Symmetric invariant bilinear forms on vertex operator algebras
A bilinear form \((-,-)\) on a vertex operator algebra \(V\) (or its module \(M)\) is said to be invariant if \((Y(a,z)u,v) = (u,Y(e^{zL(1)}\) \((- z^{-2})^{L(0)}\) \(a,z^{-1})v)\) [see \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)]. (As usual, \(Y(v,z)\) denotes the vertex operator associated with a vector \(v\) in \(V\), \(L(n)\), \(n \in \mathbb Z\), denote elements of the Virasoro algebra \(\text{Vir}\) and (in what follows) \(V = \oplus_{n \in \mathbb Z} V_{(n)}\) is a \(\mathbb Z\)-grading defined by \(L(0)\).) From the definition it easily follows (by choosing \(u\) to be a vacuum vector \textbf{1} and by taking coefficients of \(z^ 0)\) that a symmetric invariant form \((-,-)\) on \(V\) is uniquely determined by the linear functional \(f\) on \(V_{(0)}\) defined by \(f(w) = (\mathbf{1}, w)\). One of the main results of the paper under review states that a linear functional \(f\) on \(V_{(0)}\) determines a symmetric invariant bilinear form if and only if \(L(1)V_{(1)} \subset \ker f\). Moreover, the author obtains decompositions \(V_{(0)} = QP_{(0)} \oplus L(1)V_{(1)}\) in the case when \(V\) is generated by quasi-primary vectors and \(V_{(0)} = P_{(0)} \oplus L(1)V_{(1)}\) in the case when \(V\) is completely reducible as a \(\text{Vir}\)-module, and in either case defines a particular symmetric invariant bilinear form \(B(-,-)\). Then an analogue of Cartan's criterion holds: \(V\) is semisimple if and only if \(B(-,-)\) is nondegenerate.
scientific article; zbMATH DE number 701720