
TL;DR
This paper explores the structure of hom-Lie algebras, establishing their equivalence with certain differential graded algebras, and introduces omni-hom-Lie algebras to characterize regular hom-Lie algebra structures.
Contribution
It provides a new characterization of hom-Lie algebras via differential graded algebras and introduces omni-hom-Lie algebras for classifying regular hom-Lie structures.
Findings
Hom-Lie algebras are equivalent to specific differential graded algebras.
Introduction of omni-hom-Lie algebra associated with a vector space.
Regular hom-Lie algebra structures characterized by Dirac structures.
Abstract
In this paper, first we show that is a hom-Lie algebra if and only if is an -differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
