A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra

TL;DR

This paper explores the properties of the restricted universal enveloping algebra of restricted Lie-Rinehart algebras, providing an alternative proof and addressing potential gaps in the existing theory.

Contribution

It offers a new proof regarding the finite generation and projectivity of restricted universal enveloping algebras of Lie algebroids, expanding understanding in this area.

Findings

Provides an alternative proof for properties of restricted universal enveloping algebras.

Addresses potential gaps in the theoretical understanding of Lie-Rinehart algebras.

Highlights conditions under which these algebras are finitely generated and projective.

Abstract

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. This note presents an alternative proof and possibly fills a gap that might, however, only be a gap in the author's understanding.