Graded simple modules and loop modules

TL;DR

This paper establishes criteria for when a G-graded simple module can be viewed as a loop module of a simple module, explores their isomorphisms, and extends invariants like inertia group and Schur index to a broader context.

Contribution

It provides necessary and sufficient conditions for G-graded simple modules to be loop modules and generalizes invariants like the inertia group and Schur index beyond previous settings.

Findings

Criteria for isomorphism to loop modules

Conditions for complete reducibility of loop modules

Extension of invariants to a more general setting

Abstract

Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be isomorphic to each other. Under some restrictions, these loop modules are completely reducible (as ungraded modules), and some of their invariants --- inertia group, graded Brauer invariant and Schur index --- which were previously defined for simple modules over graded finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero, are now considered in a more general and natural setting.