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

We first construct two new nonsimple conformal algebras associated with the Schrödinger-Virasoro Lie algebra and the extended Schrödinger-Virasoro Lie algebra, respectively. Then we study conformal derivations and free nontrivial rank one conformal modules of these two conformal algebras. Also, we provide a computation of their universal central extensions.

In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras W(b) with b ∈ C related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over W(b) is proved to be isomorphic to M ∆,α,β with ∆ = 0 or β = 0 if b = 0, or M ∆,α with ∆ = 0 if b = 0. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of W(2, 2)-type are classified. Finally, the same thing is done for the Schrödinger-Virasoro conformal algebra.

The aim of this paper is to introduce the notions of Hom Gel'fand-Dorfman bialgebra and Hom-Lie conformal algebra. In this paper, we give four constructions of Hom Gel'fand-Dorfman bialgebras. Also, we provide a general construction of Hom-Lie conformal algebras from Hom-Lie algebras. Finally, we prove that a Hom Gel'fand-Dorfman bialgebra is equivalent to a Hom-Lie conformal algebra of degree 2.Comment: 21 pages; no figure