Stable Equivalence of Selfinjective Artin Algebras of Dynkin Type

Abstract

We prove new results on the stable equivalences of selfinjective Artin algebras of tilted Dynkin type, extending the main results of Skowroński and Yamagata to arbitrary tilted type.

Selfinjective algebras without short cycles of indecomposable modules

“…Equivalently, Riedtmann's classification can be presented as follows (see [21,Section 3]): a nonsimple selfinjective algebra A over an algebraically closed field K is of finite representation type if and only if A is a socle (geometric) deformation of an orbit algebra B/G, where B is the repetitive category of a tilted algebra B of Dynkin type A n (n 1), D n (n 4), E 6 , E 7 , E 8 , and G is an admissible infinite cyclic group of automorphisms of B. For an arbitrary field K, the problem seems to be difficult (see [3], [4], [13], [26] for some results in this direction and [27,Section 12] for related open problems). An important known result towards solution of this general problem is the description of the stable Auslander-Reiten quiver Γ s A of a selfinjective algebra A of finite representation type established by C. Riedtmann [18] and G. Todorov [31] (see also [28, Section IV.15]): Γ s A is isomorphic to the orbit quiver Z∆/G, where ∆ is a Dynkin quiver of type A n (n 1), B n (n 2), C n (n 3), D n (n 4), E 6 , E 7 , E 8 , F 4 or G 2 , and G is an admissible infinite cyclic group of automorphisms of the translation quiver Z∆.…”

Self-injective algebras with hereditary stable slice

“…In particular, it was shown that every admissible group G of the repetitive category B of a tilted algebra B is an infinite cyclic group generated by a strictly positive automorphism of B. In the series of articles [33,34,35,37,38,39] we developed the theory of self-injective algebras with deforming ideals and established necessary and sufficient conditions for a self-injective algebra A to be socle equivalent to an orbit algebra B/G, for an algebra B and an infinite cyclic group G generated by a strictly positive automorphism of B being the composition ϕν B of the Nakayama automorphism ν B of B and a positive automorphism ϕ of B.…”

“…We also mention that the class of self-injective algebras occurring in the statement (ii) is closed under stable, and hence derived, equivalences (see [24,34,39] and [26]).…”

On selfinjective algebras of finite representation type with maximal almost split sequences

“…For K algebraically closed, the problem was solved in the early 1980's by C. Riedtmann (see [7,[20][21][22]) via the combinatorial classification of the Auslander-Reiten quivers of selfinjective algebras of finite representation type over K. Equivalently, Riedtmann's classification can be presented as follows (see [26,Section 3]): a nonsimple selfinjective algebra A over an algebraically closed field K is of finite representation type if and only if A is socle equivalent to an orbit algebra B/G, where B is the repetitive category of a tilted algebra B of Dynkin type A n (n ≥ 1), D n (n ≥ 4), E 6 , E 7 , E 8 , and G is an admissible infinite cyclic group of automorphisms of B. For an arbitrary field K, the problem seems to be difficult (see [5,33] for some results in this direction and [34,Section 12] for related open problems). An important known result towards solution of this general problem is the description of the stable AuslanderReiten quiver Γ s A of a selfinjective algebra of finite representation type established by C. Riedtmann [20] and G. Todorov [36] (see also [35,Section IV.15]): Γ s A is isomorphic to the orbit quiver ZΔ/G, where Δ is a Dynkin quiver of type A n (n ≥ 1), B n (n ≥ 2), C n (n ≥ 3), D n (n ≥ 4), E 6 , E 7 , E 8 , F 4 , or G 2 , and G is an admissible infinite cyclic group of automorphisms of the translation quiver ZΔ.…”