Cohomologies and crossed modules for pre-Lie Rinehart algebras
Rings and Algebras
2022-04-06 v2 Quantum Algebra
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.
Keywords
Cite
@article{arxiv.2110.14857,
title = {Cohomologies and crossed modules for pre-Lie Rinehart algebras},
author = {Liangyun Chen and Meijun Liu and Jiefeng Liu},
journal= {arXiv preprint arXiv:2110.14857},
year = {2022}
}
Comments
23 pages, we add a reference on pre-Lie-Rinehart algebra (or left-symmetric Rinehart algebra). See [2] for more details