Towards differentiation and integration between Hopf algebroids and Lie algebroids

TL;DR

This paper develops foundational functors linking commutative Hopf algebroids and Lie-Rinehart algebras, establishing a formal differentiation and integration framework that parallels geometric concepts in Lie theory.

Contribution

It constructs and analyzes differentiation and integration functors between Hopf algebroids and Lie-Rinehart algebras, including adjunctions and conditions for Galois structures.

Findings

A contravariant differentiation functor from Hopf algebroids to Lie-Rinehart algebras.

Two integration functors from Lie-Rinehart algebras to Hopf algebroids, forming adjunctions.

Applications to geometric separability and examples illustrating the theory.

Abstract

In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.