Tannaka theory and the FRT construction over non-commutative algebras

TL;DR

This paper extends Tannaka theory to non-commutative algebras by introducing coquasitriangular bialgebroids, establishing a braiding in their comodules, and applying this to generalize the FRT construction.

Contribution

It introduces coquasitriangular bialgebroids over non-commutative algebras and develops a Tannaka type construction for them, including a Hopf algebroid version.

Findings

Category of comodules has a braiding due to coquasitriangular structure

Constructs bialgebroids from braided objects in bimodule categories

Generalizes FRT construction to non-commutative algebra setting

Abstract

Let $A$ be an algebra over a commutative ring $k$. We introduce the notion of a coquasitriangular left bialgebroid over $A$ and show that the category of left comodules over such a bialgebroid has a braiding. We also investigate a Tannaka type construction of bimonads and bialgebroids. As an application, the Faddeev-Reshetikhin-Takhtajan (FRT) construction over the algebra $A$ is established. Our construction associates a coquasitriangular bialgebroid to a braided object $(M, c)$ in the category of $A$-bimodules such that $M$ is finitely generated and projective as a left $A$-module. A Hopf algebroid version of this construction is also provided.