Existence of Gradings on Associative Algebras

Abstract

In this paper we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra $A$ does not have a non-trivial grading if and only if $A$ is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist non-trivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.

We give necessary and sufficient conditions for an associative finite dimensional algebra not to possess a non-trivial Z-grading. arXiv admin note: text overlap with arXiv:1001.1643