Dynkin diagrams, support spaces and


This survey article is an expanded version of a series of lectures given at the conference on Advances in Group Theory and its Applications which was held in Porto Cesareo in June of 2011. We are concerned with representations of finite group schemes, a class of objects that generalizes the more familiar finite groups. In the last 30 years, this discipline has enjoyed considerable attention. One reason is the application of geometric techniques that originate in Quillen’s fundamental work concerning the spectrum of the cohomology ring v[25, 26]. The subsequent developments pertaining to cohomological support varieties and representation-theoretic support spaces have resulted in many interesting applications. Here we will focus on those aspects of the theory that are motivated by the problem of classifying indecomposable modules. Since the determination of the simple modules is often already difficult enough, one can in general not hope to solve this problem in a naive sense. However, the classification problem has resulted in an important subdivision of the category of algebras, which will be our general theme. The algebras we shall be interested in are the so-called cocommutative Hopf algebras, which are natural generalizations of group algebras of finite groups. The module categories of these algebras are richer than those of arbitrary algebras:

• They afford tensor products which occasionally allow the transfer of information between various blocks of the algebra.

• Their cohomology rings are finitely generated, making geometric methods amenable to application. The purpose of these notes is to illustrate how a combination of these features with methods from the abstract representation theory of algebras and quivers provides insight into classical questions.

DOI Code: 10.1285/i15900932v33n1p49

Keywords: Dynkin diagram; support variety; representation type; small quantum groups

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