Graded Brauer groups of a groupoid with involution

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.