Hom-Lie Color Algebra Structures

Abstract

The aim of this paper is to introduce the notion of Hom-Lie color algebras. This class of algebras is a natural generalization of the Hom-Lie algebras as well as a special case of the quasi-hom-Lie algebras. In the paper, homomorphism relation between Hom-Lie color algebras are defined and studied. We present a way to obtain Hom-Lie color algebras from the classical Lie color algebras along with algebra endomorphisms and offer some applications. Besides, we introduce a multiplier σ on the abelian group Γ and provide constructions of new Hom-Lie color algebras from old ones by the σ-twists. Finally, we explore some general classes of Hom-Lie color admissible algebras and describe all these classes via G-Hom-associative color algebras, where G is a subgroup of the symmetric group S 3 .Key words: Hom-Lie algebras, Hom-Lie color algebras, Hom-Lie color admissible algebras, G-Hom-associative color algebras, Homomorphism, σ-twistsThe motivations to study Hom-Lie structures are related to physics and deformations of Lie algebras, in particular Lie algebras of vector fields. The paradigmatic examples are q-deformations of Witt and Virasoro algebras constructed in pioneering works (e.g., [3,4,5,6,8,11]). This kind of algebraic structures were initially introduced by Hartwig, Larsson and Silvestrov [8] during the process of investigating the q-deformation of Lie algebras. Later, it was further extended by Larsson and Silvestrov to quasi-hom-Lie algebras and quasi-Lie algebras [9,10], while the Hom-Lie algebra structures were more detailed studied in [12], including the Hom-Lie admissible algebras and G−Hom-associative algebras, both of which can be viewed as generalizations of Lie admissible algebras and G−associative algebras, respectively. Recently, this kind of algebras was considered in Z 2 -graded case by Ammar and Makhlouf [2] and thus is said to be Hom-Lie superalgebras. The main feature of quasi-Lie algebras, quasi-hom-Lie algebras and Hom-Lie (super)algebras is that the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps, which lead to many interesting results.The Lie admissible algebras were introduced by A. A. Albert in 1948 [1]. Physicists attempted to introduce this structure instead of Lie algebra. For instance, the validity of Lie-admissible algebras for free particles is well known. These algebras arise also in classical quantum mechanics as a generalization of conventional mechanics (see [14,15]). The study of flexible Lie admissible algebra was also initiated in [1] and has investigated in a number of papers (see e.g., [7,13]). The authors in [12] extended to Hom-Lie algebra the classical concept of Lie admissible algebras, while Hom-Lie admissible superalgebras were considered 1

Constructions and Cohomology of Hom–Lie Color Algebras

“…The following theorem generalize the result of [36]. In the following starting from a Hom-Lie color algebra and an even Lie color algebra endomorphism, we construct a new Hom-Lie color algebra.…”

“…36]). A Hom-Lie color algebra is a quadruple · · consisting of a -graded vector space , a bi-character , an even bilinear mapping · · × → (i.e., a b ⊆ a+b for all a b ∈ ), and an even homomorphism → such that for homogeneous elements x y z we have…”

“…In the following we summarize definitions of Hom-Lie and Hom-associative color algebraic structures (see [36]) generalizing the well known Lie and associative color algebras.…”

“…A cohomology of Lie color algebras were introduced and investigated in [28,30], and representations of Lie color algebras were explicitly described in [4]. Hom-Lie color algebras were studied first in [36]. In the particular case of Hom-Lie superalgebras, a cohomology theory was provided in [3], see also [28].…”

A classification of low dimensional multiplicative Hom-Lie superalgebras

“…Hom-Lie algebras were generalized to Hom-Lie superalgebras by Ammar and Makhlouf [14] and to HomLie color algebras by Yuan [15]. In particular, Cao and Luo [16] proved that there were only the trivial Hom-Lie superalgebras on the finite-dimensional simple Lie superalgebras.…”