The Brauer-Clifford Group for (S,H)-Azumaya Algebras over a Commutative Ring

Abstract

The Brauer-Clifford group BrClif(Z,G) corresponding to a finite group G and a finite-dimensional semisimple G-algebra Z was recently introduced by Alexandre Turull in the course of his work on character correspondence conjectures in group representation theory. This Brauer-Clifford group is a group of equivalence classes of Azumaya algebras over Z whose G-algebra structure agrees on restriction to the fixed (and usually nontrivial) G-algebra structure of Z. In this paper we extend the notion of the Brauer-Clifford group to the case of (S,H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is a commutative H-module algebra. These Brauer-Clifford groups turn out to be an example of the Brauer group of the symmetric monoidal category of S # H-modules, a perspective which allows one to construct a dual Brauer-Clifford group for the category of S-modules with compatible right H-comodule structure.