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Unitary quasi-finite representations of ${W}_{\infty }$. (English) Zbl 0974.17033
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 ${W}_{\infty ,p}$ ($p\in ℂ\left[x\right]$) 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 17B70 Graded Lie (super)algebras
##### Keywords:
quasi-finite highest-weight modules