Forthcoming Seminars › Seminar Series

Algebra Seminar

Tue, 27/04/2010
17:00 		Andrei Marcus (Cluj) Algebra Seminar Add to calendar L2
Hopf-Galois extensions and an exact sequence for H-Picard groups
The topic of this talk is the representation theory of Hopf-Galois extensions. We consider the following questions. Let H be a Hopf algebra, and A, B right H-comodule algebras. Assume that A and B are faithfully flat H-Galois extensions. 1. If A and B are Morita equivalent, does it follow that the subalgebras A^coH and B^coH of H-coinvariant elements are also Morita equivalent? 2. Conversely, if A^coH and B^coH are Morita equivalent, when does it follow that A and B are Morita equivalent? As an application, we investigate H-Morita autoequivalences of the H-Galois extension A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and the Picard group of A^coH.(joint work with Stefaan Caenepeel)
Tue, 11/05/2010
17:00 		Peter Cameron (Queen Mary University) Algebra Seminar Add to calendar L2
Power graphs of groups and other magmas
Tue, 18/05/2010
17:00 		Martin Bridson (Oxford) Algebra Seminar Add to calendar L2
Actions of higher-rank lattices on free groups
Tue, 25/05/2010
17:00 		Emmanuel Breuillard (Université de Paris-Sud, Orsay) Algebra Seminar Add to calendar L2
Variations on the Tits alternative(s)
Tue, 01/06/2010
17:00 		Peter Jorgensen (Newcastle) Algebra Seminar Add to calendar L2
The cluster category of Dynkin type $ A_\infty $
   The cluster category of Dynkin type $ A_\infty $ is a ubiquitous object with interesting properties, some of which will be explained in this talk.
   Let us denote the category by $ \mathcal{D} $. Then $ \mathcal{D} $ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $ D^c(C^{∗}(S^2;k)) $ of the singular cochain algebra $ C^*(S^2;k) $ of the 2-sphere $ S^{2} $. There is also a “universal” definition: $ \mathcal{D} $ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.
  Just like cluster categories of finite quivers, $ \mathcal{D} $ has many cluster tilting subcategories, with the crucial difference that in $ \mathcal{D} $, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.
   The talk will show how the cluster tilting subcategories have a rich combinatorial structure: They can be parametrised by “triangulations of the $ \infty $-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.
  This will be used to show how to obtain a subcategory of $ \mathcal{D} $ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $ \mathcal{D} $ generalises the situation from Dynkin type $ A_n $ , and how triangulations of the $ \infty $-gon are new and interesting combinatorial objects.
Tue, 08/06/2010
17:00 		Yiftach Barnea (Royal Holloway) Algebra Seminar Add to calendar L2
Abstract Commensurators of Profinite Groups
Tue, 15/06/2010
17:00 		Detlev Hoffmann (Nottingham) Algebra Seminar Add to calendar L2
Bilinear Forms and Differential Forms under Field Extensions
An important problem in algebra is the study of algebraic objects defined over fields and how they behave under field extensions, for example the Brauer group of a field, Galois cohomology groups over fields, Milnor K-theory of a field, or the Witt ring of bilinear forms over a field. Of particular interest is the determination of the kernel of the restriction map when passing to a field extension. We will give an overview over some known results concerning the kernel of the restriction map from the Witt ring of a field to the Witt ring of an extension field. Over fields of characteristic not two, general results are rather sparse. In characteristic two, we have a much more complete picture. In this talk, I will explain the full solution to this problem for extensions that are given by function fields of hypersurfaces over fields of characteristic two. An important tool is the study of the behaviour of differential forms over fields of positive characteristic under field extensions. The result for Witt rings in characteristic two then follows by applying earlier results by Kato, Aravire-Baeza, and Laghribi. This is joint work with Andrew Dolphin.

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