The Brauer Group of a Braided Monoidal Category

Abstract

Representation theory of finite groups prompted the introduction of the Brauer group of a field and also of its Schur subgroups. The theory of quadratic forms and spaces introduced the Zr2Z-graded variant of the Brauer group together with the typical Clifford algebras and related w x representations; cf. the work of Wall 29 . The extension of the Zr2Zgraded theory to gradings by other cyclic groups and consequently to abelian groups, replacing the usual Clifford algebras by the so-called generalized Clifford algebras, is then natural from the algebraic point of view. However, extension to nonabelian groups did not seem to be possible, at least not by simple modifications of the abelian theory. At the same time, the generalization of the Brauer group of graded algebras obtained w x Ž . by Long 12, 13 made use of Hopf algebras but again cocommutativity conditions were present. The recent interest in quantum groups motivated the authors to introduce the Brauer group of a quantum group as the Ž Brauer group of crossed module algebras also called quantum Yang] . w x Baxter module algebras in 4, 5 . The use of the category of crossed w x modules originated in the case of group algebras; cf. Whitehead 30 . Ž . Module algebras or coalgebras of this crossed type were introduced by w x Radford in 24 and used by Majid in the description of modules over Drinfel’d quantum doubles. Ž . The noncocommutativity aspects of the theory concerning the Brauer group in terms of crossed module algebras do not have a counterpart in classical theory. Note that a first unifying categorical theory has been w x obtained by Pareigis in 23 ; this theory deals with the Brauer group of a