Hopf Algebroids, Bimodule Connections and Noncommutative Geometry

Abstract

We construct new examples of left bialgebroids and Hopf algebroids, arising from noncommutative geometry. Given a first order differential calculus $\Omega$ on an algebra $A$, with the space of left vector fields $\mathfrak{X}$, we construct a left $A$-bialgeroid $B\mathfrak{X}$, whose category of left modules is isomorphic to the category of left bimodule connections over the calculus. When $\Omega$ is a pivotal bimodule, we construct a Hopf algebroid $H\mathfrak{X}$ over $A$, by restricting to a subcategory of bimodule connections which intertwine with both $\Omega$ and $\mathfrak{X}$ in a compatible manner. Assuming the space of 2-forms ${\Omega }^2$ is pivotal as well, we construct the corresponding Hopf algebroid $\mathcal{D}\mathfrak{X}$ for flat bimodule connections, and recover Lie-Rinehart Hopf algebroids as a quotient of our construction in the commutative case. We use these constructions to provide explicit examples of Hopf algebroids over noncommutative bases.

Minor corrections and typos fixed, Examples 4.8 and 4.18