The center of the category of bimodules and descent data for non-commutative rings

Abstract

Let $A$ be an algebra over a commutative ring $k$. We compute the center of the category of $A$-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical $A$-coring, Yetter-Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical $A$-coring $A \ot A$ is braided monoidal. We provide several applications: for instance, if $A$ is finitely generated projective over $k$ then the category of left End_k(A)$-modulesisbraidedmonoidalandwegiveanexplicitdescriptionofthebraidingintermsofthefinitedualbasisof$A$.Asanotherapplication,newfamiliesofsolutionsforthequantumYang-Baxterequationareconstructed:theyarecanonicalmaps$Ω$associatedtoanyrightcomoduleovertheSweedlercanonicalcoring$A \ot A$andsatisfythecondition$Ω^3 = Ω$. Explicit examples are provided.

arXiv comment14 pages; to appear in J. Algebra Appl

Cited by in corpus (1)

• The uniqueness of braidings on the monoidal category of non-commutative descent data