Graded Brauer groups of a groupoid with involution

TL;DR

This paper introduces a new group $RBr(G)$ that generalizes existing graded Brauer groups for groupoids with involution, linking it to Real cohomology and extensions, and providing a cohomological framework.

Contribution

It defines the group $RBr(G)$ for groupoids with involution, generalizing and connecting graded Brauer groups with Real cohomology and extensions.

Findings

$RBr(G)$ generalizes graded orthogonal Brauer groups.

$RBr(G)$ is a direct summand of the graded complex Brauer group.

$RBr(G)$ decomposes into Real cohomology and $RExt(G,U(1))$.

Abstract

We define a group $RBr(\mathcal{G})$ containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid $\mathcal{G}$ equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalization of Donovan-Karoubi's graded orthogonal Brauer group $GBrO$. More generally, it is shown to be a direct summand of the well-known graded complex Brauer goup. In addition, we prove that $RBr(\mathcal{G})$ identifies with a direct sum of a Real cohomology group and the abelian group $RExt(\mathcal{G},U(1))$ of Real graded $U(1)$-central extensions. A cohomological picture is then given.