Radical Cube Zero Selfinjective Algebras of Finite Complexity

One of our main results is a classification all the possible quivers of selfinjective radical cube zero finite dimensional algebras over an algebraically closed field having finite complexity. In the paper [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5]. Introduction This paper is a companion of [5], where all radical cube zero weakly symmetric algebras with support varieties via the Hochschild cohomology satisfying Dade’s Lemma were classified. In this paper we go half way with all selfinjective algebras with radical cube zero, in that we classify which of these have finite complexity. This is half way for the following reasons. To get a theory of support using the Hochschild cohomology ring satisfying Dade’s Lemma, for any known proof of this the Ext-algebra of all the simple modules must be a finitely generated module over the Hochschild cohomology ring, which in turn needs to be Noetherian. Denote this property by (Fg). By [4] a finite dimensional algebra satisfying (Fg) must have finite complexity. In addition the trichotomy into finite, tame and wild representation type is characterized in two different ways as (i) λmax < 2, λmax = 2 and λmax > 2 and (ii) complexity of the algebra is 1, 2 or ∞, respectively, where λmax is the eigenvalue of largest absolute value for the adjacency matrix of the algebra (λmax is a positive real number). A selfinjective algebra Λ with radical cube zero over an algebraically closed field is either of finite or infinite representation type. If Λ has finite representation type, then it satisfies (Fg) by [3]. If Λ has infinite representation type, then it is a Koszul algebra by [7, 8]. Using the results of [5] Λ has (Fg) if and only if the Koszul dual of Λ is a finitely generated module over the graded centre of the Koszul dual and this is a Noetherian ring. This was the key argument in [5], where the results were obtained through explicit calculations case by case. This approach is still available for selfinjective algebras with radical cube zero, however it seems to us that it is an almost new game to treat this class of algebras. And, as our results show, this class is seemingly much more complex than the weakly symmetric algebras with Date: September 10, 2010. 2010 Mathematics Subject Classification. 16P10, 16G20, 16L60, 16E05, 16P90; Secondary: 16S37.