Support varieties and representation type of small quantum groups

TL;DR

This paper establishes a criterion for the wildness of finite dimensional Hopf algebras using support varieties, and confirms that small quantum groups of rank at least two are wild, extending previous results to broader contexts.

Contribution

It generalizes wildness criteria to non-cocommutative Hopf algebras and proves the wildness of small quantum groups of rank at least two.

Findings

Support varieties provide a criterion for algebra wildness.

Small quantum groups of rank ≥ 2 are proven to be wild.

The approach extends previous results to more general Hopf algebras.

Abstract

In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild.