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

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B(p) may be a nontrivial extension of a finite conformal module over Vir if p = −1,where Vir is a Virasoro conformal subalgebra of B(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b(n) for n ≥ 1.

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α k -derivations of multiplicative Hom-Lie conformal algebras and study their properties.

Abstract:The purpose of this paper is to study W .2; 2/ Lie conformal algebra, which has a free COE@ -basisIn this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

Let W (b) be a class of free Lie conformal algebras of rank 2 with C[∂ ]-basis {L, H} and relationswhere b is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras W (b).