2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79311We classify the quasifinite highest weight modules over a family of subalgebras W_{\infty}^{n} of the central extension W_{1+\infty} of the Lie algebra of differential operators on the circle consisting of operators of order \geq n. We classify the unitary quasifinite highest weight modules over W_{\infty}=W_{\infty}^{1} and realize them in terms of unitary highest weight representations of the Lie algebra of infinite matrices with finitely many non-zero diagonals.Amstex, 18 pagesQuantum AlgebraRepresentation Theorytext