:

Theorem 1.1 has a Poisson counterpart. 1 In [11] is proved that the images of H under the embedding A  K pairwise commute, which obviously i implies the above statement.

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Theorem 1.4. Let A be a Poisson algebra. Assume that A is integral, and let  W A ,! Frac.A/ be its injection in its fraction field. Let B1 ; : : : ; Bn  A be Poisson subalgebras of A such that fBi ; Bj g D 0 for i ¤ j . We will write the analogue of the matrix (1.1) as a row vector: M D Œb 0 ; b 1 ; : : : ; b n  , where 0 i1 b1 Bb i C B 2C bi D B : C : @ :: A bni We set D detŒb 0 ; : : : ; bL i ; : : : ; b n : class i is nonzero we Here, as usual, we denote by bL i the i -th omitted column. Then if class 0 class class set Hiclass D class = , and the family .H / is Poisson-commutative: iD1;:::;n 0 i i fHiclass ; Hjclass g D 0 for any pair .i; j /. Remark 1. The elements bik and bjl of the matrix M belong to different subalgebras Bi and Bj if i ¤ j and hence Poisson commute. This condition reminds the classical constraints on the Poisson brackets between matrix elements which appeared in the 19th century papers of StRackel on the separation of variables of Hamilton–Jacobi systems ([45]). So the conditions of our theorem can be considered as an algebraic version of the StRackel conditions.

1.2.1 Plücker relations. Here we remind an important step of the second proof in [11] which shows the relations between the commuting elements and the Plücker-like equations. We have to prove fjclass ; class class i k g C cyclic permutation in .i; j; k/ D 0:

(1.4)

We have class i

D

n n X X

L .1/pC˛ .b ˛ /.p/ .˛;i /.1:::p:::n/ ;

pD1 ˛D0

where (if ˛ ¤ i ) p:::n/ L D .1/1˛
(which means that the p-th row in the matrix Œb 0 ; : : : ; bL ˛ ; : : : ; bL i : : : b n  should be omitted).

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We set 1˛
n n X X

L .1/˛Cˇ .fb ˛ ; b ˇ g/.p/ .˛;i ˇ;j  ˇ;i ˛;j /.1:::p:::n/ ;

pD1 ˛;ˇ D0

so identity (1.4) is a consequence of X . /˛;.i/ ˇ; .j / ;.k/ D 0

(1.5)

2Perm.i;j;k/

for all i , j , k, ˛, ˇ, . If cardf˛; : : : ; kg D 3, this identity follows from the antisymmetry relation i;j C j;i D 0. If cardf˛; : : : ; kg D 4 (resp. 5, 6), it follows from the following Plücker identities (to get (1.5), one should set V D .A˝n /˚n and ƒ some partial determinant). Let V be a vector space. Then the following holds: • if ƒ 2 ^2 .V /, and a; b; c; d 2 V  , then ƒ.a; b/ƒ.c; d /  ƒ.a; c/ƒ.b; d / C ƒ.a; d /ƒ.b; c/ D 0I

(1.6)

• if ƒ 2 ^3 .V / and a; b; c; b 0 ; c 0 2 V  , then ƒ.b; c; c 0 /ƒ.a; c; b 0 /ƒ.b; b 0 ; c 0 / C ƒ.b; c; b 0 /ƒ.c; b 0 ; c 0 /ƒ.a; b; c 0 /  ƒ.b; c; b 0 /ƒ.a; c; c 0 /ƒ.b; b 0 ; c 0 /  ƒ.b; c; c 0 /ƒ.c; b 0 ; c 0 /ƒ.a; b; b 0 / D 0I • if ƒ 2 ^4 .V / and a; b; c; a0 ; b 0 ; c 0 2 V  , then ƒ.b; c; b 0 ; c 0 /ƒ.a; c; a0 ; c 0 /ƒ.a; b; a0 ; b 0 / C ƒ.b; c; a0 ; c 0 /ƒ.a; c; a0 ; b 0 /ƒ.a; b; b 0 ; c 0 / C ƒ.b; c; a0 ; b 0 /ƒ.a; c; b 0 ; c 0 /ƒ.a; b; a0 ; c 0 /  ƒ.b; c; b 0 ; c 0 /ƒ.a; c; a0 ; b 0 /ƒ.a; b; a0 ; c 0 /  ƒ.b; c; a0 ; b 0 /ƒ.a; c; a0 ; c 0 /ƒ.a; b; b 0 ; c 0 /  ƒ.b; c; a0 ; c 0 /ƒ.a; c; b 0 ; c 0 /ƒ.a; b; a0 ; b 0 / D 0: We refer to [11] for a proof of these identities. We will need them below in some special situation arising from the commuting elements in associative and Poisson algebras which are directly connected with elliptic curves and vector bundles over them. These Plücker relations can be interpreted as a kind of Riemann–Fay identities which are in their turn related to integrable (difference) equations in Hirota bilinear form.

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2 Elliptic algebras Now we describe one of the main heroes of our story – the family of elliptic algebras. We will follow the survey [36] for notation and as our a main source of results and proofs in this section.

2.1 Definition and main properties The elliptic algebras are associative quadratic algebras Qn;k .E; / which were introduced in the papers [15], [17]. Here E is an elliptic curve and n; k are coprime integer numbers such that 1  k < n while  is a complex number and Qn;k .E; 0/ D CŒx1 ; : : : ; xn . Let E D C= be an elliptic curve defined by a lattice D Z˚ Z, 2 C, = > 0. The algebra Qn;k .E; / has generators xi , i 2 Z=nZ, subject to the relations X j i Cr.k1/ .0/ xj r xiCr D 0; j i r ./kr ./ r2Z=nZ

and has the following properties: 1) Qn;k .E; / D C ˚ Q1 ˚ Q2 ˚    such that Q˛  Qˇ D Q˛Cˇ ; here  denotes the algebra multiplication. In other words, the Palgebras Qn;k .E; / 1are Z-graded. . 2) The Hilbert function of Qn;k .E; / is ˛0 dim Q˛ t ˛ D .1t /n We consider here the set of theta functions fi .z/; i D 0; : : : ; n1g as a basis of the space of order n theta functions ‚n . /. These functions satisfy the quasi-periodicity relations p i .z C 1/ D i .z/; i .z C / D .1/n exp.2 1nz/i .z/ for i D 0; : : : ; n  1. The order one theta function .z/ p 2 ‚1 . / satisfies the conditions .0/ D 0 and .z/ D .z C / D  exp.2 1z/.z/. When E is fixed, the algebra Qn;k .E; / is a flat deformation of the polynomial ring CŒx1 ; : : : ; xn . The first order term in  of this deformation gives rise to a quadratic Poisson algebra qn;k .E/. The geometric meaning of the algebras Qn;k was explained in [18], [39], where it was shown that the quadratic Poisson structure qn;k .E/ identifies with a natural Poisson structure on P n1 D PExt 1 .E; O/, where E is a stable vector bundle of rank k and degree n on the elliptic curve E. In what follows we will denote the algebras Qn;1 .E; / by Qn .E; /.

2.2 The algebra Qn .E; / 2.2.1 Construction. For any n 2 N, any elliptic curve E D C= and any point  2 E, we construct a graded associative algebra Qn .E; / D C˚F1 ˚F2 ˚: : : , where F1 D ‚n . / and F˛ D S ˛ ‚n . /. By construction, dim F˛ D n.nC1/:::.nC˛1/ . It ˛Š is clear that the space F˛ can be realized as the space of holomorphic symmetric

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functions of ˛ variables f .z1 ; : : : ; z˛ / such that f .z1 C 1; z2 ; : : : ; z˛ / D f .z1 ; : : : ; z˛ /; f .z1 C ; z2 ; : : : ; z˛ / D .1/n e 2 i nz1 f .z1 ; : : : ; z˛ /: For f 2 F˛ and g 2 Fˇ we define the symmetric function f  g of ˛ C ˇ variables by the formula f  g.z1 ; : : : ; z˛Cˇ / X 1 D f .z1 C ˇ; : : : ; z˛ C ˇ/g.z˛C1  ˛; : : : ; z˛Cˇ  ˛/ ˛ŠˇŠ 2S˛Cˇ

Y



1i˛ ˛C1j ˛Cˇ

.zi  zj  n/ : .zi  zj /

In particular, for f; g 2 F1 we have f  g.z1 ; z2 / D f .z1 C /g.z2  /

.z1  z2  n/ .z2  z1  n/ C f .z2 C /g.z1  / : .z1  z2 / .z2  z1 /

Here .z/ is a theta function of order one. Proposition 2.1. If f 2 F˛ and g 2 Fˇ , then L f  g 2 F˛Cˇ . The operation  defines an associative multiplication on the space ˛0 F˛ . 2.2.2 Main properties of the algebra Qn .E; /. By construction, the dimensions of the graded components of the algebra Qn .E; / coincide with those for the polynomial ring in n variables. For  D 0 the formula for f  g becomes X 1 f .z1 ; : : : ; z˛ /g.z˛C1 ; : : : ; z˛Cˇ /: f  g.z1 ; : : : ; z˛C1 / D ˛ŠˇŠ 2S˛Cˇ

This is the formula for the ordinary product in the algebra S  ‚n . /, that is, in the polynomial ring in n variables. Therefore, for a fixed elliptic curve E (that is, for a fixed modular parameter ) the family of algebras Qn .E; / is a deformation of this polynomial ring. In particular, it defines a Poisson structure on this ring, which we denote by qn .E/. One can readily obtain the formula for the Poisson bracket on this polynomial ring from the formula for f  g by expanding the difference f  g  g  f in a Taylor series with respect to . It follows from semi-continuity arguments that the algebra Qn .E; / with generic  is determined by n generators and n.n1/ quadratic 2 relations. One can prove (see §2.6 in [36]) that this is the case if  is not a point of finite order on E, that is, N  62 for any N 2 N. The space ‚n . / of the generators of the algebra Qn .E; / is endowed with an action of a finite group zn that is a central extension of the group =n of points of

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order n on the curve E. It can be checked that it extends to an action of this group by algebra automorphisms of Qn .E; /. 2.2.3 Bosonization of the algebra Qn .E; /. The main approach to obtain representations of the algebra Qn .E; / is to construct homomorphisms from this algebra to other algebras with a simple structure (close to the Weyl algebras) which have a natural set of representations. These homomorphisms are referred to as bosonizations, by analogy with the known constructions of quantum field theory. LetBp;n ./ be the algebra generated by the commutative algebra of meromorphic functions f .u1 ; : : : ; up / and by the elements e1 ; : : : ; ep with defining relations e˛ f .u1 ; : : : ; up / D f .u1  2; : : : ; u˛ C .n  2/; : : : ; up  2/e˛ ; e˛ eˇ D eˇ e˛ ; f .u1 ; : : : ; up /g.u1 ; : : : ; up / D g.u1 ; : : : ; up /f .u1 ; : : : ; up /: Then Bp;n ./ is a Zp -graded algebra whose space of degree .˛1 ; : : : ; ˛p / is ˛

ff .u1 ; : : : ; up /e1˛1 : : : epp g; where f ranges over the meromorphic functions of p variables. We note that the subalgebra of Bp;n ./ consisting of the elements of degree .0; : : : ; 0/ is the commutative algebra of all meromorphic functions of p variables with the ordinary multiplication. Proposition 2.2. Let  2 E be a point of infinite order. For any p 2 N there is a homomorphism p W Qn .E; / ! Bp;n ./ defined on the generators of the algebra Qn .E; / by the formula: X f .u˛ / Q p .f / D e˛ : (2.1) .u  u / ˛ i i¤˛ 1˛p Here f 2 ‚n . / is a degree 1 generator of Qn .E; /. 2.2.4 Symplectic leaves. We recall that Qn .E; 0/ is the polynomial ring S  ‚n . /. For a fixed elliptic curve E D C= we obtain the family of algebras Qn .E; /, which is a flat deformation of the polynomial ring. We denote the corresponding Poisson algebra by qn .E/. We obtain a family of Poisson algebras depending on E, that is, on the modular parameter . Let us study the symplectic leaves of this algebra. To this end, we note that, when passing to the limit  ! 0, the homomorphism p of associative algebras gives a homomorphism of Poisson algebras. Namely, let us denote by bp;n L ˛ the Poisson structure on the algebra ˛1 ;:::;˛p 0 ff˛1 ;:::;˛p .u1 ; : : : ; up /e1˛1 : : : epp g, where f˛1 ;:::;˛p runs over all meromorphic functions defined by fu˛ ; uˇ g D fe˛ ; eˇ g D 0; where ˛ ¤ ˇ.

fe˛ ; uˇ g D 2e˛ ;

fe˛ ; u˛ g D .n  2/e˛ ;

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The following assertion results from Proposition 2.2 in the limit  ! 0. Proposition 2.3. There is a Poisson algebra homomorphism p W qn .E/ ! bp;n given by the following formula: if f 2 ‚n . /, then X f .u˛ / e˛ : p .f / D .u  u / : : : .u  u / ˛ 1 ˛ p 1˛p Let fi .u/I i 2 Z=nZg be a basis of the space ‚n . / and let fxi I i 2 Z=nZg be the corresponding basis in the space of elements of degree one in the algebra Qn .E; / (this space is isomorphic to ‚n . /). Consider the embedding of the elliptic curve E  P n1 by means of theta functions of order n: z 7! .0 .z/ W    W n1 .z//. We denote by Cp E the variety of p-chords, that is, the union of projective spaces of dimension p  1 passing through p points of E. Let K.Cp E/ be the corresponding homogeneous manifold in Cn . It is clear that K.Cp E/ consists of the points with the coordinates X i .u˛ / e˛ ; where u˛ ; e˛ 2 C. xi D .u˛  u1 / : : : .u˛  up / 1˛p Let 2p < n. Then one can show that dim K.Cp E/ D 2p and K.Cp1 E/ is the manifold of singularities of K.Cp E/. It follows from Proposition 2.3 and the fact that the Poisson bracket is non-degenerate on bp;n for 2p < n and e˛ ¤ 0 that the non-singular part of the manifold K.Cp E/ is a 2p-dimensional symplectic leaf of the Poisson algebra qn .E/. Let n be odd. One can show that the equation defining the manifold K.C n1 E/ 2 is of the form C D 0, where C is a homogeneous polynomial of degree n in the variables xi . This polynomial is a central function of the algebra qn .E/. Let n be even. The manifold K.C n2 E/ is defined by equations C1 D 0 and 2 C2 D 0, where deg C1 D deg C2 D n=2. The polynomials C1 and C2 are central in the algebra qn .E/.

3 Integrable systems There are (at least two) different ways to construct commuting families and IS associated with the elliptic algebras. We will start with general statements about the commuting elements arising from the ideas and constructions of Section 1.

3.1 Commuting elements in the algebras Qn .EI / Let us consider the following Weyl-like algebra Vn with the set of “generators” f1 ; : : : ; fn ; z1 ; : : : ; zn subject to the relations 0 D Œfi ; fj  D Œzi ; zj  D Œfi ; zj  .i ¤ j /;

fi zi D .zi  n/fi :

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Vn is spanned as a vector space by the elements of the form F .z1 ; : : : ; zn /f1m1 : : : fnmn ; where F is a meromorphic function in n variables. So we have the following commutation relations between the functions in variables zi and the elements fj : fj F .z1 ; : : : ; zn / D F .z1 ; : : : ; zj  n; : : : ; zn /fj : Remark 2. The algebra Vn looks different from the above-mentioned Weyl-like algebras Bp;n but it is isomorphic to the algebra Bn;n . We will return below to a geometric interpretation of the algebra Vn . Now we consider the following determinant ˇ ˇ ˇ0 .z1 / 1 .z1 / : : : n2 .z1 / n1 .z1 /ˇ ˇ ˇ ˇ0 .z2 / 1 .z2 / : : : n2 .z2 / n1 .z2 /ˇ ˇ ˇ D0 D ˇ : ˇ :: :: :: :: ˇ :: ˇ : : : : ˇ ˇ ˇ0 .zn / 1 .zn / : : : n2 .zn / n1 .zn /ˇ D c exp.z2 C 2z3 C    C .n  1/zn /

Y

.zi  zj /

n X  zi ;

1i<j n

iD1

where the constant c is irrelevant for us because it will be cancelled in future computations. Then we define the partial determinants Di by deleting the i -th column and adjoining an n-th column of f˛ ’s: ˇ ˇ ˇ0 .z1 / 1 .z1 / : : : i n1 .z1 / f1 ˇ ˇ ˇ ˇ0 .z2 / 1 .z2 / : : : j n1 .z2 / f2 ˇ ˇ ˇ Di D ˇ : ˇ :: :: :: :: : ˇ : ˇ : : : : ˇ ˇ ˇ0 .zn / 1 .zn / : : : j n1 .zn / fn ˇ ˇ ˇ ˇ0 .z1 / 1 .z1 / : : : n1 .z1 / ˇ ˇ ˇ ˇ0 .z2 / 1 .z2 / : : : n1 .z2 / ˇ X ˇ ˇ .1/˛Cn ˇ : D :: :: :: :: ˇ f˛ : ˇ :: ˇ : : : : 1˛n ˇ ˇ ˇ0 .zn / 1 .zn / : : : n1 .zn / ˇ ˛;i Here the subscript j j˛;i means that we omit the i -th column and the ˛-th row. The immediate corollary of (1.1) is the following Proposition 3.1. The determinant ratios form a commutative family: ŒD01 Di ; D01 Dj  D 0: The result of the proposition can be expressed in an elegant way in the form of a commutation relation of generating functions.

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Let us define a generating function T .u/ of a variable u 2 C: X T .u/ D D01 .1/j j .u/Dj : 1j n

Then we can express the function T .u/, using the formulas for the determinants of the theta functions as ˇ ˇ ˇ0 .z1 / 1 .z1 / : : : n1 .z1 /ˇ ˇ ˇ ˇ :: ˇ :: :: :: ˇ : ˇ : : : X ˇ ˇ T .u/ D D01 .1/˛ ˇˇ 0 .u/ 1 .u/ : : : n1 .u/ ˇˇ f˛ ˇ :: ˇ :: :: :: 1˛n ˇ : ˇ : : : (3.1) ˇ ˇ ˇ0 .zn / 1 .zn / : : : n1 .zn /ˇ D

X .u C

P

ˇ ¤˛

Q

zˇ /

ˇ ¤˛

1˛n

Q

1ˇ ¤˛n

.z˛  zˇ /

.u  zˇ /

fQ˛ ;

where we denote by fQ˛ the normalization fQ˛ D

.

f˛ Pn

iD1 zi /

:

We remark that the commutation relations between the variables z1 ; : : : ; zn ; fQ1 ; : : : ; fQn are the same as they were between the variables z1 ; : : : ; zn ; f1 ; : : : ; fn . In this notation the proposition now reads: Proposition 3.2. The “transfer-like” operators T .u/ commute for different values of the parameter u: ŒT .u/; T .v/ D 0: Now we will apply this result to a construction of commuting elements in the algebra Qn .EI / for even n. Let n D 2m. It is known in this case that the center Z.Q2m .EI // for  of infinite order is generated by Casimir elements f .z1 ; : : : ; zm / 2 S m .‚2m . // such that f .z1 ; : : : ; zm / D 0 for z2 D z1 C 2m. A straightforward computation shows that the space of such elements is two-dimensional and has as a basis the elements of the form Y C˛ D ˛ .z1 C : : : zm / .zi  zj  2m/; 1i¤j m

where ˛ 2 ‚2 . /, ˛ 2 Z=2Z. Let S m .‚2m . // be the space of symmetric polynomials of degree m in theta functions of order 2m. Proposition 3.3. Fix an element ‰.z/ 2 ‚mC5 . / and two complex numbers a; b 2 C. There exists a family f .u/.z1 ; : : : ; zn / 2 S m .‚2m . // indexed by u 2 C such

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that f .u/.z1 ; z1 C 2m; z2 : : : ; zm1 / (3.2) 1 .a C .m  2/b C 2m.m  2///.z1 C    C zm1 C a/ D ‰.z1 C 4m2   mC5  .z1 C z2 C b/ : : : .z1 C zm1 C b/.z2  z1  4m/ : : : .zm1  z1  4m/  .z2  z1 C 2m/ : : : .zm1  z1 C 2m/  .u C z2 C    C zm1  a  b C 2m/.u  z2 / : : : .u  zm1 / Y .zi  zj  2m/:  exp.2 i.2.m  2/z1 C z2 C    C zm1 // 2i¤j m1

Remark 3. The elements f .u/ are well defined up to a linear combination of the Casimir functions C1 ; C2 . Sketch of the proof. Consider the space F .m; / of symmetric functions defined on the subvariety of all shifted diagonals zi D zj C 2m; 1  i ¤ j  m; we have a restriction map S m .‚2m . // ! F .m; /. Formula (3.2) defines a family of functions in F .m; /, indexed by u 2 C, whose quasi-periodicity properties are the same are those of the elements of the image of the restriction map S m .‚2m . // ! F .m; /. We now show that the elements f .u/ can be lifted to a family of degree m polynomials in theta functions of order 2m. The proof is based on the existence of a certain general filtration in symmetric polynomial rings and their deformations which was introduced in [19]. The elliptic algebra Qn .E; / admits the following filtration Fr1 ;:::;rp with r1  r2      rp : an element x 2 Qn .E; / belongs to Fr1 ;:::;rp if the image p .x/ 2 Bp;n under the bosonization homomorphism p (2.1) has the following form: r

p .x/ D f .u1 ; : : : ; up /e1r1 : : : epp C elements of smaller order: Here < is the natural ordering on homogeneous components of Bn;p : .r1 ; : : : ; rp / < .s1 ; : : : ; sp / if r1 D s1 ; : : : ; r t D s t ; r t C1 < s t C1 ; for some t . For example, the subspace F1;:::;1 is generated by two central elements C1 , C2 . The most difficult part of the proof is to identify the associated graded of this filtration with some spaces of holomorphic functions satisfying certain symmetry and quasiperiodicity properties. Let us consider the highest order terms of the filtration Fp1;1;0;:::;0  Fp;0;:::;0 and take the bosonization of an element x 2 Fp;0;:::;0  Qn .E; /, p .x/ D ˛1 .u/e1p C ˛2 .u/e1p1 e2 C    :

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The element x 2 Qn .E; / belongs to Fp1;1;0;:::;0 if the coefficient function ˛1 vanishes (˛1 is a theta function of order np). Hence the elements of the quotient Fp;0;:::;0 =Fp1;1;0;:::;0 identify with theta functions of order np. Using this identification we see that the condition (3.2) on f .u/ determines the image of f .u/ in the quotient F2;1;:::;1 =F1;:::;1 . Since the space F1;:::;1 is the space generated by Casimir functions C1;2 , the element f .u/ can be lifted to an element in F2;1;:::;1  S m .‚2m . //: Now we apply these observations to establish the following result: Theorem 3.4. In the elliptic algebra Q2m .E; / the following relation holds Œf .u/; f .v/ D f .u/  f .v/  f .v/  f .u/ D 0: Proof. We will use the homomorphism Q2m .E; / ! Bm1;2m from Section 2.2.3. The element f .u/ is taken by this homomorphism to the element P Q X .u C ˇ ¤˛ zˇ / 1ˇ ¤˛m1 .u  zˇ / Q f˛ ; ˇ ¤˛ .zˇ  z˛ / 1˛m1 where we denote by f˛ the expression f˛ D ‰.z˛ C 4m2   

 m1 X

zˇ C a

ˇ D1

C

X

1 .a mC5

Y

C .m  2/b C 2m.m  2///

.z˛ C zˇ C b/ exp.2 i.2.m  2/z˛

ˇ ¤˛

zˇ //e˛ e1 : : : em1 :

ˇ ¤˛

Formula (3.1) and Proposition 3.2 then immediately imply that the images of f .u/ and f .v/ under the homomorphism m1 commute in Bm1;2m ./. It is known that the image of the algebra Q2m .E; / in Bm1;2m ./ is the quotient of Q2m .E; / by the ideal hC1 ; C2 i generated by C1 , C2 . Hence the commutator Œf .u/; f .v/ D f .u/  f .v/  f .v/  f .u/ belongs to this ideal. To show that Œf .u/; f .v/ D 0 we can consider the injective homomorphism into Bm;2m ./ and it is sufficient to verify that the coefficient before .e1 /2 : : : .em /2 equals zero, or (which is equivalent) that Œf .u/; f .v/.z1 ; z1 C 2m; z2 ; z2 C 2m; : : : ; zm ; zm C 2m/ D 0; which is a direct verification. Remark 4. We should observe that the commuting elements constructed here are parametrized by the choice of the element ‰.z/ 2 ‚mC5 . /.

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Remark 5. In [37] we constructed a family of compatible quadratic Poisson structures, a linear combination of which is isomorphic to the Poisson structure of the classical elliptic algebra qn .E/. The Magri–Lenard scheme yields a family of Poisson commuting elements (“hamiltonians” in involution) in the algebra qn .E/. These elements have degree n if n is odd and n=2 otherwise. Let n D 2m. We conjecture that the commuting family of Q2m .E/ constructed in 3.4 (for a proper choice of the parametrizing element ‰.z/) is the quantum analogue of the family of commuting “hamiltonians” in q2m .E/ generated by the Magri–Lenard scheme applied to the classical Casimir functions C0.m/ , C1.m/ of [37]. This conjecture is true in the first non-trivial case n D 6, where it is verified by direct computation.

3.2 Commuting elements in B2 ;:::;2 ./ The elliptic algebra Qn;n1 .E; / is commutative and the bosonization homomorphism yields a large class of commuting families in the corresponding Weyl-like algebra. This gives rise to a new integrable system. 3.2.1 Bosonization of the algebra Qn;n1 .E/. The map n W Qn .E; / ! Bn;n ./ from 2.2.3 (corresponding to p D n) may be generalized to the case of the elliptic algebras Qn;k .E; / ([16]). The structure of the Weyl-like algebra similar to Bn;n ./ turns out to be more complicated for k > 1. We will use this generalization in the special case k D n  1, i.e., for the commutative algebra Qn;n1 .E; /: Let us describe the generalization of n . Assume that 1  k  n  1 and expand n=k in continued fraction as follows 1 n D p1  k p2     

1 pq

:

Let Bn;p1 ;p2 ;:::;pq ./ be the associative algebra with generators z1;1 ; : : : ; zp1 ;1 ; z1;2 ; : : : ; zp2 ;2 ; z1;q ; : : : ; zpq ;q ;

e1;1 ; : : : ; ep1 ;1 ; e1;2 ; : : : ; ep2 ;2 ; e1;q ; : : : ; epq ;q ;

and t1;2 ; t2;3 ; : : : ; tq1;q ; f1;2 ; f2;3 ; : : : ; fq1;q : We will impose the following commutation relations between them: e˛; zˇ; D .zˇ;  n/e˛; ; ˛ ¤ ˇ; f˛;˛C1 t˛;˛C1 D .t˛;˛C1  n/f˛;˛C1 : The other pairs of generators commute.

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There is a “bosonization” homomorphism p1 ;:::;pq W Qn;k .E; / ! Bn;p1 ;p2 ;:::;pq ./: A generating series for elements of the image of this morphism is the “transferfunction” Tz .u/

X .u  z˛1 ;1 /.u C z˛1 ;1  z˛2 ;2 /.u C z˛2 ;2  z˛3 ;3 / : : : .u C z˛q ;q / Q .z˛ ;  zˇ; / 1˛ p

D

1

1

ˇ ¤˛ 1ˇ p 1 q

::: 1˛q pq

 .z˛1 ;1 C z˛2 ;2  t1;2 /.z˛2 ;2 C z˛3 ;3  t2;3 /    .z˛q1;q1 C z˛q;q  tq1;q /  e˛1 ;1 : : : e˛q;q f1;2 : : : fq1;q : (All these statements are contained in [17].) n The expansion of n1 in continued fraction is n 1 ; D2 n1 2      12 so q D n  1 and p1 D    D pq D 2. Proposition 3.5. The relation

e e

ŒT .u/; T .v/ D 0 holds in Bn;2;:::;2 ./. Proof. This follows from the commutativity of Qn;n1 .E/.

4 Some examples of elliptic integrable systems 4.1 Low-dimensional example: the algebra q2 .E; / The elliptic integrable system arising in 3.4 becomes transparent in the case m D 1. Then the commutative elliptic algebra Q2 .E; / has functional dimension 2 and its Poisson counterpart admits the Poisson morphism 2W

q2 .E/ ! b2 ;

where the algebra b2 consists of the elements X f˛;ˇ .z1 ; z2 /e1˛ e2ˇ ; ˛;ˇ

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where f˛;ˇ .z1 ; z2 / are meromorphic functions and the Poisson structure in b2 is given by fzi ; zj g D fei ; ej g D fei ; zi g D 0;

fei ; zj g D 2ei .i ¤ j /;

i; j D 1; 2:

The explicit formula for this mapping (for f 2 ‚2 . / a given theta function of order 2) is the following: f .z1 / f .z2 / e1 C e2 : 2 .f / D .z1  z2 / .z2  z1 / Now let 1 ; 2 be the basic theta functions of order 2, and let us compute the Poisson brackets between their images 2 .1 / and 2 .2 /: Proposition 4.1. These theta functions commute in b2 : f

2 .1 /;

2 .2 /g

D

2 .f1 ; 2 g/

D 0:

4.2 SOS eight-vertex model of Date–Miwa–Jimbo–Okado We recall some ingredients of the SOS eight-vertex model (see [9]) and the construction of its transfer operators. We then establish their relation with our operators T .u/. The 8-vertex model is an IRF (interaction round a face) statistical mechanical model; more precisely, it is a version of the Baxter model related to Felder’s elliptic quantum group E; .sl2 /. This model was studied by Sklyanin’s method of separation of variables (under antiperiodic boundary conditions) in [20] (see also [21] for the representation theory of E; .sl2 /). We will use the results of [20] in a form which we need. The antiperiodic boundary conditions of the model are fixed by the family of transfer matrices TSOS .u; /, where u 2 C is a parameter and TSOS .u; / is expressed as (twisted) traces of the L-operators LSOS .u; / defined using an “auxiliary” module over the elliptic quantum group E; .sl2 /. The L-operator acts on the tensor product of the fundamental representations of the elliptic quantum group and is twisted by the   matrix 01 10 . The L-operator is usually represented as a 2  2-matrix of the form   a.u; / b.u; / ; LSOS .u; / D c.u; / d.u; / with matrix entries meromorphic in u, and and obeys the dynamical RLL-commutation relations for the Felder elliptic R-matrix 1 0 .u C 2/ 0 0 0 .u/. C2/ .u /.2/ C B 0 0 C B . / . / R.u; / D B C: . Cu/.2/ .u/. 2/ A @ 0 0 . / . / 0 0 0 .u C 2/

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We will have to deal with the functional representations of E; .sl2 / which are described by the pairs .F; L/: Here F is a complex vector space of meromorphic functions f .z1 ; : : : ; zn ; / (or a subspace of functions which are holomorphic in a part of the variables) and L is the L-operator as above. The entries of the L-operator are acting by difference operators in the tensor product V ˝ W where dim.V / D 2 and W is an appropriate subspace in the functional space F . The Bethe ansatz method works in the case of the SOS model with periodic boundary conditions, according to Felder and Varchenko ([21]). In the antiperiodic case the family of transfer matrices TSOS .u; / D tr.KLSOS .u; //; u 2 C is commutative: ŒTSOS .u; /; TSOS .v; / D 0; as it follows from the RLL-relations by tedious computations in [41]. On the other hand it is possible to establish the explicit one-to-one correspondence between the families of antiperiodic SOS transfer-matrices and the auxiliary transfermatrices Taux .u; / (see 4.4.3 in [41]). This isomorphism is established by a version of the separation of variables. The explicit expression of the auxiliary transfer-matrix is n X .u C z˛  / Y Taux .u; / D . / ˛D1

.u C zˇ / ..z˛ C/Tz2 C.z˛ /Tz2 /; ˛ ˛ .zˇ  z˛ /

1ˇ ¤˛n

where (to compare with our “transfer”-operators in the elliptic integrable systems) we have put in the formulas of ([41], ch. 4.6) x˛ D 0;

ƒ˛ D 1;

˛ D 1; : : : ; n:

(The choice ƒ˛ D 1 corresponds to the case of separated variables (Proposition 4.36 in [41]).) are acting similarly to the generators f˛ above: The operators Tz˙2 ˛ f .z1 ; : : : ; zn / D f .z1 ; : : : ; z˛ ˙ 2; : : : ; zn /Tz˙2 : Tz˙2 ˛ ˛ Let us make the following change of the variables : f˛˙ D .z˛ /Tz˙2 ˛ Now a simple inspection of the formulas shows that Taux .u; / D TC .u; / C T .u; /; and the commutation results of Section 3.2 can be applied. So we obtain Proposition 4.2. The “transfer”-operator T .u/ coincides (up to an inessential numerical factor depending on the normalization in the definition of theta functions and rescaling of the parameter ) with the combination of the twisted traces of the auxiliary L-operator of the antiperiodic SOS-model under the following change of

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variables:

D

n X

z˛ ;

x˛ D 0;

˙2 f˛˙ D .z˛ /Tz˛ :

˛D1

Proposition 3.2 gives a simple proof of the commutation relations for the twisted traces of the SOS-model in this special case. We should observe that the commutation relations ŒTC .u/; T .v/ C ŒT .u/; TC .v/ D 0 follow from the same arguments as in Section 3.2. On the other hand, this gives additional evidence to our belief that the “integrability” condition from [11] is equivalent in some sense to a RLL-type integrability condition. We also believe that the role of the elliptic integrable systems in IRF models can be generalized and we hope to return to this point in future.

4.3 Elliptic analogs of the Beauville–Mukai systems and the Fay identity Let us describe a geometric meaning of our IS’s. We will start with the observation that the classical analogue of the Weyl-like algebra Vn given by the Poisson brackets 0 D ffi ; fj g D fzi ; zj g D ffi ; zj g .i ¤ j /;

ffi ; zi g D nfi

can be identified with the Poisson algebra of functions on the symmetric power S n .Cone.E// of a cone over an elliptic curve (more precisely, on the Hilbert scheme .Cone.E//Œn of n points on this surface.) The n commuting elements hi D D01 Di in the Poisson algebra obtained in Section 3.2 can be interpreted as an elliptic version of the Beauville–Mukai systems associated with this Poisson surface. We will develop in more details the first interesting case (n D 3) of the Poisson commuting conditions (the Plücker relations from Section 2) for this system. The Beauville–Mukai hamiltonians have the following form in this case:

H1 D

detŒe; 1 ; 2  detŒe; 0 ; 2  detŒe; 0 ; 1  ; H2 D ; H3 D ; detŒ0 ; 1 ; 2  detŒ0 ; 1 ; 2  detŒ0 ; 1 ; 2 

where the vector columns e, i , i D 0; 1; 2 have the entries 0 1 1 0 e1 i .z1 / e D @e2 A ; i D @i .z2 /A ; i D 0; 1; 2: e3 i .z3 /

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Now the integrability condition (1.6) may be expressed as a kind of 4-Riemann identity (known also as the trisecant Fay’s identity): Z A Z C   Q Q .v/ u C !C ! 

B

Z

A

D

 

Z

C



E.A; B/E.D; C / E.A; C /E.D; B/ D B Z A   Z C   E.A; D/E.C; B/ C Q u C ! Q u C ; ! E.A; C /E.D; B/ B D

D Q u C

! Q u C

!

where ! is a chosen holomorphic differential on E, and for X; Y 2 E, Jac.E/ D E, E.X; Y / D E.Y; X/ is a prime form Z B  Z C  Q  Q E.A; B/E.D; C / A D  Z C   ZDB  ; E.A; C /E.D; B/ Q Q A

(4.1)

RY X

! 2

D

Q and .u/ is an odd theta function (see [14]). Let a; b; c; d correspond to the points A; B; C; D in (4.1) and set u D a C c. Then (4.1) reads Q C c/.a Q  c/.b Q C d /Q .b  d /  Q .a C b/Q .a  b/Q .c C d /Q .c  d / .a Q C d /Q .a  d /Q .c C b/Q .c  b/ D 0; C .a where we easily recognize the commutativity conditions (1.6) for the case n D 3 and an appropriate choice of a linear relation between 4 points a, b, c, d and z1 , z2 , z3 , =3 (modulo an irrelevant exponential factor entering in the relations between the theta functions in different normalizations). Remark 6. 1. The appearance of the Riemann–Fay relations as commutativity or integrability conditions in this context looks quite natural both from the “Poisson” as well as from the “integrable” viewpoints. The “Poisson–Plücker” relations and their generalizations in the context of the Poisson polynomial structures were studied in [38]. On the other hand, the first links between integrability conditions (in the form of Hirota bilinear identities for some elliptic difference many-body-like systems) and the Fay formulas were established in [31]. 2. The Fay trisecant formulas on an elliptic curve are also related to a version of “triangle” relations known as the “associative” Yang–Baxter equation obtained by Polishchuk ([40]). This result gives additional evidence that the commutation relations (1.2), (1.3) could be interpreted as an algebraic kind of Yang–Baxter equation. An amusing appearance of NC (non-commutative) determinants in both constructions (the Cartier–Foata determinants defined above are a particular case of the quasideterminants of Gelfand–Retakh [23]) shows that the ideas of [11] might be useful

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in “non-commutative integrability” constructions which involve quasi-determinants, quasi-Plücker relations etc.

5 Discussion and future problems We have proved that the elliptic algebras (under some mild restrictions) carry families of commuting elements which become in some cases examples of integrable systems. Let us indicate some questions deserving future investigations. We plan to construct an analogue of these commuting families in the algebras Q2m .E; / which correspond to the maximal symplectic leaves of the elliptic algebras Q2mC1 .E; / using the bi-hamiltonian elliptic families. This analogue should quantize the bi-hamiltonian families in Cn in the case of an odd n. One of the main motivations for explicitly constructing the systems on Vn in terms of the determinants of theta functions of order n was the want to find a confirmation to the hypothetical “separated” form of the hamiltonians of the n-point double elliptic system proposed in the paper [5]. The relevance of our construction to this circle of problems gained additional evidence from the paper [1]. Recent discussions around integrability in Dijkgraaf– Vafa and Seiberg–Witten theories provide us with “physical” insights supporting the relation between the Beauville–Mukai and double elliptic IS (see [22], [6]). The future paper (in collaboration with A. Gorsky) which should clarify the place of the IS associated with the elliptic algebras and the integrability phenomena in SUSY gauge theories is in progress. There are some open questions about the relations of these IS with the Beauville– Mukai Lagrangian fibrations on the Hilbert scheme .P 2 nE/Œn as well as with their non-commutative (NC) counterparts proposed in [35]. It is likely that the quantization scheme for fraction fields proposed in [12] can be applied to the NC surface .PS nE/ to “extend” the deformation to the NC Hilbert scheme .PS nE/Œn introduced in [35]. Proposition 3.1 from [11] then should in principle give a NC integrable Beauville– Mukai system on .PS nE/Œn as a NC IS associated with the Cherednik algebras of [13]. Finally, we should mention an interesting question of generalization of the Beauville–Mukai IS which was introduced by Gelfand–Zakharevich on the Hilbert scheme .X9 /Œn of the Del Pezzo surface X9 obtained by the blow-up of 9 points on P 2 : These 9 points are the intersection points of two cubic plane curves. Let  be a Poisson tensor on X9 , then it can be extended to the whole P 2 so that the extension Q has 9 zeroes at these 9 points. The polynomial degree of the tensor Q is 3 and it follows that this polynomial is a linear combination of the polynomials corresponding to the initial elliptic curves. This bi-hamiltonian structure corresponds to the case n D 3 studied in the paper [37]. The open question arises: what is the algebro-geometric construction behind the bi-hamiltonian elliptic Poisson algebra of [37] in the case of arbitrary n?

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Acknowledgements. We would like to thank B. Feigin, A. Gorsky, B. Khesin, A. Levin, S. Oblezin and S. Pakuliak for helpful discussions on the subject of the paper. Our special thanks to B. Enriquez for numerous discussions and for his friendly help and support. A part of this work was reported as a talk of V. R. during the Strasbourg 72nd RCP “Quantum Groups” in June 2003. We are grateful to V. Turaev and to B. Enriquez for their invitation. We enjoyed the hospitality of IHES during the initial stage of this paper. The essential part of the work was done when V. R. was a visitor at the Section de Mathématiques de l’Université de Genève (Switzerland) and at the Erwin Schrödinger Institute for Mathematical Physics (Vienna, Austria). He is grateful to the Swiss National Science Foundation and to ESI for their invitations and support. The last revision of the paper was done during V. R.’s visit at RIMS (Kyoto). He enjoyed the hospitality of the program “Methods of Algebraic Analysis in Integrable Systems”. He is grateful to M. Kashiwara, T. Miwa and K. Takasaki for their invitation. The work of V. R. was partially supported by CNRS (six months leave in 2002– 2003) and partly by the grant RFBR 01-01-00549 and by the grant for scientific schools RFBR 00-15-96557. A. O. was partially supported by grant INTAS 00-00055, by the grant for Scientific Schools 2044-20032, and by the NATO Scientific Collaboration Program grant. He also acknowledges the hospitality of the Département de Mathématiques d’Angers (LAREMA, UMR 6093 du CNRS) and the Laboratoire de Physique Théorique, Université Paris-Sud (Orsay).

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[27] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114. [28] A. Källström, B. Sleeman, Multiparameter spectral theory, Ark. Mat. 15 (1977), 93–99. [29] B. Khesin, A. Levin, M. Olshanetskii, Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus, Commun. Math. Phys. 250 (2004), 581–612. [30] I. Krichever, A. Zabrodin, Spin generalizations of the Ruijsenaars-Schneider model, the non-abelian 2-dimensional Toda chain and representations of Sklyanin algebra, Russ. Math. Surv. 50 (1995), 1101–1150. [31] I. Krichever, P. Wiegmann, A. Zabrodin, Elliptic solutions to difference non-linear equations and related many-body problems, Commun. Math. Phys. 193 (1998), 373–396. [32] N. Markarian, On Poisson structures, privately communicated, unpublished manuscript, 2001. [33] D. Markushevich, Some algebro-geometric integrable systems versus classical ones, in The Kowalevski property (Leeds, 2000), CRM Proc. Lecture Notes 32, Amer. Math. Soc., Providence, RI, 2002, 197–218. [34] S. Mukai, Symplectic structure on the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101–116. [35] T. Nevins, T. Stafford, Sklyanin algebras and Hilbert schemes of points, Adv. Math. 210 (2007), 405–478. [36] A. Odesskii, Elliptic algebras, Russian Math. Surveys 57 (2002), 1127–1162. [37] A. Odesskii, Bihamiltonian elliptic structures, Moscow Math. J. 4 (2004), 941–946. [38] A. Odesskii, V. Rubtsov, Polynomial Poisson algebras with regular structure of symplectic leaves, Theoret. and Math. Phys. 133 (2002), 1321–1337. [39] A. Polishchuk, Poisson structures and birational morphisms associated with bundles on elliptic curves, Internat. Math. Res. Notices 13 (1998), 683–703. [40] A. Polishchuk, Triple Massey products on curves, Fay’s trisecant identity, and tangents to the canonical embedding, Moscow Math. J. 3 (2003), 105–121. [41] A. Schorr, Separation of variables for the eight-vertex SOS-model with antiperiodic boundary conditions, Diss. ETH Zürich, Zürich 2000. [42] E. Sklyanin, Some algebraic structures connected with the Yang-Baxter equations, Functional Anal. Appl. 16 (1982), 263–270. [43] E. Sklyanin, Some algebraic structures connected with the Yang-Baxter equations. Representations of a quantum algebra, Functional. Anal. Appl. 17 (1983), 273–284. [44] V. Sokolov, A. Tsyganov, Commutative Poisson subalgebras for the Sklyanin bracket and deformations of known integrable models, Theoret and Math. Phys. 133 (2002), 1730–1743. [45] P. Stäckel, Über die Integration der Hamilton-Jacobi’schen Differentialgleichung mittels Separation der Variabeln, Habilitationschrift, Halle 1891; Math. Ann. 49 (1897), 145–147 (in German). [46] V. Vakulenko, Note on the Ruijsenaars-Schneider model, arXiv:math/9909079v1 [math.QA].

On the automorphisms of UqC .g/ Nicolás Andruskiewitsch and François Dumas Facultad de Matemática, Astronomía y Física Universidad Nacional de Córdoba CIEM – CONICET .5000/ Ciudad Universitaria, Córdoba, Argentina e-mail: [email protected], URL: http://www.mate.uncor.edu/andrus Université Blaise Pascal Laboratoire de Mathématiques .UMR 6620 du CNRS/ 63177 Aubière, France e-mail: [email protected]

Abstract. Let g be a simple complex finite dimensional Lie algebra and let UqC .g/ be the positive part of the quantum enveloping algebra of g. If g is of type A2 , the group of algebra automorphisms of UqC .g/ is a semidirect product .k /2 Ì Autdiagr.g/; any algebra automorphism is an automorphism of braided Hopf algebra and preserves the standard grading [2]. This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the notion of “algebra with few automorphisms” and establish some consequences. We prove some exploratory results concerning the group of algebra automorphisms for the type B2 . We study the Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of UqC .g/.

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Braided Hopf algebra automorphisms . . . . . . . . . . . . . . . 1.1 Braided vector spaces . . . . . . . . . . . . . . . . . . . . 1.2 Automorphisms of Nichols algebras and their bosonizations 1.3 Hopf algebra automorphisms of Nichols algebras of Drinfeld–Jimbo type . . . . . . . . . . . . . . . . . . . . 2 The case where g is of type A2 . . . . . . . . . . . . . . . . . . 2.1 Graded algebras with few automorphisms . . . . . . . . . . 2.2 Algebra automorphisms of UqC .sl3 / . . . . . . . . . . . . 3 Partial results on the case where g is of type B2 . . . . . . . . . . 3.1 Some ring-theoretical properties of the algebra U C . . . . . 3.2 Automorphisms stabilizing the prime ideal.z/ . . . . . . . .

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3.3 Non permutability of the central generators z and z 0 . . . . . . . . . . . . . . 127 3.4 Action on the H -spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Introduction Let g be a finite dimensional complex simple Lie algebra and g D n ˚ h ˚ nC be a triangular decomposition of g related to a Cartan subalgebra h. The structure of the group Aut Alg Uq .g/ of algebra automorphisms of the quantum enveloping algebra Uq .g/ seems to be known only in the elementary case where g is of type A1 (see [1] or [2]). The automorphism group AutAlg UL q .bC / of the augmented form of the quantum enveloping algebra of the Borel subalgebra bC D h ˚ nC is described for any g in [11]. The groups of Hopf algebra automorphisms of Uq .g/ and UL q .bC / are determined in [10] and [11] respectively. We are concerned in this paper with the group of automorphisms of the quantum enveloping algebra UqC .g/ D Uq .nC / of the nilpotent part. Let .V; c/ be a braided vector space, see 1.1.1 below, and let B.V / be the corresponding Nichols algebra. We prove that the group AutHopf B.V / of braided Hopf algebra automorphisms of B.V / coincides with the group GL.V; c/ of automorphisms of braided vector space of .V; c/. Thus we have Aut Hopf B.V /  AutGrAlg B.V /  AutAlg B.V /; where Aut GrAlg means the group of algebra automorphisms homogeneous of degree 0, in this case with respect to the standard grading of B.V /, and Aut Alg is the group of all algebra automorphisms. The class of Nichols algebras includes symmetric algebras, free algebras, Grassmann algebras; thus there is no hope to have a round description of the group Aut Alg B.V /. In particular the computation of this group is a well-known classical open problem in the case of the symmetric algebras. Let k be the ground field and let .V; c/ be a braided vector space of diagonal type, n D dim V . Under some technical assumptions, the group GL.V; c/ reduces in this case to the semidirect product of the canonical action of the torus .k /n by the subgroup Autdiagr.c/ of the symmetric group Sn preserving the matrix of the braiding. A fundamental characterization due to Lusztig and Rosso (in different but equivalent formulations) says that UqC .g/ D B.h/, with a diagonal braiding c with matrix .q di aij / built up from q and the Cartan matrix of g, say n D dim h. Furthermore, Autdiagr c is in this case the group Autdiagr.g/ of automorphisms of the Dynkin diagram of g. Therefore, AutHopf UqC .g/ ' .k /n Ì Autdiagr.g/:

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To the best of our knowledge, the group Aut Alg UqC .g/ ha not yet been determined. However, if g of type A2 , then AutAlg UqC .g/ ' .k /2 ÌAutdiagr.g/ [2] (see also [8]). Thus, Aut Hopf B.V / D Aut GrAlg B.V / D Aut Alg B.V / in this case. This intriguing result motivates several questions: Problem 1. Is it true that AutAlg UqC .g/ ' .k /n Ì Autdiagr.g/? We conjecture that the answer is positive for any g. But even if the answer were negative, we would still ask: is it true that AutAlg UqC .g/ is an algebraic group? Problem 2. Determine the braided vector spaces .V; c/ such that AutHopf B.V / D AutGrAlg B.V / D AutAlg B.V /: Graded algebras A with the property Aut GrAlg A D AutAlg A do not seem to abound. Thus, we dare to pose: Problem 3. Classify graded algebras A with the property Aut GrAlg A D AutAlg A. In this paper we contribute mainly to Problem 1. Let us review the contents of the article. The first section is about Hopf algebra automorphisms of Nichols algebras and their bosonizations. We obtain from some general considerations the computation of the group Aut Hopf Uq .bC / ' .k /n Ì Autdiagr.g/, recovering a result from [11]. In Section 2, we briefly recall the case of type A2 . We introduce the notion of “algebras with few automorphisms”, classify gradings of these algebras and apply the results to UqC .sl3 /. Section 3 is an exploration of the case where g is of type B2 . Since there are no nontrivial diagram automorphism in this case, the question is then to determine whether or not Aut Alg UqC .g/ is isomorphic to .k /2 . A basic idea to approach the group Aut Alg UqC .g/ is to study its actions on natural sets. In [2], the study of the actions on the sets of central and normal elements is crucial. This method fails here because the center of UqC .g/ is a polynomial algebra kŒz; z 0  in two variables (which are homogeneous elements of degree 3 and 4 respectively for the canonical grading) and any normal element of UqC .g/ is automatically central. The next natural sets where our group acts are the various spectra; the investigation of these actions is the matter of this section. We begin with the ideals .z/ and .z 0 /; these are completely prime of height one and the factor domain UqC .g/=.z/ is isomorphic to the quantum Heisenberg algebra UqC .sl3 /. Using the results of Section 2, this allows to separate up to isomorphism the factor domains UqC .g/=.z/ and UqC .g/=.z 0 /. We then show, first, that automorphisms of UqC .g/ cannot exchange the ideals .z/ and .z 0 /, and, second, that the subgroup of Aut Alg UqC .g/ of automorphisms preserving the ideal .z/ reduces to the torus .k /2 . To progress further in this direction, we need a better knowledge of the prime ideals of height one. Although the stratification of the

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prime spectrum is known [12], the full classification of the prime ideals is still open. We discuss the stratification for type B2 in Section 3.4. Added in proof. After acceptance of this paper, S. Launois gave a positive answer to Problem 1 for type B2 using our Proposition 3.3. Thus our conjecture is true in this case. See [15]. Also, Problem 1 is solved for type A3 in [16]. Acknowledgements. The origin of this paper lies in conversations with Jacques Alev during visits of the first author to the University of Reims. His enthusiasm about the automorphism problem was decisive to convince us to consider this question. We also thank Gérard Cauchon and Stéphane Launois for many enlightening discussions about the third part of this paper. This joint work was partially realized during visits of the first author to the University of Clermont-Ferrand in March 2002 and June 2003, and a visit of the second author to the University of Cordoba in November 2003, in the framework of the Project ECOS conducted by J.-L. Loday and M. Ronco, of the Project CONICETCNRS “Métodos homólogicos en representaciones y álgebras de Hopf” and of the Project PICS-CNRS 1514 conducted by C. Cibils. Partial support from Agencia Córdoba Ciencia, ANPCyT-Foncyt, CONICET, ECOS, Fundación Antorchas, PICSCNRS 1514 and Secyt (UNC) is gratefully acknowledged.

1 Braided Hopf algebra automorphisms Throughout this paper N denotes the set of non-negative integers f0; 1; 2; 3; : : : g. In Sections 1.3 and 1.2 the field k is arbitrary; in 1.3, k has characteristic 0 and contains an element q not algebraic over Q.

1.1 Braided vector spaces 1.1.1 Braided vector spaces. A braided vector space is a pair .V; c/ where V is a vector space V over k and c W V ˝ V ! V ˝ V is a linear isomorphism that is a solution of the braid equation .c ˝ id/.id ˝c/.c ˝ id/ D .id ˝c/.c ˝ id/.id ˝c/. A braided vector space .V; c/ is rigid if V is finite dimensional and the map c [ W V  ˝ V ! V ˝ V  is invertible, where c [ D .evV ˝ idV ˝V  /.idV  ˝c ˝ idV  /.idV  ˝V ˝ coevV /: Here evV W V  ˝ V ! k is the usual evaluation map, and coevV W k ! V ˝ V  is the coevaluation (the transpose of the trace). 1.1.2 Automorphisms of a braided vector space of diagonal type. Let .V; c/ be a braided vector space. The braiding c W V ˝ V ! V ˝ V is said to be diagonal if there exists a basis x1 ; : : : ; xn of V and a matrix .qij /1i;j n with entries in k such

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that c.xi ˝ xj / D qij xj ˝ xi for any 1  i; j  n. In particular, V is rigid and has finite dimension n  1. Remark 1.1. If the braiding is diagonal, then the matrix .qij /1i;j n does not depend on the basis x1 ; : : : ; xn , up to permutation of the index set f1; : : : ; ng; see [5], Lemma 1.2. Let .V; c/ be a braided vector space. A linear automorphism g 2 GL.V / is said to be a braided vector space automorphism of .V; c/ if g ˝ g commutes with c. We denote by GL.V; c/ the corresponding subgroup of GL.V /. Suppose that c is of diagonal type for some basis x1 ; : : : ; xn of V and some matrix .qij /1i;j n . We consider the following subgroup of the symmetric group Sn : Autdiagr.c/ ´ f 2 Sn j qij D q .i /; .j / ; 1  i; j  ng: Any  in Autdiagr.c/ induces naturally an automorphism g 2 GL.V; c/ by g .xj / D x.j / for 1  j  n. Any g 2 GL.V; c/ of this type is called a diagram automorphism of .V; c/. Moreover it is clear that the torus .k /n acts on V by braided vector space automorphisms. The following lemma gives necessary conditions for the group GL.V; c/ to be generated by these two particular subgroups. Lemma 1.2. Let .V; c/ be a braided vector space of diagonal type, with respect to a basis x1 ; : : : ; xn of V and a matrix .qij /1i;j n with entries in k . Assume that at least one of the following conditions is satisfied: (i) For any i ¤ j , there exists h such that qih ¤ qj h . (ii) For any i ¤ j , there exists h such that qhi ¤ qhj .    q qij  is not of the form qq qq . (iii) For any i ¤ j , the matrix qji ii qjj Then we have

GL.V; c/ ' .k /n Ì Autdiagr.c/:

Proof. Let g 2 GL.V / and denote g.xi / D

P

s;i xs , 1  i  n. Then X qij r;j s;i xr ˝ xs ; .g ˝ g/.c.xi ˝ xj // D .g ˝ g/.qij .xj ˝ xi // D c.g ˝ g/.xi ˝ xj / D c

 X

s

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1i;j n

Therefore g 2 GL.V; c/ is and only if qij r;j s;i D qsr r;j s;i

for all 1  i; j; r; s  n:

(1.1)

Suppose that g 2 GL.V; c/. Since g is invertible, there exists  2 Sn such that .h/;h ¤ 0 for any 1  h  n. Then we deduce from (1.1) that qij D q .i /; .j / for all 1  i; j  n.

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Assume first that (i) is satisfied and choose i; s 2 f1; : : : ; ng such that s;i ¤ 0. Apply (1.1) with any j and r D  .j /. We obtain qs;.j / D qij D q .i /; .j / for any 1  j  n; then s D .i/. This implies that g.xi / D .i/;i x.i/ for any 1  i  n, which proves the result. Assume now that (ii) is satisfied and choose j; r 2 f1; : : : ; ng such that r;j ¤ 0. Apply (1.1) with any i and s D  .i /. We obtain q.i/;r D qij D q .i /; .j / for any 1  i  n; then r D .j /, and we conclude as in the previous case. Finally assume that (iii) holds and take i D j in (1.1). If s;i ¤ 0 then qi i D q.i/;s D qs;.i/ D q.i/;.i/ ; therefore s D  .i /, and we conclude as above. 1.1.3 Braided Hopf algebras. A non-categorical version of the concept of braided Hopf algebra is studied in [21]. A braided Hopf algebra is a collection .R; m; ; c/ such that • .R; c/ is a braided vector space; • .R; m/ is an associative algebra with unit 1; • .R; / is a coassociative coalgebra with counit "; • m, , 1, " commute with c in the sense of [21]; •  B m D .m ˝ m/.id ˝c ˝ id/. ˝ /; • the identity has an inverse for the convolution product in End R (this inverse is called the antipode and is denoted by S). Here recall that the convolution product of f; g 2 End R is given by f  g D m.f ˝ g/. A homomorphism of braided Hopf algebras is a linear map preserving m; ; c. Lemma 1.3. Let R be a braided Hopf algebra and let T W R ! R be a linear isomorphism that is an algebra and coalgebra map. Then T is a morphism of braided Hopf algebras. Proof. Let us define T  f ´ Tf T 1 , f 2 End R. Then T  .f  g/ D T .m.f ˝ g//T 1 D m.T ˝ T /.f ˝ g/.T 1 ˝ T 1 / D .T  f /  .T  g/I hence T  S D S or T S D ST . But it is shown in [20] that the braiding of a braided Hopf algebra can be expressed in terms of the product, coproduct and antipode. Thus T preserves also the braiding c. The group of braided Hopf algebra automorphisms of R is denoted by Aut Hopf R.

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1.1.4 Yetter–Drinfeld modules. Yetter–Drinfeld modules give rise to braided vector spaces and play a fundamental rôle in problems related to the classification of Hopf algebras. Let us recall that a Yetter–Drinfeld module V over a Hopf algebra H with bijective antipode S is both a left H -module and left H -comodule such that the action P and the coaction ı W V ! H ˝ V satisfy the compatibility condition: H ˝ V !V H YD ı.h  v/ D h.1/ v.1/ Sh.3/ ˝ h.2/  v.0/ for all h 2 H; v 2 V . We denote by H the category of Yetter–Drinfeld modules over H , where morphisms respect both the action and the coaction of H . H YD. It is The usual tensor product defines a structure of monoidal category on H braided, with braiding cV;W W V ˝W ! W ˝V defined by c.v˝w/ D v.1/ w˝v.0/ H for V; W 2 H YD, v 2 V , w 2 W . Then .V; cV;V / is a braided vector space for any H YD. It is known that any rigid braided vector space can be realized as a V 2 H Yetter–Drinfeld module over a (non-unique) Hopf algebra, essentially by the FRTconstruction; see [21] for references and details. H As in any braided monoidal category, there is the notion of Hopf algebras in H YD. H Hopf algebras in H YD are braided Hopf algebras by forgetting the action and the coaction. Conversely, let R be any braided Hopf algebra whose underlying braided vector space is rigid. Then there exists a (non-unique) Hopf algebra H such that R H YD [21]. can be realized as a Hopf algebra in H H YD. We recall the bosonization pro1.1.5 Bosonizations of a Hopf algebra in H cedure, or Radford biproduct, found by Radford and explained in terms of braided categories by Majid. H Let H be a Hopf algebra with bijective antipode. Let R be a Hopf algebra in H YD. The bosonization of R by H is the (usual) Hopf algebra A D R # H with underlying vector space R ˝ H , whose multiplication and comultiplication are given by:

.r #h/.s#f / D r.h.1/ s/#h.2/ f and .r #h/ D r .1/ #.r .2/ /.1/ h.1/ ˝.r .2/ /.0/ #h.2/ : The maps  W A ! H , r # h 7! .r/h, and  W H ! A, h 7! 1 # h, are Hopf algebra homomorphisms and R D fa 2 A j .id ˝/.a/ D a ˝ 1g. Conversely, let A, H be Hopf algebras with bijective antipode and let  W A ! H and  W H ! A be Hopf algebra homomorphisms such that   D idH . Then R D H fa 2 A j .id ˝/.a/ D a ˝ 1g is a Hopf algebra in H YD and the multiplication induces an isomorphism of Hopf algebras R # H ' A. 1.1.6 Nichols algebras. We recall the definition of Nichols algebra; see [4] for details and references. L H H YD. A graded Hopf algebra R D Let V 2 H n0 R.n/ in H YD is called a H YD, and if Nichols algebra of V if k ' R.0/ and V ' R.1/ in H R.1/ D P .R/; the space of primitive elements of R; R is generated as an algebra by R.1/:

(1.2)

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The Nichols L algebra of V exists and is unique up to isomorphism; it is denoted by B.V / D n0 Bn .V /. The associated braided Hopf algebra (forgetting the action and the coaction) depends only on the braided vector space .V; c/. We shall identify V with the subspace of homogeneous elements of degree 1 in B.V /. Let us recall the following explicit construction of B.V /. For any integer m  2, we denote by Bm the m-braid group. A presentation of Bm is given by generators 1 ; : : : ; m1 and relations i j D j i if ji  j j  2 and i iC1 i D iC1 i iC1 for any 1  i  m  2. There is a natural projection  W Bm ! Sm sending i to the transposition i ´ .i; i C 1/ for all i . This projection  admits a set-theoretical section s W Sm ! Bm determined by s. i / D i ; 1  i  n  1; s. !/ D s. /s.!/ if `. !/ D `. / C `.!/: Here ` denotes the length of an element of Sm with respect to the set of generators 1 ; : : : ; m1 . The map s is called the Matsumoto section. In other words, if ! D i1 : : : ij is a reduced expression of ! 2 Sm , then s.!/ D i1 : : : ij . Using the section s, the following distinguished elements of the group algebra kBm are defined: X s. /: Sm ´ 2Sm

By convention, we still denote by Sm the images of these elements in End.T m .V // D End.V ˝m / via the representation m W Bm ! Aut.V ˝m / defined by m .i / D id ˝    ˝ id ˝ c ˝ id ˝    ˝ id, with c acting on the tensor product of the copies of V indexed by i and i C 1. Let cQ be the canonical extension of c into a braiding of T .V /. Let T .V / ˝ T .V / be the algebra whose underlying vector space is T .V / ˝ T .V / with the product “twisted” by c. Q There is a unique algebra map  W T .V / ! T .V / ˝ T .V / such that .v/ D v ˝ 1 C 1 ˝ v,L v 2 V . Then T .V / is a braided Hopf algebra and B.V / D T .V /=J where J D m0 Ker Sm (see for instance [4], [18] or [19]).

1.2 Automorphisms of Nichols algebras and their bosonizations 1.2.1 Hopf algebra automorphisms of a Nichols algebra. We can now compute the group of Hopf algebra automorphisms of a Nichols algebra, cf. the notation introduced in 1.1.2. Theorem 1.4. There is a group isomorphism B W GL.V; c/ ! Aut Hopf B.V /. Proof. For any g 2 GL.V /, we denote by gQ the canonical extension of g into P an algebra automorphism of T .V /. If g commutes with c, then gQ commutes with m2 Sm , and so induces an algebra automorphism B.g/ of B.V / D T .V /=J . In order to prove that B.g/ is also a coalgebra automorphism of B.V /, we claim that gQ is a coalgebra map: ..gQ ˝ g/ Q B /.v/ D . B g/.v/ Q for any v 2 T .V /.

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It is clear that this assertion is true when v 2 V . So it is enough to prove that gQ ˝ gQ is a morphism of the algebra T .V /˝T .V /. For that, let us consider u; v; x; y 2 T .V /. Let us set c.y Q ˝ u/ D u0 ˝ y 0 in a symbolic way. By definition of the twisted product in T .V /, we have .x ˝ y/.u ˝ v/ D xu0 ˝ y 0 v. Then Q 0 /g.v/: Q .gQ ˝ g/..x Q ˝ y/.u ˝ v// D g.x/ Q g.u Q 0 / ˝ g.y Moreover, it is easy to check that the assumption g 2 GL.V; c/ implies that Q g.y/ Q ˝ g.u//, Q and so gQ 2 GL.T .V /; c/. Q Thus .gQ ˝ g/.u Q 0 ˝ y 0 / D c. .gQ ˝ g/.x Q ˝ y/.gQ ˝ g/.u Q ˝ v/ D .g.x/ Q ˝ g.y//. Q g.u/ Q ˝ g.v// Q D g.x/ Q g.u Q 0 / ˝ g.y Q 0 /g.v/: Q Hence .gQ ˝ g/..x Q ˝ y/.u ˝ v// D .gQ ˝ g/.x Q ˝ y/.gQ ˝ g/.u Q ˝ v/, as claimed. By Lemma 1.3, B.g/ preserves c. Thus, we have a well-defined map B W GL.V; c/ ! Aut Hopf B.V /, which is injective by (1.2). Conversely, let u W B.V / ! B.V / be an automorphism of braided Hopf algebras. Then u.V / D V since V D P .B.V //, and the theorem follows. 1.2.2 Automorphisms of bosonizations. Our goal is to compute the Hopf algebra automorphisms of A D R # H when R is a Nichols algebra, under suitable hypothesis. The exposition is inspired by [3], Section 6. We begin by a description of a natural class of such automorphisms for general R. H YD. Let Lemma 1.5. Let H be a Hopf algebra and let R be a Hopf algebra in H G W R ! R and T W H ! H be linear maps. Then G #T ´ G ˝T W R#H ! R#H is a Hopf algebra map if and only if the following conditions hold:

(i) T is a Hopf algebra automorphism of H , (ii) G is a Hopf algebra automorphism of R, (iii) G.h  s/ D T .h/  G.s/, s 2 R, h 2 H , (iv) ı B G D .T ˝ G/ B ı. Proof. Left to the reader. A pair .G; T / as in the lemma shall be called compatible. Lemma 1.6. Let H be a Hopf algebra and V a Yetter–Drinfeld module over H . (i) Assume that H is cosemisimple and the following hypothesis holds: (H) The types of the isotypic components of V # H under the adjoint action of H do not appear in the adjoint action of H on itself.

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Then any Hopf algebra automorphism of B.V / # H is of the form G # T , with .G; T / compatible. (ii) If in addition H is commutative, then (H) is equivalent to: (H0 ) the trivial representation does not appear as a subrepresentation of V . Proof. (i) Let ˆ be a Hopf algebra automorphism of A D B.V / # H . It is known L n that the coradical filtration of A is Am D 0nm B .V / # H [4], 1.7. Since ˆ is a coalgebra map, it preserves the coradical filtration. In particular, ˆ.H / D H and ˆ.H ˚ V # H / D H ˚ V # H . Let T W H ! H be the restriction of ˆ; this is an automorphism of Hopf algebras. Also, ˆ W H ˚ V # H ! H ˚ V # H preserves the adjoint action of H . By hypothesis (H), ˆ.V # H / D V # H . Since ˆ is an algebra map, this implies that ˆ.BnL .V / # H / D Bn .V / # H , by (1.2). Let  W A ! H be the projection with kernel n1 Bn .V / # H ; clearly, ˆ D ˆ. Hence ˆ.B.V // D B.V /, since B.V / D fv 2 A j .id ˝/.v/ D v ˝ 1g. Let G W B.V / ! B.V / be the restriction of ˆ. Since ˆ is an algebra map, ˆ D G # T . By Lemma 1.5, the pair .G; T / is compatible. (ii) If H is commutative, the adjoint action of H on itself is trivial, and the isotypic components of the adjoint action of H on V # H are of the form U # H , where U runs over the set of isotypic components of the adjoint action of H on V . This shows that (H) is equivalent to (H0 ) in this case. Hypothesis (H) is needed, as the following example shows. Let A D kŒx; g; g 1  be the tensor product of the polynomial algebra in x and the Laurent polynomial algebra in g. This is a Hopf algebra with x primitive and g group-like. The Hopf algebra automorphism T W A ! A, T .g/ D g, T .x/ D x C 1  g, does not preserve the Nichols algebra kŒx. We now consider the following particular setting. We assume that H D k is the group algebra of an abelian group. We also assume the existence of a basis x1 ; : : : ; xn y the action and of V such that, for some elements g1 ; : : : ; gn 2 , 1 ; : : : ; n 2 , coaction of  are given by h  xj D j .h/xj ;

ı.xj / D g.j / ˝ xj ;

1  j  n:

Theorem 1.7. Suppose further that 1  i  n; i ¤ "; .gi ; i / ¤ .gj ; j /; 1  i ¤ j  n:

(1.3) (1.4)

Then there is a bijective correspondence between AutHopf B.V / # H and the set of pairs . ; /, where is a group automorphism of  and W V ! V is a linear isomorphism given by .xi / D i x.i/ , 1  i  n, with i 2 k and  2 Sn , such that .gi / D g.i/ ; i D .i/ B ; 1  i  n: (1.5)

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Proof. Condition (1.3) guarantees that hypothesis (H0 ) holds. Thus any Hopf algebra automorphism of B.V / # H is of the form G # T , with .G; T / compatible, by Lemma 1.6. But T is determined by a group automorphism of , and G is of the form B. / for some 2 GL.V; c/ by Theorem 1.4. Now conditions (ii) and (iii) of Lemma 1.5 imply that 1

  .xi / D . i B

/./ .xi /;

and thus

.xi / 2

ı. .xi // D X

k xj ;

.gi / ˝ .xi /;

 2 ; 1  i  n;

1  i  n:

j W .gi /Dgj i B 1 Dj

But condition (1.4) implies that there is only one j such that .gi / D gj , i B 1 D j ; set .i/ D j . This defines , and clearly (1.5) holds. Conversely, any pair . ; / as above gives raise to a compatible pair .G; T / with G D B. / and T determined by .

1.3 Hopf algebra automorphisms of Nichols algebras of Drinfeld–Jimbo type 1.3.1 Definition and notations (cf. [6], [14], [17]). We fix q 2 k , q not algebraic over Q. Let g be a simple finite dimensional Lie algebra of rank n over k. Let g D n ˚ h ˚ nC be a triangular decomposition of g related to a Cartan subalgebra h of g. Let C D .ai;j /1i;j n the associated Cartan matrix and .d1 ; : : : ; dn / the relatively primes integers symmetrizing C . The quantum enveloping algebra of the nilpotent positive part nC of g, denoted by Uq .nC / or UqC .g/, is the algebra generated over k by n generators E1 ; : : : ; En satisfying the quantum Serre relations: 1ai;j

X   1a

D0



i;j

1ai;j 

q di

Ei

Ej .Ei / D 0

for all 1  i 6D j  n:

The quantum enveloping algebra of the positive Borel algebra bC D h ˚ nC , denoted by Uq .bC /, is the algebra generated over k by E1 ; : : : ; En ; K1˙1 ; : : : ; Kn˙1 satisfying the quantum Serre relations, the commutation between the Ki ’s and the q-commutation relations: Ki Ej D q di ai;j Ej Ki

for all 1  i; j  n:

It is well known that Uq .bC / is a Hopf algebra for the coproduct, counit and antipode defined by ".Ki / D 1; .Ki / D Ki ˝ Ki ; .Ei / D Ei ˝ 1 C Ki ˝ Ei ; ".Ei / D 0;

S.Ki / D Ki1 ; S.Ei / D Ki Ei :

We denote by H the Hopf subalgebra H D kŒK1˙1 ; : : : ; Kn˙1  ' kZn of Uq .bC /.

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1.3.2 Braided Hopf algebra structure on UqC .g/. From Corollary 33.1.5 of [17] or Theorem 15 of [19], we have UqC .g/ D B.V /, for V D kE1 ˚    ˚ kEn and c the diagonal braiding of V defined from the Cartan matrix C D .ai;j /1i;j n and the integers .d1 ; : : : ; dn / by c.Ei ˝ Ej / D q di ai;j Ej ˝ Ei : Let  W H ! Uq .bC / be the inclusion and let  W Uq .bC / ! H be the unique Hopf algebra map such that .Ki / D Ki , .Ei / D 0. Then   D idH and UqC .g/ D fa 2 Uq .bC / j .id ˝/.a/ D a ˝ 1g. Hence Uq .bC / ' UqC .g/ # H; cf. Section 1.1.5. Here the coaction is determined by ı.Ei / D gi ˝ Ei , with gi D Ki , 1  i  n. Also the action is determined by   Ei D i . /Ei for  2  D Zn , y is defined by i .Kj / D q di ai;j , 1  i; j  n. where i 2  1.3.3 Automorphisms of UqC .g/ and UqC .b/. With the notations of 1.1.2 for qi;j D q di ai;j , the subgroup Autdiagr c of Sn is the group Autdiagr.g/ of automorphisms of the Dynkin diagram of g (see for instance [10]), which acts by automorphisms on UqC .g/ and Uq .bC / by:  2 Autdiagr.g/ W Ei 7! E.i/ ; Ki 7! K.i/ by

for any 1  i  n:

The n-dimensional torus on k also acts by automorphisms on UqC .g/ and Uq .bC / .˛1 ; : : : ; ˛n / 2 .k /n W Ei 7! ˛i Ei ; Ki 7! Ki

for any 1  i  n:

Now we can prove the following theorem. Theorem 1.8. AutHopf UqC .g/ ' .k /n Ì Autdiagr.g/ ' AutHopf Uq .bC /. Proof. The first isomorphism just follows from Lemma 1.2 and Theorem 1.4. Let  D Zn , gi and i as above. By Theorem 1.7, any T 2 AutHopf Uq .bC / is determined by a pair . ; /, where is a group automorphism of  and W V ! V is a linear isomorphism given by .xi / D i x.i/ , 1  i  n, with i 2 k and  2 Sn , such that (1.5) holds. Then q di ai;j D i .Kj / D i

1

.Kj / D .i/ .K.j / / D q d.i / a.i/;.j / ;

hence  2 Autdiagr.g/. Furthermore, the second isomorphism.

is uniquely determined by . This implies

Remark 1.9. The second isomorphism in the theorem was proved previously in [11] as a corollary of the description of the group of all algebra automorphisms AutAlg Uq .bC / (in fact for the slightly different augmented algebra UL q .bC /). In fact, there exist automorphisms of the algebra UL q .bC / which are not Hopf algebra automorphisms: in

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particular some combinatorial infinite subgroups of n  n matrices with coefficients in Z, as well as the natural action of the 2n-dimensional torus on the Ei ’s and Kj ’s.

2 The case where g is of type A2 In this section the field k has characteristic 0; in Section 2.2, q 2 k is not a root of 1.

2.1 Graded algebras with few automorphisms Here we consider graded algebras with few automorphisms. To begin with, we recall the well-known equivalence between gradings and rational actions; see [6], p. 150 ff., and the references therein. Let H be an algebraic group. A representation W H ! Autk V of H on a vector space V is rational if V is union of finite dimensional rational H -modules. If H D .k /r is a torus, then there is a bijective correspondence between • rational actions of H on V , and L • gradings V D m2Zr Vm . In this correspondence, Vm is the isotypic component of type m, where Zr is identified with the group of rational characters of H . Thus, H -submodules of V are rational, and they are exactly the graded subspaces of V . Let A be an associative algebra over k. A rational action of H on A is one induced by a rational representation W H ! AutAlg A by algebra automorphisms. If H D .k /r is a torus, then there is a bijective correspondence between • rational actions of H on A, and L • algebra gradings A D m2Zr Am . Definition 2.1. An associative algebra A has few automorphisms if the following conditions hold. (i) There exists a finite dimensional Aut Alg A-invariant subspace V such that the restriction Aut Alg A ! GL.V / is injective; we identify AutAlg A with its image in GL.V /. (ii) Aut Alg A is an algebraic subgroup of GL.V /. (iii) The action of Aut Alg A on A is rational. (iv) The connected component .Aut Alg A/0 of the identity of Aut Alg A is isomorphic to a torus .k /r . Let A be an algebra with few isomorphisms. Then A has a canonical grading induced by the rational action of .Aut Alg A/0 ' .k /r : M A.m/ : AD m2Zr

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P Let j j W Zr ! Z be the function jmj D 1j r mj if m D .m1 ; : : : ; mr / 2 Zr . The Z-grading induced L by the canonical grading via j j is called the standard grading and is denoted by A D M 2Z AŒM  . Thus M AŒM  D A.m/ ; M 2 Z: m2Zr WjmjDM

Lemma 2.2. Let A be an algebra with few automorphisms. (i) Any algebra automorphism preserves the canonical grading, and those in .Aut Alg A/0 are homogeneous of degree 0. L (ii) Assume that the canonical grading is non-negative: A D m2Nr A.m/ . Then any algebra automorphism is homogeneous of degree 0 with respect to the standard grading. Proof. (i) Let  2 Aut Alg A, let inn W AutAlg A ! Aut Alg A be the inner automorphism defined by  and let inn W Zr ! Zr be the induced group homomorphism. r Then .A.m/ / D A.inn c .m// for any m 2 Z , and this implies the claim.

b

b



(ii) Under this hypothesis, P the matrix of inn in the canonical basis has non-negative entries. Thus .AŒM  / s0 AŒM Cs ; but some power of  belongs to .Aut Alg A/0 , hence only s D 0 survives. Starting from the canonical grading, new gradings of A can be constructed by means of morphisms of groups Zr ! Zt ; we show now that no other grading arises in this case. Theorem 2.3. Let A be an algebra with few automorphisms and let M An AD

(2.1)

n2Zt

be any algebra grading of A. Then there is a morphism of groups ' W Zr ! Zt such that M A.m/ (2.2) An D m2Zr W'.m/Dn

for all n 2 Zt . Proof. Let T be the torus Zt and let be the representation of T induced by the grading (2.1). Let V be the vector subspace as in Definition 2.1; since V is stable under Aut Alg A, it is also clearly stable under T . Consider the commutative diagram: 

/ Aut Alg A T EE EE ss EE sss s E s res jV EE " ysss GL.V / .

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The map jV is a homomorphism of algebraic groups; then is homomorphism of algebraic groups, say by [13], Ex. 3.10, p. 21. Since T is connected, .T / .Aut Alg A/0 . Thus, the transpose of induces a morphism of groups ' W Zr ! Zt . But then A.m/  A'.m/ . Since  X X M X A.m/ D A.m/ An D A; AD m2Zr

n2Zt m2Zr W'.m/Dn

n2Zt

we get the equality (2.2).

2.2 Algebra automorphisms of UqC .sl 3 /

 2 1  2.2.1 Notations. We suppose here that g D sl3 . Then we have n D 2, C D 1 2 , C d1 D d2 D 1, and Uq .g/ is the algebra generated over k by E1 and E2 satisfying the relations E12 E2  .q 2 C q 2 /E1 E2 E1 C E2 E12 D E22 E1  .q 2 C q 2 /E2 E1 E2 C E1 E22 D 0: The algebra UqC .sl3 / is usually named the quantum Heisenberg algebra. In the following we will denote it by H. Setting E3 D E1 E2  q 2 E2 E1 , it is easy to check (see for instance [2]) that H is the iterated Ore extension generated over k by the three generators E1 , E2 , E3 with relations E1 E3 D q 2 E3 E1 ;

E2 E3 D q 2 E3 E2 ;

E2 E1 D q 2 E1 E2  q 2 E3 :

The center of H is the a polynomial algebra in one indeterminate Z.H/ D kŒ where the quantum Casimir element  is given by  D .1  q 4 /E3 E1 E2 C q 4 E32 D E3 E3 with E3 D E1 E2  q 2 E2 E1 : L Let H D Hm;n be the canonical N2 -grading of H defined by putting E1 on degree .1; 0/ and E2 on degree .0; 1/. In particular E3 ; E3 2 H1;1 and  2 H2;2 . 2.2.2 Automorphisms and gradings of UqC .sl 3 /. For all ˛; ˇ 2 k , there exists one automorphism z˛;ˇ of H such that z˛;ˇ .E1 / D ˛E1 and z˛;ˇ .E2 / D ˇE2 . z ´ f z˛;ˇ j ˛; ˇ 2 k g ' .k /2 and the diagram automorphism We introduce G ! of H defined by !.E1 / D E2 and !.E2 / D E1 . Studying the action of any algebra automorphism of H on the center and on the set of normal elements of H, z and the Proposition 2.3 of [2] gives that Aut Alg H is the semi-direct product of G subgroup of order 2 generated by ! (see Proposition 4.4 of [8] for another proof). So we have for the type A2 the following positive answer to Problem 1. Theorem 2.4. For g of type A2 , the algebra H D UqC .g/ satisfies AutAlg H ' .k /2 Ì S2 . Here is a consequence of this result which will be useful in the next section.

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L 2 Corollary L 2.5. Let H D 2 Hm;n be the canonical N -grading of H. Let H D Tm;n be another N -algebra grading of H. Then there exists a matrix .pr qs / 2 M.2; Z/ with non-negative entries such that Hm;n  TpmCq n;rmCsn for all .m; n/ 2 N2 : Proof. This follows from Theorem 2.3, since H has few automorphisms by Theorem 2.4.

3 Partial results on the case where g is of type B2 A natural step in the study of Problem 1 would be to consider the other Lie algebras g of rank 2. We summarize in this section some partial results concerning the case B2 . In this section, k is an algebraically closed field of characteristic 0 and q 2 k a quantization parameter not a root of 1.

3.1 Some ring-theoretical properties of the algebra U C 3.1.1 Notations. Let g be the complex simple Lie algebra over k of type B2 . We denote U C D UqC .g/. We recall all notations of 1.3.1, but denote now by ei the  2 1 generators Ei ; we have here n D 2, C D 2 2 , and .d1 ; d2 / D .2; 1/. Then the quantum Serre relations take the form: 2 X 2 D0

 q2

3 X 3 D0

 q

ei2 ej .ei / D 0

for i D 1; j D 2; ai;j D 1; di D 2;

ei3 ej .ei / D 0

for i D 2; j D 1; ai;j D 2; di D 1:

We compute the quantum binomial coefficients:  2 2 3 3 0 q 2 D 2 q 2 D 1; 0 q D 3 q D 1;  2 3 3 2 2 2 2 ; : 1 q2 D q C q 1 q D 2 q Dq C1Cq We conclude that U C is the algebra generated over k by two generators e1 and e2 with commutation relations: (S1) e12 e2  .q 2 C q 2 /e1 e2 e1 C e2 e12 D 0, (S2) e23 e1  .q 2 C 1 C q 2 /e22 e1 e2 C .q 2 C 1 C q 2 /e2 e1 e22  e1 e23 D 0.

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3.1.2 U C as an iterated Ore extension. From the natural generators e1 and e2 of U C , we introduce following [22] the q-brackets: e3 D e1 e2  q 2 e2 e1

and

z D e2 e3  q 2 e3 e2 :

Relations (S1) and (S2) imply: e1 e3 D q 2 e3 e1 , e1 z D ze1 and e2 z D ze2 . In particular z is central in U C . From [22], the monomials .z i e3j e1k e2l /.i;j;k;l/2N4 form a PBW basis of U C . So U C is the algebra generated over k by e1 ; e2 ; e3 ; z with relations: e3 z D ze3 ; e1 z D ze1 ;

e1 e3 D q 2 e3 e1 ;

e2 z D ze2 ;

e2 e3 D q 2 e3 e2 C z;

e2 e1 D q 2 e1 e2  q 2 e3 :

In other words U C is the iterated Ore extension (cf. [6]): U C D kŒe3 ; zŒe1 I  Œe2 I ; ı D S Œe2 I ; ı with S D kŒe3 ; zŒe1 I . Here  is the automorphism of kŒe3 ; z defined by  .z/ D z, .e3 / D q 2 e3 , is the automorphism of kŒe3 ; zŒe1 I   defined by .z/ D z, .e3 / D q 2 e3 , .e1 / D q 2 e1 , ı is the -derivation of kŒe3 ; zŒe1 I   defined by ı.z/ D 0, ı.e3 / D z, ı.e1 / D q 2 e3 , and S is the subalgebra of U C generated by e3 , z and e1 . L 3.1.3 Grading of U C . We consider the canonical grading U C D n0 Un putting the natural generators e1 and e2 in degree 1 (and then e3 and z are of degree 2 and 3 respectively) defined from the basis .z i e3j e1k e2l /.i;j;k;l/2N4 of U C by: Un D L L i j k l 3iC2j CkClDn kz e3 e1 e2 for any n  0. We denote by I D n1 Un the ideal generated by e1 , e2 , e3 , z. 3.1.4 A localization of U C . The subalgebra of U C generated over k by e1 and e3 is a quantum plane (with e3 e1 D q 2 e1 e3 ); we will denote it by kq 2 Œe3 ; e1 . Its localization at the powers of e3 and e1 is the quantum torus kq 2 Œe3˙ ; e1˙ . The automorphism and the -derivation ı extend to kq 2 Œe3˙ ; e1˙ , and denoting by V the algebra kq 2 Œe3˙ ; e1˙ ŒzŒe2 I ; ı, we obtain the embedding: U C D kŒe3 ; zŒe1 I  Œe2 I ; ı D kq 2 Œe3 ; e1 ŒzŒe2 I ; ı  V D kq 2 Œe3˙ ; e1˙ ŒzŒe2 I ; ı: Let us introduce in U C the bracket: w D e2 e3  e3 e2 D z C .q 2  1/e3 e2 2 U3 . It follows from commutation relations in 3.1.2 that: e1 w D we1 C .1  q 2 /e32 , e2 w D q 2 we2 , and e3 w D q 2 we3 . Then the element z 0 D e1 w  q 4 we1 2 U4

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satisfies z 0 e1 D e1 z 0 and z 0 e2 D e2 z 0 , and so is central in U C . A straightforward computation shows that its development in the PBW basis of 3.1.2 is z 0 D .1  q 4 /.1  q 2 /e3 e1 e2 C q 4 .1  q 2 /e32 C .1  q 4 /ze1 : In particular, z 0 D s1 e2 C s0 , with s0 D q 4 .1  q 2 /e32 C .1  q 4 /ze1 2 kq 2 Œe3 ; e1 Œz, and s1 D .1  q 4 /.q 2  1/e1 e3 non-zero in kq 2 Œe3 ; e1 Œz. So we have in V the identity e2 D s11 z 0  s11 s0 , with s11 and s11 s0 in kq 2 Œe3˙ ; e1˙ Œz. Explicitly: e2 D

1 .1 

q 4 /.q 2

We conclude that

 1/

e31 e11 z 0 C

q4

1 1 e31 z  2 e 1 e3 : 1 q 1 1

(3.1)

U C  V D kq 2 Œe3˙ ; e1˙ Œz; z 0 :

Observe that z and z 0 being central in V , the only relation between the generators of V which is not a commutation is the q 2 -commutation e3 e1 D q 2 e1 e3 . 3.1.5 Conjugation in U C . We have introduced in 3.1.2 the homogeneous element e3 of degree 2 defined from natural generators e1 and e2 by e3 D e1 e2  q 2 e2 e1 , which satisfy e1 e3 D q 2 e3 e1 and e2 e3  q 2 e3 e2 D z. Conjugating q in q 1 , we can also consider e3 D e1 e2  q 2 e2 e1 D .1  q 4 /e1 e2 C q 4 e3 and prove that e1 e3 D q 2 e3 e1 and e2 e3 q 2 e3 e2 D q 4 z. We obtain in particular the relations e3 e3 D .1q 4 /q 2 e1 e3 e2 Cq 4 e32 and z 0 D .1q 2 /.e3 e3 C.1Cq 2 /ze1 /, which will be used in the sequel. 3.1.6 Center and normalizing elements of U C . Denote by Z.U C / the center and N.U C / the set of normalizing elements in U C . Lemma 3.1. N.U C / D Z.U C / D kŒz; z 0 . Proof. The calculation of Z.U C / can be deduced from general results of [7]. The equality N.U C / D Z.U C / for the type B2 was observed in [8], Remark 2.2 (iii). We give here a short direct proof using the embedding U C  V . Take f 2 N.U C / nonzero. From Proposition 2.1 of [8], f is q-central in U C , that is, there exist m; n 2 Z such that f e1 D q m e1 f and f e2 D q n e2 f , and so f e3 D q mCn e3 f . In V , the element f is a finite sum: X fi;j .z; z 0 /e1i e3j with fi;j .z; z 0 / 2 kŒz; z 0 : f D i;j 2Z

Since the polynomials fi;j .z; z 0 / are central in V , the identities f e1 D q m e1 f and f e3 D q mCn e3 f give by identification 2j D m and 2i D mn for all i; j such that fi;j .z; z 0 / 6D 0. Then f D fi;j .z; z 0 /e1i e3j , where i D  mCn and j D m . It follows 2 2

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from relation (3.1) and from the q-commutation f e2 D q n e2 f that 2j C 2i D 2i D 2j  2i D n. So i D j D n D 0, and f D f0;0 .z; z 0 / 2 kŒz; z 0 . We have proved that N.U C / kŒz; z 0 . The inverse inclusions kŒz; z 0  Z.U C / N.U C / are clear. Remark 3.2. For g of type B2 , Autdiagr.g/ is trivial and so Problem 1 must be here reformulated as follows: do we have Aut Alg U C ' .k /2 ? The method used in [2] for the case A2 was based on the fact that any automorphism of H preserves the center Z.H/, which is a polynomial algebra in one variable, and the non-empty set of normal but non central elements of H. It follows from Lemma 3.1 that the second argument fails for the case B2 , and that the first one is much more complicated to use in view of the structure of the automorphism group of a commutative polynomial algebra in two variables. A natural idea to determine the group AutAlg U C is to study the action of an automorphism on the prime spectrum of U C . The structure of this spectrum remains widely unknown as far as we know (see further final remark of 3.3) and we only present some partial results in the following. In particular the two central generators z and z 0 do not play symmetric rôles (see Proposition 3.4), and we conjecture that any automorphism of U C stabilizes the prime ideal .z/, which would be sufficient to solve the problem in view of Proposition 3.3.

3.2 Automorphisms stabilizing the prime ideal.z/ 3.2.1 The factor algebra U C =.z/. It is clear that the ideal .z/ generated in U C by the central element z is completely prime, and the factor domain U C =.z/ is the quantum enveloping algebra H D UqC .sl3 / considered in 2.2.1. The canonical map  W U C ! U C =.z/ induces an isomorphism between U C =.z/ and H defined by .e1 / D E1 and .e2 / D E2 , where E1 and E2 are the natural generators of H introduced in 2.2.1. Observe that .e3 / D E3 and .z 0 / D .1  q 2 /. 3.2.2 The group Autz .U C /. We introduce the subgroup Autz .U C / of all automorphisms of the algebra U C which stabilize the ideal .z/. In particular Autz .U C / contains the subgroup G ´ f ˛;ˇ j ˛; ˇ 2 k g ' .k /2 , where ˛;ˇ is the automorphism defined for any ˛; ˇ 2 k by ˛;ˇ .e1 /

D ˛e1

and

˛;ˇ .e2 /

D ˇe2 :

It is clear that any  2 Autz U C induces an automorphism Q 2 Aut H defined by Q 1 / D ..e1 // and .E Q 2 / D ..e2 //. We denote by ˆ the group morphism .E C z for the ˆ W Aut U ! Aut H,  7! Q . The notation is coherent since ˆ.G/ D G  2 z ' .k / introduced in 2.2.1. The next proposition shows that ˆ defines an group G z isomorphism between Aut z U C and G. Proposition 3.3. The subgroup Autz .U C / of algebra automorphisms of U C stabilizing the ideal .z/ is isomorphic to .k /2 .

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Proof. Step 1. We prove that, for any  2 Autz U C , there exist ;  2 k and p.z/ 2 kŒz such that .z/ D z and .z 0 / D z 0 C p.z/. Take  2 Autz U C . There exists u 2 U C , u 6D 0 such that .z/ D uz. But .z/ 2 Z.U C / because z 2 Z.U C /, and so u 2 Z.U C /. Similarly  1 .z/ D vz for some v 2 Z.U C /, v 6D 0. Thus z D . 1 .z// D .v/uz in Z.U C / D kŒz; z 0  (see 3.1.6), which implies that u 2 k . Denoting u D , we conclude that .z/ D z with  2 k . The restriction of  to Z.U C / is a k-automorphism of kŒz; z 0  such that .z/ D z. By surjectivity, the z 0 -degree of .z 0 / is necessarily 1. Denote .z 0 / D r.z/z 0 C p.z/ with r.z/; p.z/ 2 kŒz, r.z/ 6D 0. Using an analogue expression for  1 .z 0 / in the equality z 0 D . 1 .z 0 // we obtain that r.z/ D  2 k . z Let  2 Aut U C and Q D ˆ. /. From Step 2. We prove that Im ˆ D G.  Theorem 2.4, there exist ˛; ˇ 2 k and i 2 f0; 1g such that Q D z˛;ˇ ! i . Suppose that i D 1. Then ! D ˆ. 0 / for  0 D ˛1 ;ˇ 1  , which satisfies . 0 .e1 // D !.E1 / D E2 and . 0 .e2 // D !.E2 / D E1 . In other words, there exist a; b 2 U C such that  0 .e1 / D e2 C za and  0 .e2 / D e1 C zb. Applying  0 to the first Serre relation (S1) in U C , we obtain .e2 Cza/2 .e1 Cbz/.q 2 Cq 2 /.e2 Cza/.e1 Cbz/.e2 Cza/C.e1 Cbz/.e2 Cza/2 D 0: Using the grading of 3.1.3, this identity develops into an expression s C t D 0 with L s D e22 e1  .q 2 C q 2 /e2 e1 e2 C e1 e22 2 U3 and the rest t in n5 Un . Then s D 0. But the relations of 3.1.5 allow to compute: s D e2 .e2 e1  q 2 e1 e2 /  q 2 .e2 e1  q 2 e1 e2 /e2 D q 2 e2 e3 C e3 e2 D q 2 z 6D 0. This is a contradiction, and we z conclude that i D 0 and so Q D z˛;ˇ 2 G. Step 3. We prove that ˆ is injective. We fix an automorphism  2 Autz .U C / such that  2 Ker ˆ. By definition of Ker ˆ, there exist a; b 2 U C such that .e1 / D e1 C za

and

.e2 / D e2 C zb:

The elements e3 and e3 being defined as q-brackets of e1 and e2 , we deduce that .e3 / D e3 C zc; 2

where c D .e1 b  q be1 / C .ae2  q 2 e2 a/ C z.ab  q 2 ba/, and N .e3 / D e3 C z c; where cN D .e1 b  q 2 be1 / C .ae2  q 2 e2 a/ C z.ab  q 2 ba/. Applying  to the relations e1 e3 D q 2 e3 e1 (see 3.1.2) and e3 D .1  q 4 /e1 e2 C 4 q e3 (see 3.1.5) we obtain by identification .e1 c  q 2 ce1 / C .ae3  q 2 e3 a/ C z.ac  q 2 ca/ D 0; .1  q 4 /ae2 C .1  q 4 /e1 b C q 4 c C .1  q 4 /zab D c: N Consider now z D e2 e3  q 2 e3 e2 . By Step 1, there exists some  2 k such that z D z C z.e2 c  q 2 ce2 / C z.be3  q 2 e3 b/ C z 2 .bc  q 2 cb/. Simplifying by z, we obtain   1 D .e2 c  q 2 ce2 / C .be3  q 2 e3 b/ C z.bc  q 2 cb/. The right member lies in the ideal I defined in 3.1.3 and the left member is a scalar in k D U0 . Therefore

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 D 1. We conclude that .z/ D z

and .e2 c  q 2 ce2 / C .be3  q 2 e3 b/ C z.bc  q 2 cb/ D 0:

Similar calculations for the other central generator z 0 D .1  q 2 /.e3 e3 C .1 C q /ze1 / (see 3.1.5), using the equality .z 0 / D z 0 C p.z/ from Step 1, give  D 1 and p.z/ D .1  q 2 /zs.z/ for some s.z/ 2 kŒz, which satisfies 2

s.z/ D .1 C q 2 /za C zc cN C e3 cN C q 4 ce3 C .1  q 4 /ce1 e2 : From 3.1.2 we can consider the degree function deg at the indeterminate e2 in the polynomial algebra U C D S Œe2 I ; ı. Introduce in particular the three positive integers d D deg .e1 /, d 0 D deg .e2 / and d 00 D deg .e3 /. Comparing the degree of the two members of all the equalities obtained above, we obtain by very technical considerations whose details are left to the reader that we have necessarily d D d 00 D 0 and d 0 D 1. In other words there exist a; b1 ; b0 ; c 2 S such that .e1 / D e1 C za;

.e2 / D e2 C z.b1 e2 C b0 /;

.e3 / D e3 C zc;

.z/ D z:

In particular the restriction S of  to S is an automorphism of S fixing z. Consider the field of fractions K D k.z/, the algebra T D KŒe3 Œe1 I   S , and the extension T of S to T . Since T is a quantum plane over k (with e1 e3 D q 2 e3 e1 ), we can apply Proposition 1.4.4 of [1] and deduce that the K-automorphism T satisfies .e1 / D f e1 and .e3 / D ge3 for some f; g 2 k . Because .e1 / and .e3 / are in the subalgebra S D kŒzŒe3 Œe1 I  of T D k.z/Œe3 Œe1 I  , we have in fact that f , g are non-zero in kŒz. By the same argument for the automorphism  1 , there exist f 0 , g 0 non-zero in kŒz such that  1 .e1 / D f 0 e1 and  1 .e3 / D g 0 e3 . By composition of  and  1 it follows that ff 0 D gg 0 D 1, and hence f; g 2 k . Denoting f D ˛ and g D , we obtain ˛e1 D e1 C za and e3 D e3 C zc. These equalities in S imply that .˛  1/e1 2 zS and .  1/e3 2 zS with ˛;  2 k , and so ˛ D  D 1 and a D c D 0. To summarize, we have .e1 / D e1 , .e3 / D e3 and .z/ D z. It is then easy to check that we have also .e2 / D e2 . We conclude that  D idU C .

3.3 Non permutability of the central generators z and z0 3.3.1 The prime ideal .z0 /. The ideal .z 0 / generated in U C by the central element z 0 is completely prime. A direct proof consists in checking by computations in the iterated Ore extension U C and its localization V D kq 2 Œe3˙1 ; e1˙1 Œz; z 0  that any element v 2 V satisfying z 0 v 2 U C is necessarily an element of U C ; then the ideal .z 0 / D z 0 U C is no more than the contraction z 0 V \ U C of the completely prime ideal z 0 V of V . The complete primeness of .z 0 / can also be proved by the algorithmic method of [9], or can be deduced from the description of the H -prime spectrum of U C (see 3.4) following the method of [12]. Proposition 3.4. There are no algebra automorphisms of U C sending .z/ to .z 0 / or .z 0 / to .z/.

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Proof. Recall that H denotes the factor algebra U C =.z/ and  the canonical map U C ! H (see 2.2.1 and 3.2.1). Denote H0 D U C =.z 0 / and  0 the canonical map U C ! H0 . We have observed that the center of H is the polynomial algebra Z.H/ D kŒ; note that  equals .z 0 / up to a multiplication by a non-zero scalar. By direct calculations in H0 one can prove similarly that the center of H0 is L the polynomial 0 0 2 C Um;n putting algebra Z.H / D kŒ .z/. Consider the standard N -grading U D e1 on degree .1; 0/ and e2 on degree .0; 1/. The central generator z is homogeneous 2 C 2 of degree L .1; 2/ and the N -grading induced on H D U =.z/ is0 just the N -grading H D Hn;m considered in 2.2.1. The central generator z is homogeneous of degreeL .2; 2/, and the N2 -grading induced on H0 D U C =.z 0 / will be denoted by H0n;m . H0 D Now suppose that there exists some algebra automorphism  of U C such that .z 0 / D .z/. Then  induces an isomorphism O W H0 !LH defined by O  0 D  . Tn;m defined by Tn;m D In particular H can be graded by the N2 -grading H D 0 O .Hm;n /. Since E3 ; E3 2 H1;1 and  2 H2;2 , it follows from Corollary 2.5 that there exist two integers r; s  1 with E3 ; E3 2 Tr;s and  2 T2r;2s . Set t1 D O 1 .E3 /, t2 D O 1 .E3 /, and t D O 1 ./ D O 1 .E3 E3 / D t1 t2 . By construction we have t1 ; t2 2 H0r;s and t 2 H02r;2s . Because Z.H/ D kŒ and O is an isomorphism of algebras, t is a generator of the polynomial algebra Z.H0 / D kŒ 0 .z/. Then there exist  2 k ,  2 k such that t D  0 .z/ C . Since t 2 H02r;2s , we have necessarily  D 0. We obtain in H0 the equality  0 .z/ D 1 t1 t2 where  2 k and t1 ; t2 2 H0r;s . C Choosing u1 ; u2 2 Ur;s such that  0 .u1 / D 1 t1 and  0 .u2 / D t2 , we deduce in LU 0 C C an equality z D u1 u2 C z u for some u 2 U . Back to the N-grading U D Un introduced in 3.1.3, we have clearly u1 u2 2 U2.rCs/ whereas z 2 U3 and z 0 2 U4 . It follows that u D 0, and then z D u1 u2 gives the contradiction. Remark 3.5. Suppose that we can prove similarly that U C =I is not isomorphic to H ' U C =.z/ for any height one prime ideal I of U C . Then we could deduce that any algebra automorphism of U C necessarily stabilizes .z/, thus, by Proposition 3.3, answering Problem 1 in the affirmative (see 3.2). For example, if I D .z  ˛/ with ˛ 2 k , it is possible to separate U C =I from H up to isomorphism (by technical considerations on the q-brackets in both factor algebras, which are not developed here). Unfortunately the complete description of height one prime ideals in U C is not known as far as we know. In the last section we restrict to the graded prime ideals.

3.4 Action on the H -spectrum 3.4.1 H -spectrum of UqC .g/. Here g is an arbitrary simple finite dimensional Lie algebra. We consider all data and notations of 1.3.1. Denote by H the n-dimensional torus .k /n . The canonical Zn -grading on UqC .g/ given by deg Ei D i , the i -th term of the canonical basis of Zn , induces a rational action of H on UqC .g/. The H spectrum of UqC .g/ is the set of graded prime ideals. We denote it by H -Spec UqC .g/.

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It determines a stratification of the whole prime spectrum Spec UqC .g/. We refer to [6], Section II, for more details on stratifications of iterated Ore extensions and summarize here some results obtained in [12] concerning the case of UqC .g/. More generally, [12] determines a stratification of Spec S w for a family of algebras w S (also denoted by R0w , see [12], l. 7, p. 217, for the equivalence between the two notations), where w is an element of the Weyl group W of g. The space of graded prime ideals of S w is indexed by the set W }! W ´ f.x; y/ 2 W  W j x  w  yg; where  is the Bruhat order (see for example [14], App. 1). The algebra UqC .g/ is the particular case w D e, and thus the index set of H -Spec UqC .g/ is just W . The author introduces in [12] an ideal Q.y/ for any y 2 W (this is the ideal Q.e; y/e with the notations of [12], p. 231 and p. 236) and proves (see [12], 7.1.2 (ii), p. 239) that H  Spec UqC .g/ D fQ.y/gy2W : By [12], 6.11, p. 236, we have Q.y/  Q.y 0 / if and only if y 0  y. Furthermore, [12], 5.3.3, p. 225 (with notation Y .y/ D Ye .e; y/), asserts that a Spec UqC .g/ D Y .y/; (3.2) y2W

where Y .y/ denotes the set of prime ideals containing Q.y/. By [12], 6.12, p. 237, each subset Y.y/ has a unique minimal element, namely Q.y/, and coincides with the H -strata of Spec UqC .g/ corresponding to the graded prime ideal Q.y/. That is: T Y.y/ D SpecQ.y/ UqC .g/ D fP 2 Spec UqC .g/ j Q.y/ D h2H h  P g: 0 Applying [12], 6.13, ` p. 238, the 0 closure of each Y .y/ is a union of other Y .y /’s, namely Y.y/ D y 0 2W;y 0 y Y.y /; and the disjoint union (3.2) is then a stratification of Spec UqC .g/.

3.4.2 H -spectrum of U C for the type B2 . Suppose now that g is of type B2 and keep all notations of 3.1. The Weyl group W is of order 8. Its elements can be described by their action on the roots f"1 ; "2 g and as words on the two generators s1 and s2 as follows: 7 "1 ; s2 W "1 7! "1 ; "2 ! 7 "2 ; s1 W "1 7! "2 ; "2 ! s 1 s2 W " 1 ! 7 "2 ; "2 ! 7 "1 ; s2 s1 W "1 ! 7 "2 ; "2 ! 7 "1 ; 7 "1 ; "2 ! 7 "2 ; e W "1 ! 7 "1 ; "2 ! 7 "2 : s1 s2 s1 s2 D s2 s1 s2 s1 W "1 ! Using the results of [12] recalled in 3.4.1, the H -prime spectrum of U C D UqC .g/ has exactly 8 elements. The ideals .0/, .e1 /, .e2 / and .e1 ; e2 / are clearly graded prime ideals of U C . This is also the case for the ideals .e3 / and .e3 /, the factor algebras being in both cases domains isomorphic to a quantum plane. Finally the prime ideals .z/ and .z 0 / considered in 3.2 and 3.3 are also graded prime ideals of U C . So the

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poset .W; / and the H -spectrum of U C are the following. s1 s2 s1 sJ2 JJJ JJJ JJJ J s1 s2 s1 U s s s i 2 1 2 UUUU UUUU iiiiiii UiUi iiii UUUUUUU i i i UU iii s1 s2 UU s s ii 2 1 UUUU UUUU iiiiiii UiUi iiii UUUUUUU i i i UUU iii s1 Ji s JJ t 2 JJ tt t JJ t JJ tt JJ tt t J tt e tt ttt t t t ttt

.0/ H HH vv HH v v HH v v HH v vv 0 .z/ SSS k .z / k SSS k k SSS kkkk SSk kkkk SSSSSS k k k SS kkk .e3 / SSS .e3 / SSSS kkk SSSkSkkkkk kk SSSS SSSS kkkk kkkk .e1 / H .e2 / HH vv HH v HH vv HH vv vv .e1 ; e2 /

Proposition 3.6. The subgroup of algebra automorphisms of U C stabilizing the H spectrum of U C is isomorphic to .k /2 . Proof. Any automorphism of U C stabilizing the H -spectrum of U C preserves the set of height one prime graded ideals of U C , that is, f.z/; .z 0 /g. Then the result follows from Propositions 3.3 and 3.4. 3.4.3 Automorphisms of some subalgebras of U C indexed by W . Take all data and notations of 1.3.1 and consider for any w in the Weyl group W of g and for e D ˙1 the subalgebras U C .w; e/ of UqC .g/ defined in [17]. For any reduced expression w D si1 si2 : : : sin of an element w 2 W , the elements Ei.c1 1 / Ti01 ;e .Ei.c2 2 / / : : : Ti01 ;e Ti02 ;e : : : Ti0n1 ;e .Ei.cn n / / for various .c1 ; c2 ; : : : ; cn / 2 Nn form a basis of a subspace U C .w; e/ of UqC .g/ which does not depend of the reduced expression of w ([17], 40.2.1, p. 321). The 0 appearing in this definition are the symmetries of UqC .g/ Lusztig automorphisms Ti;e related to the braid group action defined in [17], 37.1.2. From [17], 40.2.1 (d), we have: if `.si w/ D `.w/  1, then Ei U C .w; e/  U C .w; e/. In particular, for w0 the element of maximal length in W , we have U C .w0 ; e/ D UqC .g/ (see the remark after 40.2.2 of [17]). The subspaces U C .w; e/ can be identified with the subalgebras Uq .nw / of Lemma 1.5 of [8] and also appear in [14], p. 123. Suppose now that g is of type B2 , take all notations of 3.1 and denote by Aw the subalgebra U C .w; 1/ of U C , for any w 2 W expressed as a word in the generators s1 and s2 of 3.4.2. Using the above definition of the basis of Aw and the properties of the 0 (in particular [17], 39.2.3), straightforward calculations allow to automorphisms Tj;1

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obtain the following description by generators of the eight subalgebras Aw of U C :

mmm mmm m m mm mmm

U C D As1 s2 s1 s2 D As2 s1 s2 s1

khe2 ; w; e3 i D As1 s2 s1

As2 s1 s2 D khe1 ; e3 ; wi x

khe2 ; wi D As1 s2

khe2 i D As1 TTTT TTTT TTTT TTT

QQQ QQQ QQQ QQQ Q

As2 s1 D khe1 ; e3 i

As2 D khe1 i

Ae D k

j jjjj j j j jjj jjjj

with the commutation relations 8 2 ˆ < e2 w D q we2 ; left side e3 w D q 2 we3 ; e3 e2 D e2 e3  w; ˆ : e1 w D we1 C .1  q 2 /e32 ; e1 e2 D q 2 e2 e1 C e3 ; e1 e3 D q 2 e3 e1 ; 8 2 ˆ < e1 e3 D q e3 e1 ; right side we x we x 1 D e1 w x C .q 2  1/e3 2 ; x 3 D q 2 e3 w; ˆ : x e2 e1 D q 2 e1 e2  q 2 e3 ; e2 w x D q 2 we x 2: e2 e3 D e3 e2 C w; The eight algebras are iterated Ore extensions. At level one, As1 and As2 are just commutative polynomial algebras in one variable. At level two, As1 s2 and As2 s1 are isomorphic to a same quantum plane (with parameter q 2 ) and so by [1] their group of algebra automorphisms is isomorphic to .k /2 . The third level introduces some asymmetry in the diagram. One can prove by direct calculations (which are left to the reader) similar to the proof of [2], Lemme 2.2 and Proposition 2.3, that: • the center of the algebra As1 s2 s1 is Z D kŒz with z D .1  q 2 /e3 e2 C w, S • its set of normal elements is N D n0 kŒzw n , • its automorphism group is .k /2 acting by .˛; ˇ/ W e2 7! ˛e2 ; e3 7! ˇe3 , w 7! ˛ˇw; and 2 x 1/e3 2 , • the center of the algebra As2 s1 s2 is Z 0 D kŒu with u D .1q 4 /e1 wC.q S • its set of normal elements is N 0 D n0 kŒue3 n ,

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• its automorphism group is .k /2 acting by .˛;  / W e1 7! ˛e1 , e3 7! e3 , w x 7! x ˛ 1  2 w. Using the fact that an isomorphism from As1 s2 s1 to As2 s1 s2 must map N to N 0 and Z to Z 0 , one can prove by direct computations and identifications using the basis of monomials in the natural generators that the algebras As1 s2 s1 and As2 s1 s2 are not isomorphic.

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