Support Varieties and Representation Type of Self-Injective Algebras

TL;DR

This paper develops a new criterion for determining when a finite-dimensional self-injective algebra is wild, based on module complexity and Hochschild cohomology, with applications to various algebra types.

Contribution

It introduces an alternative wildness criterion using Hochschild cohomology and module varieties, extending the theory to Hopf algebras and diverse algebra classes.

Findings

Algebra is wild if a module has complexity ≥ 3 and certain Hochschild cohomology conditions.

Established connections between module varieties and algebra representation type.

Applied the criterion to Hecke algebras, quantum groups, and Nichols algebras.

Abstract

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.