Periodicity of self-injective algebras of polynomial growth

Abstract

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.