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

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.