Unitary quasi-finite representations of \(W_\infty\)
\textit{H. Awata, M. Fukuma, Y. Matsuo} and \textit{S. Odake} [J. Phys. A 28, 105-112 (1995; Zbl 0852.17025)] developed a theory of quasi-finite highest-weight representations of the subalgebras \(W_{\infty, p}\) (\(p\in\mathbb C[x]\)) of \(W_{1+\infty}\), the most important being \(W_{\infty}=W_{\infty, x}\). In the paper under review the authors develop a general approach to these problems by following the basic ideas in [\textit{V. Kac} and \textit{A. Radul}, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17026)]. The main result is the classification and construction of all unitary irreducible quasi-finite highest-weight modules over \(W_{\infty}\). These modules are realized in terms of unitary highest-weight representations of the Lie algebra of infinite matrices with finitely many nonzero diagonals.
scientific article; zbMATH DE number 1552987