Abelian extensions and crossed modules of Hom-Lie algebras

2018/02/12 by Casas, José-Manuel, García-Martínez, Xabier
#17A30 #17B55 #18G55 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1802.04061

Abstract

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat¹-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Related

• Crossed modules for Hom-Lie antialgebras
• Non-Abelian Cohomology and Extensions of Lie Algebras
• The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras
• Crossed extensions of Lie algebras
• Cohomology and crossed modules extension of Hom-Leibniz-Rinehart algebras