Symmetric invariant pairings in vertex operator super algebras and Gramians
type: Item
instance of: scholarly article
title: Symmetric invariant pairings in vertex operator super algebras and Gramians
zbMATH Open document ID: 0958.17010
DOI: 10.1016/S0022-4049(97)00206-5
author: Hirotaka Tamanoi
published in: Journal of Pure and Applied Algebra
publication date: 2001-04-11T00:00:00Z
review text: 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.
reviewed by: Dražen Adamović
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 1330820
zbMATH Keywords: vertex operator superalgebra
zbMATH Keywords: symmetric invariant pairings
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: quasi-primary vectors
zbMATH Keywords: hermitian pairings
zbMATH Keywords: Gramian determinants
MaRDI profile type: Publication
full work available at URL: s0022-4049(97)00206-5
OpenAlex ID: W2093867405
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description: scientific article; zbMATH DE number 1330820