Mackey's obstruction map for discrete graded algebras

TL;DR

This paper extends Mackey's obstruction theory to discrete, non-saturated graded algebras, defining a cohomological obstruction map that classifies simple graded modules and relates to the graded Artin-Wedderburn decomposition.

Contribution

It introduces an obstruction map for discrete, non-saturated graded algebras, generalizing Mackey's theory and connecting it to the structure of graded simple algebras.

Findings

Obstruction map assigns second cohomology classes to simple graded modules.

The map is a homomorphism of abelian monoids under graded product.

The obstruction class influences the graded Artin-Wedderburn decomposition.

Abstract

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles. Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely simple graded modules. The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules. The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.