Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvature

TL;DR

This paper generalizes a theorem about crossed products from Hopf algebras to Hopf algebroids and applies it to universal enveloping algebras of Lie-Rinehart algebras, linking algebraic structures with geometric examples.

Contribution

It extends the theorem to left Hopf algebroids and provides a new decomposition of universal enveloping algebras based on connections.

Findings

Universal enveloping algebras decompose as crossed or smash products depending on connections.

Application to invariant vector fields on principal bundles as crossed products.

Generalization of the Blattner-Cohen-Montgomery theorem to Hopf algebroids.

Abstract

We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connection, that is, a linear (resp. Lie-Rinehart) splitting of a Lie-Rinehart algebra extension, we provide a crossed (resp. smash) product decomposition of the associated universal enveloping algebra, and vice versa. As a geometric example, we describe the associative algebra generated by the invariant vector fields on the total space of a principal bundle as a crossed product of the algebra generated by the vertical ones and the algebra of differential operators on the base.