On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-3/2 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V-(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V-(3/2) is naturally a V-(2)-module equipped with a V-(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A, U) and a simple vertex operator superalgebra L-L(A,L- (U)) (l,0) for every nonzero number l such that L-L(A,L-U) (l,0)((2)) = A and L-L(A,L-U)(l,0)((3/2)) = U. (C) 2022 Elsevier Inc. All rights reserved.