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Journal of Lie Theory 32 (2022), No. 2, 313--326
Copyright Heldermann Verlag 2022

Crossed Extensions of Lie Algebras

Apurba Das
Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh, India


It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. We show that such extensions represent elements in the $(n+1)$-th Chevalley-Eilenberg cohomology group $H^{n+1}_{CE} (\mathfrak{g}, M)$.

Keywords: Lie algebras, Chevalley-Eilenberg cohomology, crossed modules, crossed extensions.

MSC: 17B56, 17B55, 17A32.

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