Tame Minimal Non-Polynomial Growth Simply Connected Algebras

this article is to introduce and classify (by quivers and relations) a class of tame minimal non-polynomial growth simply connected algebras, which we call (generalized) pg-critical algebras. Moreover we describe basic properties of pg-critical algebras. In particular, we prove that the Euler (respectively Tits) quadratic form of any pg-critical algebra A is positive semi-definite of corank 2 and the eigenvalues of the Coxeter transformation of A are roots of unity. We also describe completely the structure of non-regular connected components of the Auslander-Reiten quivers of pg-critical algebras. The paper is organized as follows. In section 2 we fix the notations and recall the definitions needed. In section 3 we introduce the polynomial growth critical algebras and classify them by quivers and relations. In particular, we prove that all such algebras are simply connected. In section 4 we determine the tilting classes of polynomial growth critical algebras and show that the Euler form of any such algebra is positive semi-definite with radical of rank 2. Section 5 is devoted to the spectral properties of polynomial growth critical algebras. We determine the Coxeter polynomial of any polynomial growth critical algebra and show that the spectral radius of its Coxeter matrix is 1. In the final section 6 we investigate the module category of polynomial growth critical algebras. Here, we describe the structure of all non-regular components of their Auslander Reiten quivers and discuss the behaviour of non-regular components in the category of indecomposable modules. The authors gratefully acknowledge support from the Polish Scientific Grand KBN No. 2 PO3A 020 08 and the Sonderforschungsbereich 343 (Universit at Bielefeld). 2 Preliminarie