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 cohomology groups.

中文翻译：

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$$\varvec{n}$$ n -Lie–Rinehart 代数的扩展和交叉模

我们引入了n -Lie-Rinehart 代数的概念，作为将 Lie-Rinehart 代数推广到n元情况。这个概念也是n -Lie 代数的代数模拟。我们发展了表示论并描述了一个n -Lie-Rinehart 代数的上同调复数。此外，我们通过 2-cocycle研究了n -Lie-Rinehart 代数的可拓理论。最后，我们引入了n -Lie-Rinehart 代数的交叉模，以更好地理解它们的第三上同调群。