Cette rencontre aura lieu au CIRM Centre International de Rencontres Mathématiques du 25 au 29 septembre 2006.
La Rencontre est articulée autour de 4 cours de 3h, 11 conférences de 45 mm et 7 exposés de 30mm.
Les cours seront assurés par: Anton Alekseev (Université de Genève): La série de Campbell-Hausdorff et la conjecture de Kashiwara-Vergne , Nicolás Andruskiewitsch (Universidad Nacional de Córdoba, Argentine): On finite-dimensional Hopf algebras, Ibrahim Assem (Université de Sherbrooke, Québec): les parties gauche et droite d'une catégorie de modules. et Bernhard Keller (Université de Paris 7): Cluster algebras and quiver representations.
Les participants suivants ont confirmés leur venue:
Université de Genève, Suisse.
La série de Campbell-Hausdorff et la conjecture de Kashiwara-Vergne
Dans ce mini-cours, je présenterai la solution récente de la conjecture de Kashiwara-Vergne sur les propriétés de la série de Campbell-Hausdorff. Je commencerai par le rappel du théoreme de Duflo sur l'isomorphisme entre le centre de l'algèbre enveloppante et l'anneau des polynomes invariants. Je vais ensuite donner plusieurs énoncés équivalents de la conjecture et l'idée de la preuve.
Ce mini-cours est basé sur l'article: A. Alekseev, E. Meinrenken, On the Kashiwara-Vergne conjecture, math.QA/0506499, et sur les discussions avec C. Torossian.
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Universidad Nacional de Córdoba, Argentina.
On finite-dimensional Hopf algebras
In this series of talks I will present recent results on the classification of finite-dimensional Hopf algebras. This first talk is about pointed Hopf algebras. The classification of finite-dimensional complex pointed Hopf algebras with abelian group of group-likes is now more or less understood. If all prime divisors of the order of the group are greater than 7, then the classification is known. For the remaining cases, the key step is the classification of finite-dimensional Nichols algebras of diagonal type, but this was recently achieved by I. Heckenberger. I will recall known results and explain new techniques to attack the next case, namely pointed Hopf algebras with non-abelian group. I will report on joint work with S. Zhang and F. Fantino on pointed Hopf algebras with group S_n. I will also comment on work in progress with Heckeberger and Schneider on the Weyl groupoid of a semisimple Yetter-Drinfeld module. The second talk is about semisimple Hopf algebras. I will discuss the main open conjectures on this class of Hopf algebras. I will then explain recent results with S. Natale on doulbe groupoids and their relation with fusion categories and throught them with semisimple Hopf algebras. The third talk is about general finite-dimensional Hopf algebras. I will briefly recall the main results on Hopf algebras of fixed dimension. Then I will report on recent work with G. Garcia. Let G be a complex simple algebraic group and let Gamma be a finite subgroup of G. We construct under some mild assumptions an infinite family of Hopf algebras, neither pointed nor dual to pointed nor semisimple, of dimension equal to order of Gamma times the dimension of G. This generalizes a result of E. Muller for SL(2).
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Université de Sherbrooke, Québec.
Les parties gauche et droite d'une catégorie de modules
Soient A une algèbre de dimension finie sur un corps, et mod-A la catégorie des A-modules de type fini. La partie gauche de mod-A, introduite par Happel, Reiten et Smalo, est la sous-catégorie pleine de mod-A formée des A-modules indécomposables dont tous les prédécesseurs sont de dimension projective au plus un. La partie de droite est définie dualement. L'objectif du mini-cours est de décrire ces sous-catégories, puis de montrer comment elles permettent de définir des classes d'algèbres dont la théorie des représentations est (dans une grande mesure) prévisible.
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Université de Paris 7, France.
Cluster algebras and quiver representations
Cluster algebras were invented by S. Fomin and A. Zelevinsky in the spring of the year 2000. Their initial motivation was to find a combinatorial approach to the objects constructed by Lusztig in his work on total positivity in algebraic groups and canonical bases in quantum groups. We are still far from these aims but it has turned out that cluster algebras have strong connections with many other subjects, notably combinatorics, Teichmuller spaces, integrable systems, Poisson geometry and last not least the representation theory of quivers and finite-dimensional algebras. One of the most important classes of cluster algebras which have been linked to representation theory of finite-dimensional algebras are the so-called acyclic cluster algebras. After pioneering work by Marsh-Reineke-Zelevinsky, acyclic cluster algebras were studied by many authors, building on fundamental contributions notably by Buan-Marsh-Reineke-Reiten-Todorov, Caldero-Chapoton-Schiffler and Caldero-Chapoton. In this series of lectures, we will concentrate on acyclic cluster algebras. The first lecture will be devoted to definitions and examples. In the second lecture, we will establish the link with quiver representations. In the third and final lecture, we will introduce the cluster category and show how its triangulated structure allows one to obtain relations for the cluster algebra.
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Université de Saint-Etienne, France.
Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras
It is a joint work with Rachel Taillefer. Recently, R. Bocklandt proved a conjecture by M. Van den Bergh stating that a graded quiver algebra A (with relations) Calabi-Yau of dimension 3 is defined from a homogeneous superpotential W. In this talk, we prove that if we add to W any superpotential of smaller degree, we get a PBW deformation of A. Such PBW deformations are Calabi-Yau and are characterised among all PBW deformations of A. Various examples are presented.
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Université de Montpellier 2, France.
Hopf monads: an introduction
The notion of a Hopf algebra has been generalized in various ways, which may be loosely termed 'quantum groupoids'. In this talk, I will present the notion of Hopf a monad (introduced in a joint work with A. Virelizier), which provides a very general setting for such objects (at least in the finite dimensional case). I will illustrate this notion with a few examples, including the Drinfeld double of a Hopf algebra (with a new approach to Drinfeld-Yetter modules), and other examples of a different nature, and show that a substantial part of the (basic) theory of finite dimensional Hopf algebras carries over to Hopf monads.
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Université de Montpellier 2, France.
N-complexes as functors, amplitude cohomology and fusion rules
We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined. This work is in collaboration with Andrea Solotar and Robert Wisbauer.
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Universidade de Sao Paulo, Brasil.
Composition of irreducible morphisms in quasi-sectional paths
This is a joint work with C.Chaio and S.Trepode (Universidad Nacional de Mar del Plata, Argentina). Let A be an artin algebra. Using the so-called Auslander-Reiten theory, one can assign to A a quiver Q called the ,Auslander-Reiten quiver of A which "represents" the indecomposable finitely generated A-modules together with some morphisms between them, called irreducible. Unfortunately, Q does not give all the informations on the category modA of finitely generated A-modules because not all the morphisms can be re-constructed form the irreducible ones. However, (sum of) composition of irreducible morphisms can give important informations on modA.
A morphism f from X to Y is called irreducible provided it does not split and whenever f=gh, then either h is a split monomorphism or g is a split epimorphism. It is not difficult to see that such an irreducible morphism f belongs to the radical R=R(X,Y) but not to its square. Consider now a non-zero composition g of n irreducible morphisms, where n is at least 2. It is not always true that g belongs to R^n and not to R^{n+1}. We shall discuss here the situation of a composition of n irreducible morphisms lying in quasi-sectional paths. We say that a non-sectional path $X_1 \to X_2 \to \dots \X_n \to X_{n+1}$ is a left quasi-sectional path provided $X_1 \to X_2 \to \dots \X_n$ is sectional. Dually, one can define a right quasi-sectional path. We shall look, particularly, at components of type $\mathbb Z A_{\infty}$ or stable tubes and we shall give necessary and sufficient conditions for the existence of n irreducible morphisms between modules lying in such components with non-zero composition and belonging to R^{n+1}.
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Zurich, Suisse.
Lagrangian operad
Given a symplectic manifold $M$, we may define an operad structure on the the spaces $\op^k$ of the Lagrangian submanifolds of $(\overline{M})^k\times M$ via symplectic reduction. If $M$ is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semi-classical part of Kontsevich's deformation of $C^\infty({\bf R}^d)$ is a deformation of the trivial symplectic groupoid structure of $T^*{\bf R}^d$.
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Université de Paris 11, France.
Algèbres graduées et formes multilinéaires
On montre que les algèbres homogènes qui sont Koszul de dimension globale finie et qui sont Gorenstein sont complètement caractérisées par des formes multilinéaires. Ceci permet de décrire l'espace des modules de telles algèbres comme un espace de modules de formes multilinéaires modulo l'action du groupe linéaire. L'analyse de plusieurs exemples caractéristiques permettra d'illustrer les concepts introduits.
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Université de Reims, France.
The Poisson trace group of the variety $( h+ h^*)/W$ for a Weyl group W of rank 2
We consider a simple Lie algebra $g$ of rank 2 over $\bf C$, with Cartan subalgebra $ h$ and Weyl group $W$. Then $W$ acts by symplectomorphisms on $ h+h^*$. Hence, the algebra of regular functions of the variety $( h+ h^*)/W$, namely $S=S( h+ h^*)^W$, inherits a Poisson algebra structure. We compute here $dim(HP_0(S))$ and show that it is equal to the dimension of $HH_0(A^W)$, where $A$ is the second Weyl algebra. These two numbers are $1$ for $A_2$, $2$ for $B_2$ and $3$ for $G_2$. The main tool is an action of the Lie algebra ${sl}(2)$ on $S$, which allows to decompose $S$ into several isotypical components. We shall also give examples of groups for which these two numbers are different.
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Université de Magdeburg, Allemagne et Leeds, Royaume-Uni.
Bilinear forms, Hochschild homology and invariants of derived module categories (joint work with Christine Bessenrodt and Alexander Zimmermann; ArXiv:math.RT/0603189)
Let $A$ be a symmetric algebra over a perfect field $k$ of positive characteristic $p$. For such algebras, B. K\"ulshammer introduced, for any $n$, spaces $T_n(A)$ as those elements of $A$ whose $p^n$-th power lies in the commutator subspace $K(A)$. He then considered the orthogonal spaces with respect to the symmetrizing bilinear form on the symmetric algebra $A$. These $T_n(A)^{\perp}$ are ideals of the center of $A$, i.e. of the degree 0 Hochschild cohomology of $A$. It has recently been shown by A. Zimmermann that the sequence of these ideals is invariant under derived equivalences of the symmetric $k$-algebras. In the talk we explain how to extend this theory to arbitrary finite-dimensional, not necessarily symmetric, $k$-algebras. The way to achieve this is by passing to the trivial extension algebras. In this way we obtain new invariants of the derived module categories of finite-dimensional $k$-algebras. In some sense it can be seen as an extension of the well-known fact that the degree 0 Hochschild homology $A/K(A)$ is invariant under derived equivalence. We also present various applications of the above results.
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Université de Poitiers, France.
Grupoides et géométrie de Poisson
A une variété lisse de Poisson peut etre, sous certaines conditions, associé ¿un groupo"ide symplectique appel "réalisation symplectique". Nous utilisons ce groupo"ide pour désingulariser certaines variétés symplectiques (admettant des singularités). En un mot, nous donnerons une construction qui permet d'obtenir des désingularisations symplectiques à partir d'une réalisation symplectique.
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Université de Prague, République Tchéque.
Natural differential operators and graph complexes
We explain how the Invariant Tensor Theorem together with an elementary representation theory reduces the problem of classifying natural differential operators into a problem f ormulated in terms of graph complexes. This reduced problem can then be grasped by powerfu l methods of homological theory of graph complexes. We believe that this combination of so far separated areas could lead to many deep and unexpected results, or at least simplify existing proofs. As an example, we give a simple proof of the fact that all natural operators on linear connections are generated by the t orsion, curvature and covariant derivative. We also prove an apparently new theorem charac terizing natural operations on vector fields.
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Universidad Nacional de Córdoba, Argentina.
Fusion categories coming from vacant double groupoids
This talk is about a joint work with Sonia Natale. Ostrik introduced a fundamental family of fusion categories, constructed from finite group data, later studied by Etingof, Nikshych and Ostrik, the so called group-theoretical tensor categories. Andruskiewitsch and Natale introduced another class of tensor categories constructed from finite double groupoids. To a certain double groupoid $\Tc$ and a perturbation data $\vartheta$ they assign a semisimple weak Hopf algebra $\ku^{\vartheta}\Tc$ and, a fortiori, a tensor category $\Rep(\ku^{\vartheta}\Tc)$. In this talk I will show that when category $\Rep(\ku^{\vartheta}\Tc)$ is fusion and $\Tc$ is a vacant double groupoid then $\Rep(\ku^{\vartheta}\Tc)$ is group-theoretical.
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Université de Poitiers, France.
Cohomologie de Poisson et déformations en petite dimension
Les surfaces dans ${\bf C}^3$, données par une équation algébrique, sont naturellement munies d'une structure de Poisson, symplectique en dehors de ses singularités éventuelles. Pour cette structure de Poisson, et son prolongement à l'espace ambiant ${\bf C}^3$, je calcule la cohomologie de Poisson, dans le cas ou la surface à une singularité isolée. La cohomologie de Poisson intervient dans l'étude des déformations de structures de Poisson et les résultats que j'obtiens en dimensions deux et trois me permettent d'écrire toutes les déformations des structures de Poisson étudiées, à équivalence près.
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Universidad Nacional de Sur, Bahía Blanca, Argentina.
Hochschild cohomology via incidence algebras
For each presentation (Q,I) of a finite dimensional k-algebra A, we consider an incidence algebra associated and compare their Hochschild cohomology groups.
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Université de Edinburgh, Ecosse.
Poisson cohomology for semi-classical limits of quantum alg ebras
A theorem from Van den Bergh provides a Poincar¿duality between the Hochschild cohomology of an algebra and its homology with value in a twisting¿¿ bimodule. For quantum algebras, the semiclassical limit of this bimodule gives rise to a Poisson module over the semiclass ical limit of the algebra. In this talk, after explaining the general ideas of this process, we will focus on the cas e of the quantum affine space. We will show that the Poisson module associated restores a Poincaré duality between Poisson homology and cohomology for the semiclassical limit, allow ing us to compute the cohomology. This is a joint work with Stephane Launois.
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Université de Saint-Etienne, France.
Algèbres quantiques avec loi de redressement et applications
La théorie des groupes quantiques fournit des déformations non commutatives de certains groupes algébriques et de leurs espaces naturels de représentation, sous la forme d'algèbres de Hopf co-agissant sur certaines algèbres quantiques. On est donc en presence d'un contexte naturel de théorie des invariants quantique, ou les anneaux d'invariants du cas classique sont remplaces par des anneaux de co-invariants. C'est dans le cadre de l'étude de ces anneaux de co-invariants que l'on est amené a introduire la notion d'algèbre quantique avec loi de redressement. En effet, de nombreux anneaux de co-invariants sont des exemples de telles algèbres. On s'attachera a présenter les propriétés homologiques des algèbres en question et plus particulièrement la propriété de Cohen-Macaulay au sens de la géometrie algébrique non commutative.
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Université de Montréal, Québec.
Les catégories amassées et les algèbres répliquées
Soit A une algèbre héréditaire. Sa catégorie amassée $C_A$ (cluster category) ainsi que sa catégorie m-amassée $C^m_A$ (m-cluster category) sont par définition des quotients de la catégorie dérivée $D^b(mod A)$. Nous présenterons une construction qui réalise un domaine fondamental de $C_A$ comme la partie gauche de l'algèbre dupliquée $A^{(1)}$ de $A$; ainsi qu'un domaine fondamental de $C^m_A$ comme la m-partie gauche de l'algèbre m-répliquée $A^{(m)}$ de $A$.
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Université de Sherbrooke, Québec.
Algèbres de groupes gauches héréditaires par morceaux (travail conjoint avec J. Dionne et M. Lanzilotta)
Étant donnés une algèbre A, un groupe G et une action de G sur A, on d é finit l'algèbre de groupe gauche A[G]. Les algèbres de groupes gauches ont fait leur apparition au cours des années 80 avec les travaux de Reiten, Riedtmann et de la Pe~na, et il a depuis été montré que les algèbres A et A[G] partagent plusieurs propriétés. Dans cet exposé, nous nous intéressons à la propriété d'être héréditaire par morceaux, c'est-à-dire qu'il existe une catégorie abélienne héréditaire H et une équivalence entre les catégories dérivées de complexes bornées de A et H. Les algèbres héréditaires par morceaux ont été fortement étudiées par Happel, notamment. Nous montrons que sous certaines hypothèses, le fait que A soit héréditaire par morceaux entraîne qu'il en est de même pour A[G].
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Universidad de Buenos Aires, Argentina.
Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology
I'll present two applications of the Cartan-Eilenberg spectral sequence for a change of rings to the problem of computing Hochschild cohomology. The description of the complete cohomology structure of an algebra with one generator and some extensions of known results on the cohomology of quotients by homological ideals.
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Université de Montpellier 2, France.
Double of Hopf monads
Hopf monads generalize Hopf algebras to a non-braided setting. I will explain how to construct double of such structures and their links with spherical categories (and low-dimensional topology). This is a joint work with A. Bruguières.
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