Nicolás Andruskiewitsch (Córdoba, Argentina)
Title: On the finite generation of the cohomology of abelian extensions of Hopf algebras
Abstract: A finite-dimensional Hopf algebra is called quasi-split if it is Morita equivalent to a split abelian extension of Hopf algebras. Combining results of Schauenburg and Negron, it is shown that every quasi-split finite-dimensional Hopf algebra satisfies the finite generation cohomology conjecture of Etingof and Ostrik. This is applied to a family of pointed Hopf algebras in odd characteristic introduced by Angiono, Heckenberger and the first author, proving that they satisfy the aforementioned conjecture.
Michel Brion (Grenoble, France)
Title: Vector fields on algebraic varieties in positive characteristics
Abstract: In characteristic zero, it is known that a vector field on an algebraic variety lifts uniquely to its normalization, and admits a reduction of singularities in low dimension. Both results fail in positive characteristic. The talk will present a remedy to this, based on notions of equivariant normality and regularity.
Stephen Donkin (York, UK)
Title: Endomorphism algebras of Young permutation modules for Hecke algebras
Abstract: We consider endomorphism algebras of Young permutation modules for Hecke algebras of type A. In certain cases, including quantisations of the partition algebras, we describe the block distribution of the simple modules. We also describe certain equivalences between blocks, The methods involve descent from the representation theory of quantum general linear groups.
Vyacheslav Futorny (SUSTech, China)
Title: Twisted localization for Affine Lie algebras
Abstract: Twisted localization is a key tool in the representation theory of vertex algebras. We will discuss the state of the art of the theory.
Stéphane Gaussent (Saint-Etienne, France)
Title: On the tensor product of irreducible representations of Kac-Moody Lie algebras
Abstract: The representation theory of Kac-Moody Lie algebras over the complex numbers is a very active area of research since Kac and Moody introduced these Lie algebras. A lot of tools can be used in this context: crystal graphs, geometry of flag manifolds, KLR 2-categories... But some natural conjectures still remain open, for instance the Kostant conjecture or the Schur positivity conjecture. In this talk, I will report on a joint work with Rekha Biswal on the description of some irreducible components in the tensor product of two irreducible integral highest weight representations of a symmetrisable Kac-Moody Lie algebra.
Anne Moreau (Paris-Saclay, France)
Title: Isomorphisms between W-algebras
Abstract: To any vertex algebra one can attach invariants of different nature: its character (a formal series), its associated variety (a Poisson variety), etc... In this talk, I will explain how to exploit these invariants to obtain nontrivial isomorphisms between W-algebras at admissible levels. To study a more general setting, one can use totally different techniques developed more recently that we will also mention.
Simon Riche (Clermont-Ferrand, France)
Title: Semiinfinite sheaves on affine flag varieties
Abstract: We will explain how, generalizing a construction of Gaitsgory, one can define and study a category of sheaves on the affine flag variety of a complex reductive group that "models" sheaves on the corresponding semiinfinite flag variety, with coefficients in a field of positive characteristic, and which should provide a geometric model for a category of representations of the Langlands dual Lie algebra over the given coefficient field. As an application, we use this construction to compute the dimensions of stalks of the intersection cohomology complex on Drinfeld's compactification, with coefficients in any field of good characteristic. This is joint work with Pramod Achar and Gurbir Dhillon.
Marc Rosso (IMJ, France)
Title: On Feigin's homomorphisms and quantum shuffle algebras
Abstract: "Quantum upper triangular subalgebras" of quantum groups are known to be the subalgebras generated in degree 1 of certain quantum shuffle algebras, and Feigin's maps are morphisms from them to some quantum polynomial algebras. Feigin's homomorphisms have been extended by D. Rupel to the whole quantum shuffle algebras, using the explicit expression for the product in terms of braid group action.
We will introduce some quantum quasi-shuffle algebras to provide another construction, avoiding explicit computations. We show that quantum polynomial algebras can be realized naturally as quotients of suitable quantum quasi-shuffle algebras. The result then follows from a universal property of quantum quasi-shuffle algebras relative to Hopf algebra morphisms.
Susan Sierra (Edinburgh, UK)
Title: Ideals of enveloping algebras of infinite-dimensional Lie algebra
Abstract: If L is an infinite-dimensional Lie algebra, it is believed that the enveloping algebra U(L) of L is never Noetherian. However, it appears that if L is ``not too big'', then the two-sided ideals of U(L) may be more tractable. We survey what is known about two-sided ideals of U(L) if L is either an affine Lie algebra or the Virasoro algebra. For affine Lie algebras, we can say a great deal; for the Virasoro algebra, there are few results but many conjectures.
If g is a finite-dimensional simple Lie algebra, the affine Lie algebra \hat{g} is an extension of the loop algebra Lg = g[s, s^{-1}] by a central element c. The rings U(\hat{g})/(c-\lambda) are simple for any nonzero scalar \lambda, but the two-sided structure of U(Lg) = U(\hat{g}/(c)) is more complicated. We show that U(Lg) does not satisfy the ascending chain condition on two-sided ideals, but that the two-sided ideals still have a nice structure: there is a canonical collection of ideals I_n, parameterised by positive integers, so that any two-sided ideal of U(Lg) contains some I_n. The ideals I_n can be thought of as universal annihilators of classes of finite-dimensional representations of Lg.
The Virasoro algebra Vir is a central extension of the Witt algebra of vector fields on the punctured affine line. Much less is known about two-sided ideals of U(Vir), but intuition can be drawn both from affine Lie algebras and from results on Poisson ideals of the symmetric algebra of Vir.
Wolfgang Soergel (Freiburg, Germany)
Title: Six functors and their coherence
Abstract: I want to explain a somewhat down-to-earth formalism detailing in which way the various natural isomorphisms of the six functors formalism are compatible. I also want to discuss a natural setting for proper pushforward along "locally proper separated maps".
Shaobin Tan (Xiamen, China)
Title: Representations for extended affine Lie algebras and vertex algebras
Abstract: Extended affine Lie algebras are generalization of the finite dimensional Lie algebras and affine Kac-Moody algebras. The elliptic Lie algebras are extended affine Lie algebras of nullity two. We knew that the restricted modules for any untwisted affine Kac-Moody algebra are isomorphic to the modules for the associated affine vertex algebra, while the restricted modules for the twisted affine Kac-Moody algebra are isomorphic to the twisted modules for the affine vertex algebra. In this talk we will deal with the integrable representations of elliptic toroidal Lie algebras, and the vertex algebras associated with elliptic Lie algebras of maximal type.
Efim Zelmanov (SUSTech, China)
Title: On Growth and Complexity Functions
Abstract: I will discuss the history and recent results concerning (i) growth functions of groups, algebras, monoids, and languages and (ii) complexity functions of infinite sequences.
