Conformal Super-Biderivations on Lie Conformal Superalgebras

Abstract

In this paper, the conformal super-biderivations of two classes of Lie conformal superalgebras are studied. By proving some general results on conformal super-biderivations, we determine the conformal super-biderivations of the loop super-Virasoro Lie conformal superalgebra and Neveu–Schwarz Lie conformal superalgebra. Especially, any conformal super-biderivation of the Neveu–Schwarz Lie conformal superalgebra is inner.

1. Introduction

Lie conformal superalgebras, introduced by Kac in [1], encode the singular part of the operator product expansion of chiral fields in conformal field theory. The conformal super-algebras play important roles in quantum field theory, vertex algebras, integrable systems, and so on and have drawn much attention in the branches of physics and mathematics. Finite simple Lie conformal superalgebras were classified by Fattori and Kac in [2], and their representation theories were developed in [3–5]. Moreover, some infinite Lie conformal superalgebras were also studied, such as loop super-Virasoro Lie conformal superalgebra [6] and Lie conformal superalgebras of Block type [7]. Other results on Lie conformal superalgebras can be seen in [8, 9].

In recent years, biderivations have been extensively studied for various algebra structures [10–14]. The authors in [15, 16] generalized biderivations of Lie algebras to the concept of super-biderivations of superalgebras independently. The authors in [17] studied super-biderivations on the super Galilean conform algebra. The conformal biderivations of the loop Lie conformal algebra and loop Virasoro Lie conformal algebra are determined in [18].

As a generalization of conformal biderivations of Lie conformal algebras and a parallel concept of super-biderivations of Lie superalgebras, we introduce the concept of conformal super-biderivations on Lie conformal superalgebras. We hope that biderivations would contribute to the development of structure theories of Lie conformal superalgebras. This is our motivation to present this paper.

In this paper, we concentrate on the loop super-Virasoro Lie conformal superalgebra (see [6]), which is defined as a -module with a -basis and λ-brackets given byfor any In section 2, we recall definition of Lie conformal superalgebras. Some general results about conformal super-biderivations are obtained in section 3. In section 4, we determine the conformal super-biderivations of and Neveu–Schwarz Lie conformal superalgebra , and all conformal super-biderivations of the are inner.

Throughout this paper, all vector spaces are over the complex field

2. Preliminaries

In this section, we recall definition about Lie conformal superalgebra in [1, 4, 7, 8].

A vector space is called ₂-graded algebra if and is called ₂-homogenous and we write

Definition 1. A Lie conformal superalgebra is a ₂-graded -module with a -linear map called the λ-bracket satisfying and the following axioms:(1)Conformal sesquilinearity: , .(2)Skew-supersymmetry: .(3)Jacobi identity: , for all where is an indeterminate and is a derivation of the λ-bracket.From conformal sesquilinearity, we can conclude thatBy Definition 1, the loop super-Virasoro Lie conformal superalgebra with and andfor

Example 1. (see [9]). Neveu–Schwarz Lie conformal superalgebra is a free ₂-graded -module with the following conditions:where and

3. Conformal Super-Biderivations

Definition 2. Let be a Lie conformal superalgebra. We call a conformal bilinear map a conformal super-biderivation of if it satisfies the following equations:for all homogeneous .

Remark 1. Equation (6) is equivalent to equation (7).

Proof. We suppose equation (7) satisfies. On the one hand, using equation (5), we haveOn the other hand, using conformal sesquilinearity, we obtainThen Replace by , respectively, and by conformal sesquilinearity, we haveThis implies that equation (6) is satisfied. The reverse conclusion follows similarly.
We call the conformal super-biderivation the inner conformal super-biderivation if there exists a fixed complex number such that
To avoid lengthy notations, we let

Lemma 1. Let be a conformal super-biderivation of Lie conformal superalgebra Then,for any homogeneous

Proof. Firstly, from the definition of conformal super-biderivation, we haveUsing the Jacobi identity of Lie conformal superalgebras, we obtain Therefore, we getwhich impliesThus, for any homogeneous

Lemma 2. Let be a conformal super-biderivation of Lie conformal superalgebra Then,(i), for all homogeneous (ii)If , then , where is the center of

Proof. By skew-supersymmetry and conformal sesquilinearity, we haveNote that On the other hand, we have Thus, this implies that for any homogeneous This implies (ii) directly follows from (i).

4. Conformal Super-Biderivations of

Theorem 1. Every conformal super-biderivation on the has the following forms:for all where for any are complex numbers.

Proof. Let be a conformal super-biderivation on the We shall complete the proof by verifying the following four claims.

Claim 1. There exist some complex numbers for any such thatfor all .
For any we may assume thatwhere By Lemma 2(i), we haveFurthermore, we note thatThat is,Therefore, we have It follows that which implies From (32), we haveBy comparing the degrees of on both sides of (33), we can suppose thatfor some Substituting this formula to (33), we can obtainComparing the coefficients of one can deduce Thus, which we denote by Therefore, (35) can be written asWe get Considering the coefficients of we havewhich implies and
By (31), we obtainWe suppose thatfor some . Thus,By comparing the coefficients of on both sides of (40), one can deduceHence, , and we denote it by We also get . Therefore, We denote by . Thus,where
Furthermore, by Lemma 2(i), we haveTherefore,That is,We getwhich shows that . Hence, denoted by Thus,Then, we conclude thatfor all where for any are complex numbers.

Claim 2. for all
For any we suppose thatwhere By Lemma 2(i), we haveTherefore,That is,We getThenHence,which implies andWe getThen,We obtainfor all where for any are complex numbers.

Claim 3. for all
For any we suppose thatwhere By Lemma 2(i), we haveThus,That is,Hence,We getThen,which implies andIt followsThis shows thatHence,for all where for any are complex numbers.

Claim 4. for all
For any we suppose thatwhere By Lemma 2(i), we haveTherefore,That is,Hence,Thus, we haveIt followswhich implies andhenceWe can get and for all where for any are complex numbers.

Theorem 2. Every conformal super-biderivation on has the following forms:where are complex numbers. Therefore, any conformal super-biderivation on is inner.

The proof of this theorem is similar to that of Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Education and Scientific Research Project for Middle-Aged and Young Teachers in Fujian Province (no. JT180559) of PR China.