Existence of Gradings on Associative Algebras

TL;DR

This paper characterizes when finite dimensional associative algebras admit non-trivial gradings, linking algebraic properties to automorphism groups, and applies results to blocks of group algebras with specific defect groups.

Contribution

It provides a complete criterion for the existence of non-trivial gradings on connected algebras, connecting algebra structure, quiver properties, and automorphism groups.

Findings

Connected algebra has no non-trivial grading iff it is basic with one vertex and unipotent outer automorphism group

No non-trivial gradings exist on blocks of group algebras with quaternion defect groups and one simple module

Characterization links algebra structure to automorphism group properties

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.