Semi-conformal structure on certain vertex superalgebras associated to vertex superalgebroids

Abstract

In this paper, we first give the definiton of a vertex superalgebroid. Then we construct a family of vertex superalgebras associated to vertex superalgebroids. As a main result, we find a sufficient and necessary condition that this vertex superalgebras are semi-conformal. In addition, we give an concrete example of this vertex superalgebras and apply our results to this superalgebra. 1 and necessary condition that the associated vertex superalgebras are semi-conformal. In addition, we give an concrete example of such vertex superalgebra and apply our results to this superalgebra. This paper is organized as follows. In Section 2, we review some basic notations, formulas and properties for Lie superalgebras and vertex superalgebras. We give the definitions of 1-truncated conformal superalgebra, vertex superalgebroid and Lie superalgebroid, which are generalization of these algebras in non-super version. We also review tools from [GMS] and [LY1], then we construct an N-graded vertex superalgebra for a given vertex superalgebroid. In Section 3, we recall the definition of semi-conformal vertex superalgebra and find a sufficient and necessary condition that the given vertex superalgebras are semi-conformal. In Section 4, we give an example of this family of vertex superalgebras. PreliminariesWe use the usual symbols Z, Z + and N for the set of integers, the positive integers, and the nonnegative integers respectively.Let M = M0⊕M1 be any superalgebra, i.e., (Z/2Z)-graded algebra. Any element u in M0 (resp. M1) is said to be even (resp. odd). For any homogeneous element u, we define |u| = 0 if u is even, |u| = 1 if u is odd. We define ε u,v = (−1) |u||v| , for any homogeneous elements u, v ∈ M . We note that, the space of endomorphisms of M , denoted by End(M ) is a superalgebra.Throughout this paper, when we write |u| for an element u ∈ M , we will always implicitly assume that u is a homogeneous element. Definition 2.1. A Lie superalgebra is a superalgebra A = A0⊕A1 with multiplication [·, ·] satisfying the following two axioms: for homogeneous a, b, c ∈ A, skew − supersymmetry : [a, b] = −ε a,b [b, a]. super Jacobi identity : [a, [b, c]] = [[a, b], c] + ε a,b [b, [a, c]].Example 2.2. Let A = A0 ⊕ A1 be an associative superalgebra. A becomes a Lie superalgebra with(2.1)Let A be a Lie superalgebra. Then End(A) is an associative superalgebra, and hence it carries a structure of Lie superalgebra by (2.1).Definition 2.4. Let A = A0 ⊕ A1 be an associative superalgebra. An endomorphism D ∈ End(A) s is called a derivation of degree s, if it satisfies the identity D(ab) = D(a)b + (−1) s|a| aD(b), a, b ∈ A.(2.2)Denote by Der(A) s the space of derivations on A of degree s.