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

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...

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Bibliographic Details
Published in:arXiv.org 2022-12
Main Authors: Hou, Shuai, Hu, Meiyan, Song, Lina, Zhou, Yanqiu
Format: Article
Language:English
Subjects:
Algebra
Deformation
Homology
Homomorphisms
Lie groups
Operators (mathematics)
Online Access:Get full text
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Summary: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.
ISSN:2331-8422