On δ-Hom-Jordan Lie conformal superalgebras

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Abstract

In this paper, we introduce the representation theory of δ-Hom-Jordan Lie conformal superalgebras and discuss the case of adjoint representations. Furthermore, we develop the cohomology theory of δ-Hom-Jordan Lie conformal superalgebras and discuss some applications to the study of deformations of regular δ-Hom-Jordan Lie conformal superalgebras. Finally, we introduce derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras and study their properties.

Introduction

Lie conformal superalgebras were introduced by Kac in [3], in which he gave an axiomatic description of the singular part of the operator product expansion of chiral fields in conformal field theory. It is a useful tool to study vertex superalgebras and has many applications in the theory of Lie superalgebras. Moreover, Lie conformal superalgebras have close connections to Hamiltonian formalism in the theory of nonlinear evolution equations. Zhao, Yuan and Chen developed the deformation of Lie conformal superalgebras and introduced derivations of Lie conformal superalgebras and studied their properties in [9].

As a generalization of Lie superalgebras and Jordan Lie algebras, the notion of δ-Jordan Lie superalgebra (L,[,],δ) was introduced in [5], which is intimately related to both Jordan-super and antiassociative algebras. The case of δ=1 yields the Lie superalgebra, and the other case δ=1 is called a Jordan-Lie superalgebra because it turns out to be a Jordan superalgebra. It is convenient to consider both cases δ=±1, and called δ-Jordan Lie superalgebras. However, the results on δ-Jordan Lie algebras and δ-Jordan Lie superalgebras are few in the present period.

In [1], Ammar and Makhlouf introduced the notion of Hom-Lie superalgebras, they gave a classification of Hom-Lie admissible superalgebras and proved a graded version of Hartwig–Larsson–Silvestrov Theorem. 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 superalgebras. In [8], Yuan introduced the notion of Hom-Lie conformal superalgebra and proved that a Hom-Lie conformal superalgebra is equivalent to a Hom-Gel’fand–Dorfman superbialgebra.

Recently, the Hom-Lie conformal algebra was introduced and studied in [7], where it was proved that a Hom-Lie conformal algebra is equivalent to a Hom-Gel’fand-Dorfman bialgebra. Zhao, Yuan and Chen [7] developed the cohomology theory of Hom-Lie conformal algebras and discussed some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, they introduced derivations of multiplicative Hom-Lie conformal algebras and studied their properties in [10], which is different from [9]. Ma, Chen and Zhang introduced the notions of δ-Hom-Jordan Lie superalgebras and discussed the concepts of αk-derivations, representations and T-extensions of δ-Hom-Jordan Lie superalgebras in detail, and established some cohomological characterizations in [4].

The following questions arise naturally: 1. How do we introduce the notion of δ-Hom-Jordan Lie conformal superalgebra? 2. How do we give a representation theory of δ-Hom-Jordan Lie conformal superalgebras? 3. How do we give derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras?

The aim of this article is to answer these questions.

Let us briefly describe the setup of the present article. In Section 2, we introduce a representation theory of δ-Hom-Jordan Lie conformal superalgebras, in particular, we discuss the cases of adjoint representations. In Section 3, we develop the cohomology theory of δ-Hom-Jordan Lie conformal superalgebras and discuss some applications to the study of deformations of regular δ-Hom-Jordan Lie conformal superalgebras. In Section 4, we introduce the notion of derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras and prove the direct sum of the space of derivations is a δ-Hom-Jordan Lie conformal superalgebra.

Throughout the paper, all algebraic systems are supposed to be over a field . In addition to the standard notations Z and R, and denote by Z+ the set of all nonnegative integers.

Section snippets

Representations of δ-Hom-Jordan Lie conformal superalgebras

In this section, we introduce the representation theory of δ-Hom-Jordan Lie conformal superalgebras and discuss the case of adjoint representations.

Definition 2.1

A δ-Hom-Jordan Lie conformal superalgebra R=R0̄R1̄ is a Z2-graded []-module equipped with an even linear endomorphism α such that α=α, and a -linear map RR[λ]R,ab[aλb]such that [RϕλRφ]Rϕ+φ[λ], ϕ,φZ2, and the following axioms hold for a,b,cR [aλb]=λ[aλb],[aλb]=(+λ)[aλb],[aλb]=δ(1)|a||b|[bλa],δ=±1,[α(a)λ[bμc]]=δ[[aλb]λ+μα(c)]+δ

Nijenhuis operators of δ-Hom-Jordan Lie conformal superalgebras

In this section, we introduce the notions of Nijenhuis operators and deformations of δ-Hom-Jordan Lie conformal superalgebras and show that the deformation generated by a 2-cocycle Nijenhuis operator is trivial.

Definition 3.1

Let (R,δ,α) be a δ-Hom-Jordan Lie conformal superalgebra. An n-cochain (nZ+) of a regular δ-Hom-Jordan Lie conformal superalgebra R with coefficients in a module (M,β) is an even -linear map γ:RnM[λ1,,λn],(a1,,an)γλ1,,λn(a1,,an), where M[λ1,,λn] denotes the space of polynomials

Derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras

In this section, we study derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras and prove the direct sum of the space of derivations is also a δ-Hom-Jordan Lie conformal superalgebra.

Definition 4.1

Let (R,δ,α) be a multiplicative δ-Hom-Jordan Lie conformal superalgebra. Then a Hom-conformal linear map Dλ:RR is called an αk-derivation of (R,δ,α) if Dλα=αDλ,Dλ([aμb])=δk[Dλ(a)λ+μαk(b)]+δk(1)|a||D|[αk(a)μDλ(b)], for any a,bR.

Denote by Derαk(R) the set of αk-derivations of the multiplicative δ

Acknowledgments

The authors are grateful to the referee for carefully reading the manuscript and for many valuable comments which largely improved the article. The work of S. J. Guo is supported by the NSF of China (No. 11761017) and the Youth Project for Natural Science Foundation of Guizhou provincial department of education (No. KY[2018]155). The work of S. X. Wang is supported by the Anhui Provincial Natural Science Foundation (Nos. 1908085MA03 and 1808085MA14).

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