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G. Bobinski also wrote up some reports about the seminar lectures from November 2007 until September 2009.
Hall algebras associated to triangulated categories with homological finiteness have been constructed as an anologue of Ringel-Hall algebras for abelian categories. In this talk, we recall it and define a new type of Hall algebras from triangulated categories with odd periodic translation. In particular, we characterize the Hall algebra from the 3-periodic orbit category of the derived category of a hereditary abelian category.
Preprojective algebras and Kac-Moody Lie algebras are just two sides of the same coin. In an extensive project with Geiss and Leclerc we are trying to transfer information from one side to the other.
Associated to any reduced expression of an element in the Weyl group of the Kac-Moody algebra one can associate a set of generalized minors on the Lie Theory side, and a set of indecomposable rigid modules on the preprojective side. These module are uniquely filtered by bricks. (A brick is a module without selfextensions having trivial endomorphism ring.) We are going to explain some basic properties of these bricks and we will explain their role in the categorification of certain cluster algebras.
Using dimer models, first studied in physics, we can produce non-commutative crepant resolutions (NCCRs) of all Gorenstein 3-fold affine toric singularities. In this talk I will introduce via examples, dimers and their corresponding toric varieties. I will then talk about a ‘consistency’ condition that underlies the key step in the proof of the NCCR property.
My talk complements the series of talks by Dirk Kussin on the relationship between weighted projective lines of type (2,3,p) and the invariant subspace problem for graded nilpotent operators of degree p.
We highlight the role of pattern recognition
We will illustrate the pattern recognition process, by using Ringel's and Schmidmeier's paper “Invariant subspaces of nilpotent linear operators” as a starting point.
We will further point out failing and successful proof strategies, when trying to link nilpotent operators to weighted projective lines, and in this context also discuss and evaluate the largely complementary role of the two major tasks:
We establish a close link between the category of vector bundles over a weighted projective line of weight type (2,3,p) and the invariant subspace category of linear operators of nilpotency index p studied by Ringel and Schmidmeier.
We work over a field k of prime characteristic p and consider the algebraic group Gn=Aut(k[x1,…,xn]/(x1p,…,xnp)). The aim of the talk is to give a classification and computation of all simple Gn-representations. For the classification we will establish a 1–1 correspondence between the simple Gn-representations and the simple GLn-representations. This correspondence uses the fact, that GLn is canonically a closed subgroup of Gn, and it is inspired by the machinery for classifying simple representations of reductive groups. Then we will give a complete computation of the simple Gn-representations with respect to their associated GLn-representations.
We work over an infinite field K of arbitrary characteristic. Given a quiver Q, we denote by I(Q,n) (SI(Q,n), respectively) the algebra of invariants (semi-invariants, respectively) of representations of Q for dimension vector n. The generators for SI(Q,n) are known. In this talk we will explicitly describe a minimal (by inclusion) generation system for SI(Q,n) for n=(2,…,2).
Note that the known generating system for I(Q,n) is “simpler” than that for SI(Q,n). Moreover, the ideal of relations between generators for invariants is known in contrast to semi-invariants. Nevertheless, a minimal generating system for I(Q,n) is not known and it is unclear how it can be explicitly described.