71st Algebra Day – April 6, 2013

From Yangians to quantum loop algebras via abelian difference equations

From Yangians to quantum loop algebras via abelian difference equations

Abstract: The finite-dimensional representations of the Yangian Y_h(g) and quantum loop algebra U_q(Lg) of a complex, semisimple Lie algebra have long been known to share many similar features. Assuming that q is not a root of unity, I will explain how to construct an equivalence of categories between finite-dimensional representations of U_q(Lg) and an appropriate subcategory of finite-dimensional representations of Y_h(g). This equivalence is governed by the monodromy of an additive, abelian difference equation. This is joint work with Sachin Gautam.
 

On the cluster category of a marked surface

On the cluster category of a marked surface

Abstract: The main character of this talk is the cluster category C(S,M) of a marked surface (S,M). In the puncture-free case, we give an explicit description of its objects, and we relate objects without self-extensions in C(S,M) to curves in (S,M) without self-intersections. Moreover, and even if there are punctures, we describe the suspension functor of C(S,M) geometrically, as an element in the marked mapping class group of (S,M).
 

Topological properties of torsors and homogeneous spaces over valued fields

Topological properties of torsors and homogeneous spaces over valued fields

Abstract: This is a report on work in progress with Ofer Gabber and Laurent Moret-Bailly. Let K be the fraction field of an henselian ring of positive characteristic p, e.g. Fp((t)). Let Y be a K-variety, H an algebraic group over K, and f: X -> Y an H -torsor over Y. We consider the induced map X(K) -> Y(K) which is continuous for the topologies deduced from the valuation. If I denotes the image of this map, we investigate the following questions:
(a) Is I locally closed (resp. closed) in Y(K) ?
(b) Is the continuous bijection X(K)/H(K) -> I a homeomorphism ?