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arXiv · 2212.02729

Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras

Shuai Hou·Meiyan Hu·Lina Song·Yanqiu Zhou
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Abstract

In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_\infty$-algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.

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Shuai Hou, Meiyan Hu, Lina Song, Yanqiu Zhou. 2022-12-06. Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras. https://arxiv.org/abs/2212.02729

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