Derived equivalence classification of symmetric algebras of polynomial growth

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.Comment: 11 page

Deformed preprojective algebras of type L: Kuelshammer spaces and derived equivalences

Bialkowski, Erdmann and Skowronski classified those indecomposable self-injective algebras for which the Nakayama shift of every (non-projective) simple module is isomorphic to its third syzygy. It turned out that these are precisely the deformations, in a suitable sense, of preprojective algebras associated to the simply laced ADE Dynkin diagrams and of another graph L_n, which also occurs in the Happel-Preiser-Ringel classification of subadditive but not additive functions. In this paper we study these deformed preprojective algebras of type L via their Kuelshammer spaces, for which we give precise formulae for their dimensions. These are known to be invariants of the derived module category, and even invariants under stable equivalences of Morita type. As main application of our study of Kuelshammer spaces we can distinguish many (but not all) deformations of the preprojective algebra of type L up to stable equivalence of Morita type, and hence also up to derived equivalence.Comment: 24 page

Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

It is an important aspect of cluster theory that cluster categories are "categorifications" of cluster algebras. This is expressed formally by the (original) Caldero-Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras. Let \tau c --> b --> c be an Auslander-Reiten triangle. The map X has the salient property that X(\tau c)X(c) - X(b) = 1. This is part of the definition of a so-called frieze. The construction of X depends on a cluster tilting object. In a previous paper, we introduced a modified Caldero-Chapoton map \rho depending on a rigid object; these are more general than cluster tilting objects. The map \rho sends objects of sufficiently nice triangulated categories to integers and has the key property that \rho(\tau c)\rho(c) - \rho(b) is 0 or 1. This is part of the definition of what we call a generalised frieze. Here we develop the theory further by constructing a modified Caldero-Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A. We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers. The new map is a proper generalisation of the maps X and \rho.Comment: 16 pages; final accepted version to appear in Bulletin des Sciences Math\'ematique