# Christian Kassel

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

# Notes d'exposés et transparents

"The Hilbert scheme of n points on a torus and modular forms", Oberseminar Zahlentheorie, Universität zu Köln (24 October 2016)

Abstract: We compute the zeta function of the Hilbert scheme of n points on a two-dimensional torus and show it satisfies a remarkable functional equation. To this end we establish an explicit formula for the number Cn(q) of ideals of codimension n of the algebra Fq[x,y,x-1, y-1] of Laurent polynomials in two variables over a finite field of cardinality q. This number is a palindromic polynomial of degree 2n in q. Moreover, Cn(q) = (q-1)2 Pn(q), where Pn(q) is another palindromic polynomial, whose integer coefficients turn out to be non-negative and which is a q-analogue of the sum of divisors of n. We also give arithmetical interpretations of the values of Cn(q) and of Pn(q) at q = -1 and at roots of unity of order 3, 4, 6.

"On combinatorial zeta functions", Colloquium, Universität Potsdam (13 May 2015)

Abstract: Zeta functions appear in many fields of mathematics and in various guises. I will give examples of zeta functions obtained from counting points in a variety, loops in a graph, or words in an alphabet. In some emblematic cases such a function is rational. I will next concentrate on zeta functions constructed from matrices with entries in group rings. As Kontsevich was the first to prove, some of the latter zeta functions, though not rational, are algebraic. For such zeta functions I will explain where the algebraicity comes from and show examples where they can be computed explicitly.

"Le multiplicateur de Schur de 1900 à nos jours", Colloquium, LAMFA, Amiens (1er octobre 2014)

Résumé : Il y a un peu plus de cent ans Schur introduisait le "multiplicateur" d'un groupe fini pour en classifier les représentations projectives. Plus récemment, les travaux de Drinfeld sur les groupes quantiques ont permis de faire apparaître un avatar intéressant du multiplicateur de Schur. Dans cet exposé je rappelle la définition du multiplicateur de Schur, puis je présente sa variante quantique.

"Algebraicity of zeta functions associated to matrices", Conference "Words, Codes and Algebraic Combinatorics", Cetraro, Italy (1 July 2013)

Abstract: Following and generalizing a construction by Kontsevich, we associate a zeta function to any matrix with entries in a ring of noncommutative Laurent polynomials with integer coefficients. We show that such a zeta function is an algebraic function. Reference: C. Kassel, C. Reutenauer, "Algebraicity of the zeta function associated to a matrix over a free group algebra", arXiv:1303.3481.

"Non-commutative torsors", Algebra seminar, Universitetet i Oslo, Norway (2 September 2011)

Abstract: Based on joint work with Pierre Guillot, "Cohomology of invariant Drinfeld twists on group algebras", IMRN 2010 (2010), 1894-1939 (arXiv:0903.2807).

"Cohomological invariant for groups coming from quantum group theory", Oberseminar Topologie, Universität Bonn, Germany (23 November 2010)

Abstract: Report on joint work with Pierre Guillot, "Cohomology of invariant Drinfeld twists on group algebras", IMRN 2010 (2010), 1894-1939 (arXiv:0903.2807).

"Tate-Vogel cohomology", Journées en l'honneur de Pierre Vogel, Institut Henri Poincaré, Paris (27 October 2010)

Abstract: This is a short report on unpublished work by Pierre Vogel, in which he extended the Tate cohomology to any group.

"Any Hopf algebra fibers over an affine variety of the same dimension", Conference "Quantum Groups", Université Blaise Pascal, Clermont-Ferrand (30 August 2010)

Abstract: In this talk I show how to attach an affine algebraic variety V to any Hopf algebra H. This variety is the base space of a quantum principal fiber bundle with "structural group" H. If H has finite dimension d, then V is of dimension d as a variety. These results have been obtained in joint work with Eli Aljadeff (Advances in Math. 208 (2008), 1453-1495) and with Akira Masuoka (Rev. Un. Mat. Argentina 51:1 (2010), 79-94).

"Polynomial identities", Colloquium, University of Southern California, Los Angeles, USA (30 September 2009)

Abstract: After an introduction to the classical theory of polynomial identities, we present some recent developments involving group gradings and Hopf algebras. All results will be illustrated with matrix algebras.

"Invariant Drinfeld twists on group algebras", International Workshop/Special Session of CMS Summer Meeting 2009 "Groups and Hopf Algebras", St. John's, Newfoundland, Canada (5 June 2009)

Abstract: Drinfeld twists were introduced by Drinfeld in his work on quasi-Hopf algebras; they have been used to classify certain classes of (co)semisimple Hopf algebras. In joint work with Pierre Guillot (arXiv:0903.2807), after observing that the invariant Drinfeld twists on a Hopf algebra form a group, we determine this group when the Hopf algebra is the algebra of a finite group G. The answer involves the group of class-preserving outer automorphisms of G as well as all abelian normal subgroups of G of central type. In my lecture I shall also present several examples for which the group of invariant twists has been completely computed by us.

"Homology (of) Hopf algebras", Rencontre franco-britannique et Journées en l'honneur des soixante ans de Jacques Alev, Reims (28 novembre 2008) et Quantum Group Workshop, Toulouse (16 mai 2009)

Abstract: Report on "The lazy homology of a Hopf algebra" by Julien Bichon and the speaker, arXiv:0807.1651. To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresentations of H. We also compute the lazy homology of the Sweedler algebra.

"Quantum principal fiber bundles and polynomial identities", Conference on Noncommutative Structures in Mathematics and Physics, Brussels, Belgium (23 July 2008)

Abstract: In recent years there has been an increasing interest in quantum principal fiber bundles. For such bundles the structural group is a Hopf algebra. Classifying quantum principal fiber bundles is rather a difficult task. In joint work with Eli Aljadeff we made a step towards their classification by constructing certain versal deformations. Our construction is based on an adapted theory of polynomial identities.

"Polynomial identities in noncommutative geometry", Colloque d'algèbre non commutative, Université de Sherbrooke, Québec, Canada (12 and 13 June 2008)

Abstract: These are the slides of a minicourse I gave on joint work with Eli Aljadeff. Starting from an algebra obtained by twisting a Hopf algebra with a cocycle, I define the generic cocycle cohomologous to the given cocycle and out of it I define a generic Hopf Galois extension over an explicit commutative algebra. The construction is illustrated with the help of the four-dimensional Sweedler algebra. I also define a theory of polynomial identities for comodule algebras and show how the universal comodule algebras obtained from these identities is related to the generic Hopf Galois extension. This gives a description of the universal comodule algebra after some central localization.

Plan of the slides: 1. Hopf Galois extensions, twisted algebras, and the classification problem. 2. The generic cocycle and the corresponding parameter space. 3. An example: the Sweedler algebra. 4. The generic Galois extension. 5. Polynomial identities and the universal comodule algebra.

"The free group F2, the braid group B3, and palindromes", Deuxième Congrès Canada-France, UQAM, Montréal, Québec, Canada (2 juin 2008)

Abstract: (Joint work with Christophe Reutenauer) We define a self-map Pal: F2 --> F2 of the free group on two generators a, b, using automorphisms of F2 that form a group isomorphic to the braid group B3. The map Pal restricts to de Luca's right iterated palindromic closure on the submonoid generated by a, b, and is continuous for the profinite topology on F2. The values of Pal are palindromes and coincide with the elements g of F2 such that abg is conjugate to bag.

"Sobre palabras y trenzas" (in Spanish), Coloquio, Departamento de Matemáticas, Universidad de Chile, Santiago, Chile (30 de Enero 2008)

"Some universal constructions on Hopf algebras", Conference "Shuffles, descents and representations'' in memory of Manfred Schocker, Nice (10-14 September 2007)

Abstract: (Joint work with Eli Aljadeff) This lecture is devoted to several universal constructions leading to interesting identities for each Hopf algebra. It includes a generalization of Dedekind's group determinant. We shall work out these constructions on the simple example of the Sweedler algebra, which is the smallest non-commutative non-cocommutative Hopf algebra.

"Des mots et des tresses" [Of words and braids], Séminaire des Doctorants, IRMA, Strasbourg (31 mai 2007)

"From Christoffel words to braids", Colloquium, Rutgers, New Brunswick, NJ, USA (13 April 2007)

Abstract: In 1875 Christoffel published a paper in Latin (sic!) in which he associated a word w on a two-letter alphabet to the sequence formed by the multiples of an integer a considered modulo a coprime integer b. He then uses the word w to recover the expansion of a/b as a continued fraction. This may be one of the first systematic uses of words in mathematics. In this talk we'll start from Christoffel words and discuss various aspects of an interesting class of substitutions, the so-called Sturmian morphisms. In particular we'll show how Sturmian morphisms can be naturally identified with braids on four strings. This talk is based on joint work with C. Reutenauer, UQAM, Montreal.

"Hopf Galois extensions up to homotopy", Second joint meeting of AMS, DMV, ÖMG, Mainz, Germany (16-19 June 2005)

Abstract: (Joint work with Hans-Jürgen Schneider) Hopf Galois extensions are noncommutative analogues of principal fibre bundles with structural group replaced by a Hopf algebra. I discuss a concept of homotopy for Hopf Galois extensions and show how it allows a certain classification of such extensions. In particular, we determine all Hopf Galois extensions up to homotopy in the case when the Hopf algebra is a Drinfeld-Jimbo quantum enveloping algebra.

"Les variétés de Schubert et leurs lieux singuliers" [Schubert varieties and their singular loci], Colloquium, Département de Mathématiques, Montpellier (16 décembre 2004)

Résumé : Les variétés de Schubert ont été introduites au 19e siècle pour résoudre des problèmes de géométrie énumérative. De nombreuses propriétés de ces variétés s'expriment de manière combinatoire à l'aide des permutations. Je rappelle la définition des variétés de Schubert et j'explique comment des graphes planaires simples ont permis en 2001 de résoudre un problème resté longtemps ouvert, celui de la détermination de leurs lieux singuliers.

"From Sturmian morphisms to the braid group B_4", Workshop "Braid groups and applications", Banff International Research Station, Canada (16--21 October 2004)

"Surjectivité de la norme : problèmes effectifs en cohomologie des groupes" [Surjectivity of the norm map: effective problems in group cohomology], Séminaire de Topologie algébrique, Université Paris 13 (12 mars 2004)

Résumé : Soit G un groupe fini opérant par automorphismes sur un anneau R. Considérons l'application norme N_G : R --> R^G de R vers le sous-anneau R^G des éléments G-invariants. Un très joli résultat d'Aljadeff et Ginosar dit que N_G : R --> R^G est surjective si et seulement si pour tout sous-groupe abélien élémentaire E de G, la norme N_E : R --> R^E est surjective. De manière équivalente, il existe un élément x_G dans R tel que N_G(x_G) = 1 si et seulement si pour tout sous-groupe abélien élémentaire E de G il existe un élément x_E de R tel que N_E(x_E) = 1. Lorsque R est non commutatif, c'est un problème ouvert que de trouver une formule explicite pour x_G en termes des éléments x_E. Dans mon exposé je présente une méthode permettant de résoudre ce problème pour tous les groupes finis. C'est un travail commun avec Eli Aljadeff.

"Recent developments on Artin's braid groups", Colloquium, Rutgers University, NJ, USA (25 October 2002)

Abstract: There have been two recent interesting developments on braid groups: one is the proof by Bigelow and Krammer that braid groups have finite-dimensional faithful representations; another one is the construction by Dehornoy of an invariant linear ordering on these groups (no such ordering was known before). In my lecture I concentrate on Dehornoy's ordering and explain how it came up unexpectedly in the study of a large cardinal axiom in set theory. I also give an alternative topological construction of this ordering.

"Explicit norm one elements for ring actions of finite abelian groups", Algebra Seminar, Rutgers University, NJ, USA (25 October 2002)

Abstract: It is known that the norm map N_G for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map N_U is surjective. Equivalently, there exists an element x_G in R satisfying N_G(x_G) = 1 if and only for every elementary abelian subgroup U there exists an element x_U in R such that N_U(x_U) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for x_G in terms of the elements x_U. Eli Aljadeff and the speaker solved this problem when the group G is abelian.

"Action of the absolute Galois group of the rationals on knot invariants" Abel Bicentennial Conference, Oslo, Norway (5 June 2002)

"Recent developments on Artin's braid groups" Journées mathématiques franco-russes, Lomonosov State University (MGU) , Moscow, Russia (15 December 2000)

(9 novembre 2016)