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

TL;DR

This paper investigates deformed preprojective algebras of type L, analyzing their Kuelshammer spaces to distinguish various deformations up to derived and stable equivalences, advancing understanding of their invariants.

Contribution

It provides explicit formulas for Kuelshammer space dimensions of these algebras and uses these invariants to distinguish many deformations up to derived equivalence.

Findings

Kuelshammer spaces are invariants under stable equivalences of Morita type.

Explicit formulas for Kuelshammer space dimensions are derived.

Many deformations of preprojective algebras of type L are distinguishable up to derived equivalence.

Abstract

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.