Hom-Lie 2-algebras

Abstract

In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, $H{L}_{\infty }$-algebras, which is the hom-analogue of $L_{\infty }$-algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term $H{L}_{\infty }$-algebras are equivalent. We give a detailed study on skeletal hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We also proved that there is a one-to-one correspondence between strict hom-Lie 2-algebras and crossed modules of hom-Lie algebras. We give the construction of strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic hom-Lie algebras.

21 pages, Journal of Algebra, 376 (2013) 174-195

References in corpus (5)

• Hom-algebras and homology
• Representations and Cohomology of n-ary multiplicative Hom-Nambu-Lie algebras
• Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras
• Omni-Lie 2-algebras and their Dirac structures
• Hom-algebras via PROPs

Cited by in corpus (17)

• Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids
• $\mathcal{O}$-Operators on Hom-Lie algebras
• Cohomology and 1-parameter formal deformations of Hom-Lie triple systems
• One-parameter formal deformations of Hom-Lie-Yamaguti algebras
• Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation
• Another approach to Hom-Lie bialgebras via Manin triples
• Non-degenerate Killing forms on Hom-Lie superalgebras
• A complement on representations of Hom-Lie algebras
• Complex and Kahler structures on hom-Lie algebras
• Sweedler duality for Hom-(co)algebras and Hom-(co)modules
• Quadratic and symplectic structures on 3-(Hom)-$\rho$-Lie algebras
• Cohomology characterizations of non-abelian extensions of Hom-Lie algebras
• Matching Hom-Setting of Rota-Baxter algebras, dendriform algebras and pre-Lie algebras
• Derivation Hom-Lie 2-algebras and non-abelian extensions of Hom-Lie algebras
• Some properties of factor set in regular Hom-Lie algebras
• On hom-Lie algebras
• On the capablility of Hom-Lie algebras