Equivariant Brauer groups and braided bi-Galois objects

Abstract

Introduction excerpt: The definition of the Brauer group of a field, introduced by Richard Brauer, goes back to 1929. The Brauer group of a field is an abelian group classifying central simple algebras. The Brauer group of a field was generalized to the Brauer group of a commutative ring by Auslander and Goldman in 1960. The Brauer group of a commutative ring consists of equivalence classes of central seperable algebras. Central separable algebras are also called Azumaya algebras. Since then, many generalizations have been made in several directions. For example, Wall introduced the Brauer group of Z2-graded algebras, called the Brauer-Wall group, which in his turn has been generalized to gradings by other groups. Another generalization is the so-called Brauer-Long group, that is the Brauer group of dimodule algebras over a finitely generated, projective, commutative and cocommutative Hopf algebra, introduced by Long. A dimodule is simultaneously a module and a comodule satisfying a certain compatibility condition. One can note that the Brauer-Wall group can be seen as a subgroup of the Brauer-Long group over the group Hopf algebra kZ2. Caenepeel, Van Oystaeyen and Zhang have generalized the construction by Long and introduced the Brauer group of Yetter-Drinfel’d module algebras, hereby disposing the restriction on the Hopf algebra to be finitely generated, projective, commutative and cocommutative. The only requirement is that the antipode is bijective. All the variations of Brauer groups mentioned above can be seen as particular cases of the Brauer group of a braided monoidal category, which is introduced by Van Oystaeyen and Zhang in 1998. Note that earlier, before the concept of a braided monoidal category came to life, Pareigis had introduced the Brauer group of a symmetric monoidal category.

De Brauer groep van een veld is een abelse groep, bestaande uit equivalentieklassen van centrale enkelvoudige algebras. De groep is ge¨ıntroduceerd in 1929, maar kent ondertussen verschillende veralgemeningen. De Brauer groep van een braided mono¨ıdale categorie, ge¨ıntroduceerd door Van Oystaeyen en Zhang, omvat bijna alle veralgemeningen. Vaak probeert men Brauer groepen te beschrijven met behulp van exacte rijen. Met name de groep van (bi-)Galois objecten keert regelmatig terug in de berekening van een Brauer groep. Dit zullen de twee voornaamste ingredi¨enten vormen in deze thesis: Brauer groepen en (braided) bi-Galois objecten.