Support Varieties and Representation Type of Small Quantum Groups

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.

Auslander-Reiten theory of Frobenius-Lusztig kernels

“…In 2009, Drupieski [13] gave a generalization to positive characteristic that gives also quantum analogues of higher Frobenius kernels, which he called (higher) Frobenius-Lusztig kernels. Building on results by Feldvoss and Witherspoon [30] in a recent paper [39], the author has shown that the non-simple blocks of a Frobenius-Lusztig kernel are of wild representation type in all but two cases. Therefore, the class of Frobenius-Lusztig kernels can also serve as an example for understanding the Auslander-Reiten theory of wild self-injective algebras.…”

“…The proofs for the group algebra case (see, for example, [6]) generalize without much difficulty. Some of them may be found in [10] or [30]. Proposition 1.1.…”

Varieties for Modules of Finite Dimensional Hopf Algebras

“…If M is an A-module, consider Ext * A (M, M) to be an H * (A, k)-module via − ⊗ M followed by Yoneda composition. We make the following assumptions, as in [23]:…”

“…Applications of varieties for modules abound, and are well developed for some of the classes of Hopf algebras described above. For example, one can construct modules with prescribed support (see [2,7,23,28]). Representation type can be seen in the varieties (see [22,23,33]).…”

Examples of support varieties for Hopf algebras with noncommutative tensor products

“…Our definition of support variety is equivalent to that of Snashall and Solberg [8] since we have assumed G is a p-group. It differs from that of Feldvoss and the second author [5], since the cohomology H * (A, k) := Ext * A (k, k) is isomorphic to H * (G, k), and not to H * (G, k)⊗k [L]. It has an advantage over the latter in that it remembers more information about an A-module.…”