On selfinjective artin algebras having generalized standard quasitubes

Abstract

Dedicated to Daniel Simson on the occasion of his seventieth birthday MSC: 16D50; 16G10; 16G70 a b s t r a c t We give a complete description of the Morita equivalence classes of all connected selfinjective artin algebras for which the Auslander-Reiten quiver admits a family of quasitubes having common composition factors, closed under composition factors, and consisting of modules not lying on infinite short cycles.

Selfinjective algebras having a generalized standard family of quasi-tubes maximally saturated by simple and projective modules

“…Namely, every algebra Λ is a quotient algebra of a selfinjective algebra A with Γ A having a generalized standard stable tube (see [28], [29]). We refer to [6], [13], [14], [16] for some work on the structure of selfinjective algebras having generalized standard families of quasi-tubes.…”

On Auslander-Reiten components of algebras without external short paths

“…We also note that the problem of describing the self-injective algebras whose Auslander-Reiten quiver admits a stable tube without external short paths is more difficult, because the stable tubes occur in families of quasi-tubes. We refer to [18] for a wide class of self-injective algebras having infinitely many stable tubes without external short paths.…”

Self-injective artin algebras without short cycles in the component quiver

“…This will be a consequence of Propositions 4.5 and 4.6. Note that, by [18,Corollary 1.4], k is a field, and therefore A is a finite-dimensional algebra over a field.…”

“…Then, since the family C is closed under composition factors, we conclude that there is r ∈ Q such that C contains all quasi-tubes C A r,x , x ∈ X r , of C A r . This forces, by [18,Proposition 6.4], g to be of the form g = ϕν 2 B for some positive automorphism ϕ of B. Suppose that ϕ is a rigid automorphism of B.…”