Unitary quasi-finite representations of \(W_\infty\). (English) Zbl 0974.17033

Kac, Victor G.; Liberati, José I.

Lett. Math. Phys. 53, No. 1, 11-27 (2000).

H. Awata, M. Fukuma, Y. Matsuo and 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 [V. Kac and 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.

Reviewer: M.Primc (Zagreb)

MSC:

17B68	Virasoro and related algebras
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B70	Graded Lie (super)algebras

Keywords:

quasi-finite highest-weight modules