Graded simple modules and loop modules

Source abstract

Necessary and sufficient conditions are given for a G G -graded simple module over a unital associative algebra, graded by an abelian group G G , to be isomorphic to a loop module of a simple module, as well as for two such loop modules (associated to a subgroup H H of G G ) 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.