Support varieties and representation type of self-injective 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 [7]: 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 in [23]. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.

Erratum to “Support Varieties and Representation Type of Small Quantum Groups”

“…We note that our results in [4] almost exclusively use the Hochschild support variety theory, and do not require any additional hypotheses. This theory is more general, and does not take advantage of the tensor products of modules one has at hand for a Hopf algebra.…”

“…However, it does not necessarily factor through its action on Ext The proof of [3, Theorem 4.3] is incorrect, or at least incomplete; the Hopf algebras u + q (g) are not in general quasitriangular. However, Hochschild support variety theory [2,4] provides an alternative proof of this statement: By [4,Lemma 6.3] or [5, §2], the representation type of u + q (g) is the same as that of u >0 q (g), and we may apply [4,Theorem 5.4] to the latter algebra. The remainder of the proofs of the statements in [3,Section 4] are correct, as the Hopf algebras u q (g) are quasitriangular.…”

Auslander-Reiten theory of small half quantum groups

“…of the form Z[A ∞ ]/τ m for some m ∈ N). The case of finite components does not occur in our context since finite components correspond to representation-finite blocks by a classical theorem of Auslander, these do not appear as shown in [FW09,FW11].…”

“…In 1990 Lusztig defined a quantum analogue of the restricted enveloping algebra, called the small quantum group. Its Borel and nilpotent parts were shown to have wild representation type by Feldvoss and Witherspoon for g sl 2 in [FW09,FW11] (a generalization of a result by Cibils [Cib97]). In this paper we give an analogue of Erdmann's Theorem and prove: ζ (g)-module.…”

“…In [FW09,FW11] Feldvoss and Witherspoon have shown that for g sl 2 the connected algebras u ζ (b) and u ζ (n) are wild.…”