Past Representation Theory Seminar

E.g., 2020-01-26
E.g., 2020-01-26
E.g., 2020-01-26
24 January 2013
14:00
Yuri Bazlov
Abstract
Central extensions of a finite group G correspond to 2-cocycles on G, which give rise to an abelian cohomology group known as the Schur multiplier of G. Recently, the Schur multiplier was defined in a much more general setting of a monoidal category. I will explain how to twist algebras by categorical 2-cocycles and will mention the role of such twists the theory of quantum groups. I will then describe an approach to twisting rational Cherednik algebras by cocycles, and will discuss possible applications of this new construction to the representation theory of these algebras.
  • Representation Theory Seminar
17 January 2013
14:00
Matthias Krebs
Abstract
It has been shown that the Auslander-Reiten-quiver of an indecomposable algebra contains a finite component if and only if A is representation finite. Moreover, selfinjective algebras are representation finite if and only if the tree types of the stable components are given by Dynkin Diagrams. I will present similar results for the Auslander-Reiten-quiver of a functorially finite resolving subcategory Ω. We will see that Brauer-Thrall 1 and Brauer-Thrall 1.5 can be proved for these categories with only little extra effort. Furthermore, a connection between sectional paths in A-mod and irreducible morphisms in Ω will be given. Finally, I will show how all finite Auslander-Reiten-quivers of A-mod or Ω are related to Dynkin Diagrams with a notion similar to the tree type that coincides in a finite stable component.
  • Representation Theory Seminar
29 November 2012
14:00
Alexandre Bouayad
Abstract
<p>We introduce a deformation process of universal enveloping algebras of Borcherds-Kac-Moody algebras, which generalises quantum groups' one and yields a large class of new algebras called coloured Borcherds-Kac-Moody algebras. The direction of deformation is specified by the choice of a collection of numbers. For example, the natural numbers lead to classical enveloping algebras, while the quantum numbers lead to quantum groups. We prove, in the finite type case, that every coloured BKM algebra have representations which deform representations of semisimple Lie algebras and whose characters are given by the Weyl formula. We prove, in the finite type case, that representations of two isogenic coloured BKM algebras can be interpolated by representations of a third coloured BKM algebra. In particular, we solve conjectures of Frenkel-Hernandez about the Langland duality between representations of quantum groups. We also establish a Langlands duality between representations of classical BKM algebras, extending results of Littelmann and McGerty, and we interpret this duality in terms of quantum interpolation.</p>
  • Representation Theory Seminar

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