Extensions and crossed modules of $n$-Lie Rinehart algebras

Abstract

We introduce a notion of n-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie Rinehart algebras. Furthermore, we investigate extension theory of n-Lie Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules of n-Lie Rinehart algebras to gain a better understanding of their third dimensional cohomology groups.