Symmetric invariant pairings in vertex operator super algebras and Gramians
Symmetric invariant bilinear forms on vertex operator algebras were studied in [\textit{I. B. Frenkel}, \textit{Y.-Z. Huang} and \textit{J. Lepowsky}, On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc. 494 (1993; Zbl 0789.17022)] and [ \textit{H. Li}, J. Pure Appl. Algebra 96, No. 3, 279-297 (1994; Zbl 0813.17020)]. In the paper under review the author considers a vertex operator superalgebra \(V= \displaystyle\bigoplus_{ 0 \leq n \in {1 \over 2} {\mathbb Z}_+ } V_n\) such that \(V_0\) is one dimensional generated by the vacuum vector. Such a vertex operator superalgebra is called connected. Let \(D(n), n \in {\mathbb Z}\), denote the elements of the Virasoro algebra. The author defines a symmetric bilinear pairing on \(V\) and shows that this pairing is graded symmetric, \(D(-1)\)-invariant and \(V\)-invariant with respect to a bracket product \([a,b] = a_0 b\). He proves that \(D(-n)\) and \(D(n)\) are adjoint to each other on the subspace of quasi-primary vectors. The author also investigates connected vertex operator superalgebras generated by vectors of weight one half. Then hermitian pairings associated to certain conjugate linear involutions are positive definite. Moreover, the pairings of canonical basis vectors can be expressed in terms of Gramian determinants of matrices of binomial coefficients.
scientific article; zbMATH DE number 1330820