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

TL;DR

This paper characterizes the center of the category of bimodules over a non-commutative algebra, revealing its multiple equivalent descriptions and applications to braided monoidal categories and solutions to the quantum Yang-Baxter equation.

Contribution

It provides six isomorphic descriptions of the center of bimodule categories, connecting noncommutative descent data, corings, and differential geometry, with applications to quantum algebra.

Findings

The center equals the weak center and can be described via six isomorphic categories.

The category of comodules over the Sweedler canonical coring is braided monoidal.

New solutions to the quantum Yang-Baxter equation are constructed from coring comodules.

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)$-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of $A$. As another application, new families of solutions for the quantum Yang-Baxter equation are constructed: they are canonical maps $\Omega$ associated to any right comodule over the Sweedler canonical coring $A \ot A$ and satisfy the condition $\Omega^3 = \Omega$. Explicit examples are provided.