On Hom–Lie algebras

Abstract

In this paper, first we show that (g, [·, ·], α) is a Hom-Lie algebra if and only if ( g * , α * , d) is an (α * , α * ) 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.

Ado theorem for nilpotent Hom–Lie algebras

“…Proof. Straightforward computation, see [SX,Proposition 4.1], or [GMMP,Lemma 4.3] for a more general BiHom-associative case. The nondegeneracy is evident.…”

“…Proof. Straightforward computation, see [SX,Theorem 4.2] for a similar computation in the Hom-Lie case, or [GMMP,Proposition 4.4] for a more general BiHom-associative case.…”

Hom-Lie-Rinehart algebras

“…A coboundary operator for a hom-Lie algebroid with coefficients in it's representation on a vector bundle can be defined since any hom-Lie algebroid is a hom-Lie-Rinehart algebra. Now, let us first recall the definition of (σ, τ )-differential graded commutative algebra from [20]. Definition 7.3.…”

On Hom-Gerstenhaber algebras, and Hom-Lie algebroids

“…In this section, we recall some basic definitions from [8], [11], and [13]. Let R be a commutative ring with unity and Z + be the set of all non-negative integers.…”