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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2012
- DOI:
- 10.48550/arXiv.1202.2057
- arXiv:
- arXiv:1202.2057
- Bibcode:
- 2012arXiv1202.2057M
- Keywords:
-
- Mathematics - Functional Analysis;
- Mathematics - General Topology;
- Mathematics - K-Theory and Homology;
- Mathematics - Operator Algebras;
- 22-XX;
- 46KXX;
- 22A22
- E-Print:
- 47 pages, minor corrections