A Semi-equivariant Dixmier-Douady Invariant

TL;DR

This paper introduces a semi-equivariant Dixmier-Douady invariant as a second-degree cohomology class in a new cohomology theory, aiding the study of semi-equivariant principal bundles and complex vector bundles with linear or anti-linear actions.

Contribution

It develops a novel semi-equivariant cohomology framework and constructs a generalized Dixmier-Douady invariant for semi-equivariant principal bundles.

Findings

Constructed a semi-equivariant Dixmier-Douady invariant.

Applied the invariant to complex vector bundles with linear/anti-linear actions.

Provided tools for studying liftings of semi-equivariant principal bundles.

Abstract

A generalisation of the equivariant Dixmier-Douady invariant is constructed as a second-degree cohomology class within a new semi-equivariant \v{C}ech cohomology theory. This invariant obstructs liftings of semi-equivariant principal bundles that are associated to central exact sequences of structure groups in which each structure group is acted on by the equivariance group. The results and methods described can be applied to the study of complex vector bundles equipped with linear/anti-linear actions, such as Atiyah's Real vector bundles.