Forms of algebras stably equivalent to self-injective algebras of polynomial growth
Let \(K\) be an algebraically closed field. By an algebra we mean a finite dimensional \(K\)-algebra. Two algebras \(A\) and \(B\) are said to be stably equivalent if there is an equivalence of the factor categories \(\underline{\text{mod}}(A)\) and \(\underline{\text{mod}}(B)\) of finite dimensional right modules \(\text{mod}(A)\) and \(\text{mod}(B)\) modulo the ideal consisting of all homomorphisms factorizing through projective modules. One of the main results of the paper is the following theorem. Let \(A\) be a self-injective standard representation-infinite algebra which is of tame representation type and of polynomial growth. If \(B\) is an algebra stably equivalent to \(A\) then: (a) \(B\) is self-injective standard representation-infinite of tame representation type and of polynomial growth, or (b) there is a degeneration algebra \(C\) of \(B\) which is of tame representation type and of polynomial growth, the Auslander-Reiten quivers of \(B\) and \(C\) coincide, and there exists an equivalence of categories \(\underline{\text{mod}}(B)\cong\underline{\text{mod}}(C)\) preserving simple modules.
standard representation-infinite algebras
scientific article; zbMATH DE number 1218712
scientific article