Abstract. We classify (up to Morita equivalence) all symmetric special biserial algebras of Euclidean type, by algebras arising from Brauer graphs.Introduction and the main result. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finitedimensional K-algebra with identity, which we shall assume (without loss of generality) to be basic and connected. For an algebra A, we denote by mod A the category of finite-dimensional right A-modules and by D the standard duality Hom K (−, K) on mod A. The Cartan matrix C A of A is the matrix (dim K Hom A (P i , P j )) 1≤i,j≤n for a complete family P 1 , . . . , P n of pairwise nonisomorphic indecomposable projective A-modules.An algebra A is called selfinjective if A ∼ = D(A) in mod A, that is, the projective A-modules are injective. Further, A is called symmetric if A and D(A) are isomorphic as A-bimodules. For a selfinjective algebra A, we denote by Γ s A the stable Auslander-Reiten quiver of A, obtained from the Auslander-Reiten quiver Γ A of A by removing all projective modules and the arrows attached to them. We also note that if A is symmetric then the Auslander-Reiten translation τ A = D Tr in mod A is the square Ω 2 A of the Heller syzygy operator Ω A . An important class of selfinjective algebras is formed by the algebras of the form B/G, where B is the repetitive algebra [8] (locally finite-dimensional, without identity)
We give a complete derived equivalence classiÿcation of all weakly symmetric algebras of Euclidean type. As a consequence, a complete stable equivalence classiÿcation of these algebras is obtained.
A finite dimensional algebra A over an algebraically closed field is called a selfinjective algebra of Euclidean type if A is the orbit algebraB B=G, whereB B is the repetitive algebra of a tilted algebra B of Euclidean type and G is an admissible group of automorphisms ofB B. It is known that the class of selfinjective algebras of Euclidean type coincides with the class of tame selfinjective algebras having simply connected Galois coverings and a finite (nonempty) family of generic modules. We classify all weakly symmetric algebras of Euclidean type. Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/28/15 12:39 PM Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/28/15 12:39 PM Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/28/15 12:39 PM Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/28/15 12:39 PM
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