Cohomologies and crossed modules for pre-Lie Rinehart algebras

TL;DR

This paper develops a cohomology theory for pre-Lie-Rinehart algebras, classifies their extensions and crossed modules via cohomology groups, and introduces related 2-algebra structures.

Contribution

It introduces the notion of crossed modules for pre-Lie-Rinehart algebras and relates them to third cohomology groups, expanding the algebraic framework.

Findings

Abelian extensions classified by second cohomology.

Crossed modules classified by third cohomology.

Construction of pre-Lie-Rinehart algebras from r-matrices.

Abstract

A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules for (pre-)Lie Rinehart algebras.