Conformal Super-Biderivations on Lie Conformal Superalgebras

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.


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. e 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][4][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][11][12][13][14]. e authors in [15,16] generalized biderivations of Lie algebras to the concept of super-biderivations of superalgebras independently. e authors in [17] studied super-biderivations on the super Galilean conform algebra. e conformal biderivations of the loop W(a, b) 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. is is our motivation to present this paper.
In this paper, we concentrate on the loop super-Virasoro Lie conformal superalgebra cls (see [6]), which is defined as a for any i, j ∈ Z. 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 cls and Neveu-Schwarz Lie conformal superalgebra NS, and all conformal super-biderivations of the NS are inner. roughout this paper, all vector spaces are over the complex field C. 1 , and x ∈ V i is called Z 2 -homogenous and we write |x| � i.
and the following axioms: where λ is an indeterminate and z is a derivation of the λ-bracket.

Conformal Super-Biderivations
Definition 2. Let R be a Lie conformal superalgebra. We call a conformal bilinear map φ λ : for all homogeneous x, y, z ∈ R. (6) is equivalent to equation (7).

Remark 1. Equation
Proof. We suppose equation (7) satisfies. On the one hand, using equation (5), we have On the other hand, using conformal sesquilinearity, we obtain Replace λ, μ by − z − λ − μ, μ, respectively, and by conformal sesquilinearity, we have is implies that equation (6) is satisfied. e reverse conclusion follows similarly.
We call the conformal super-biderivation φ a λ the inner conformal super-biderivation if there exists a fixed complex number a such that φ a λ (x, y) � a[x λ y]. To avoid lengthy notations, we let □ Lemma 1. Let ϕ λ be a conformal super-biderivation of Lie conformal superalgebra R. en, for any homogeneous x, y, w, v ∈ R.
Proof. Firstly, from the definition of conformal superbiderivation, we have Using the Jacobi identity of Lie conformal superalgebras, we obtain erefore, we get which implies us, for any homogeneous x, y, w, v ∈ R. □ Lemma 2. Let ϕ λ be a conformal super-biderivation of Lie conformal superalgebra R. en, Proof. By skew-supersymmetry and conformal sesquilinearity, we have Note that On the other hand, we have for any homogeneous x, y, w, v ∈ R. is implies (ii) directly follows from (i).

Conformal Super-Biderivations of cls
Theorem 1. Every conformal super-biderivation ϕ λ on the cls has the following forms: for all i, j ∈ Z, where a k for any k ∈ Z are complex numbers.
Proof. Let ϕ λ be a conformal super-biderivation on the cls. We shall complete the proof by verifying the following four claims.
□ Claim 1. ere exist some complex numbers a k for any k ∈ Z such that for all i, j ∈ Z. For any i, j ∈ Z, we may assume that where f k ij (z, λ), g k ij (z, λ) ∈ C[z, λ]. By Lemma 2(i), we have Furthermore, we note that at is, erefore, we have It follows that which implies From (32), we have By comparing the degrees of z on both sides of (33), we can suppose that for some c 0 (λ), c 1 (λ) ∈ C[λ]. Substituting this formula to (33), we can obtain Comparing the coefficients of c 3 , one can deduce c 1 (μ) − c 1 (λ) � 0. us, c 1 (λ) ∈ C which we denote by c 1 . erefore, (35) can be written as We get c 0 (μ) � 2c 1 μ. Considering the coefficients of z, we have which implies c 1 � 0 and g k ij (z, λ) � 0. By (31), we obtain We suppose that for some a 0 (λ), a 1 (λ) ∈ C[λ]. us, By comparing the coefficients of z on both sides of (40), one can deduce Hence, a 1 (μ) ∈ C, and we denote it by a 1 . We also get a 0 (λ) � 2λa 1 . erefore, f k ij (z, λ) � a 1 (z + 2λ). We denote a 1 by a k ij . us, Journal of Mathematics 5 where a k ij ∈ C. Furthermore, by Lemma 2(i), we have at is, en, we conclude that for all i, j ∈ Z, where a k for any k ∈ Z are complex numbers.
We get en z + μ + c + 3 2 λ k∈Z a k− i− j L k μ+c G i+j � k∈Z d k ij (z + μ + c, λ) L i+j μ+c L k + k∈Z h k ij (z + μ + c, λ) L i+j μ+c G k .