Tannaka theory and the FRT construction over non-commutative algebras

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.

v2, 56 pages, 118 picture files. Fully revised. The invertibility of the lax braiding of FRT bialgebroids is discussed. A relation to Hayashi's construction of face algebras is added

References in corpus (1)

• Fiber functors, monoidal sites and Tannaka duality for bialgebroids

Cited by in corpus (3)

• Skew Braces as Remnants of Co-quasitriangular Hopf Algebras in $\mathrm{SupLat}$
• Pivotal Objects in Monoidal Categories and Their Hopf Monads
• Isotopy Quotients of Hopf Algebroids and the Fundamental Groupoid of Digraphs