Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

Abstract

The purpose of this paper is to introduce and study super Hom-Gel'fand-Dorfman bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing super Hom-Gel'fand-Dorfman bialgebras and obtain some infinite-dimensional Hom-Lie superalgebras from affinization of super Hom-Gel'fand-Dorfman bialgebras. Also, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. Finally, we characterize one-dimensional central extensions of quadratic Hom-Lie conformal superalgebras by using certain bilinear forms of super Hom-Gel'fand-Dorfman bialgebras.In Section 2 of this paper, we summarize the main Hom-type (super)algebras and recall the notions of super Gel'fand-Dorfman bialgebra and Lie conformal superalgebra. In Section 3, we introduce super Hom-Gel'fand-Dorfman bialgebras and provide five different ways for constructing super Hom-Gel'fand-Dorfman bialgebras by extending some constructions of Hom-Gel'fand-Dorfman bialgebras given in [23] and Hom-Novikov algebras given in [21]. We also construct affinization of super Hom-Gel'fand-Dorfman bialgebras, which leads us to a class of infinite-dimensional Hom-Lie superalgebras. In Section 4, we introduce the notion of Hom-Lie conformal superalgebra, which is a superanalogue of Hom-Lie conformal algebras introduced in [23] as well as a Hom-version of Lie conformal superalgebras. In particular, we introduce quadratic Hom-Lie conformal superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. This generalizes the equivalent theorem in the ungraded case obtained in [23], and in classical Lie conformal superalgebra case given in [4,13]. Section 5 is devoted to central extensions of Hom-Lie conformal superalgebras. We characterize one-dimensional central extensions of quadratic Hom-Lie conformal superalgebras by using certain bilinear forms of super Hom-Gel'fand-Dorfman bialgebras, generalizing a construction due to Hong [6]. We should point out that all these results could naturally be extended to more general graded cases.Throughout this paper, all linear spaces and tensor products are over the complex field C. In addition to the standard notation Z, we use Z + to denote the set of nonnegative integers. §2. PreliminariesLet us begin with some definitions, including Hom-Lie superalgebra, Hom-associative algebra, Hom-Novikov superalgebra, super Gel'fand-Dorfman bialgebra, Hom-Gel'fand-Dorfman bialgebra, Hom-Lie conformal algebra and Lie conformal superalgebra. For detailed discussions we refer the reader to the literatures (e.g. [1,7,21,23,24] and references therein).Let V be a superspace that is a Z 2 -graded linear space with a direct sum V = V 0 ⊕ V 1 . The elements of V j , j = {0, 1}, are said to be homogenous and of parity j. The parity of a homogeneous element x is denoted by |x|. Throughout what follows, if |x| occurs in an...

Some characterizations of color Hom-Poisson algebras

“…According to our denition, we could see Hom-Poisson algebras [18] (resp. Hom-Poisson superalgebra [28]) as color Hom-Poisson algebras with G = {0} (resp. G = Z2 and ε(x, y) = (−1) xy for all homogeneous elements x, y).…”

“…The notion of color algebras is extended to the Hom-setting by studying Hom-Lie superalgebras, Hom-Lie admissible superalgebras [1], color Hom-Lie algebras [20] and color Hom-Novikov algebras [3]. For further informations on other graded Hom-type algebras, on may refer to, e.g., [12], [19], [21], [22].…”

“…For any x 0 , x 1 , y 0 , y 1 in A, define the operation onà by (x 0 , x 1 ) (y 0 , y 1 ) = (x 0 • x 1 , 0)and two even linear mapsα :à −→à and ∂ :à −→à byα(x 0 , x 1 ) = (α(x 0 ), α(x 1 )) and ∂(x 0 , x 1 ) = (d(x 0 )), x 1 ). then (Ã, ,α) is a commutative Hom-associative superalgebra (an ε-commutative Hom-associative color algebra with G = Z 2 ) and ∂ is an even derivation of (Ã, ,α) such that ∂ •α =α • ∂[21]. By Theorem 3.1 (Ã, , * , α) is a color Hom-Novikov-Poisson algebra where…”

Some results on $δ$-Hom-Jordan Lie conformal superalgebras

“…(1) If δ = 1 in Definition 2.1, then the δ-Hom-Jordan Lie conformal superalgebra (R, δ, α) is just the Hom-Lie conformal superalgebra in Yuan [8].…”

“…Later, Ammar, Makhlouf and Saadaoui [2] studied the representation and the cohomology of Hom-Lie superalgebras, and calculated the derivations and the second cohomology group of q-deformed Witt superalgebra. In [8], Yuan introduced Hom-Lie conformal superalgebra and proved that a Hom-Lie conformal superalgebra is equivalent to a Hom-Gel'fand-Dorfman superbialgebra.…”