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

Abstract

Białkowski, Erdmann and Skowroński classified those indecomposable selfinjective 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 n via their Külshammer 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 Külshammer spaces we can distinguish many (but not all) deformations of the preprojective algebra of type L n up to stable equivalence of Morita type, and hence also up to derived equivalence.