Oscillator Realization of Higher-U(N+, N−)-Spin Lie Algebras of W∞-type and Quantized Symplectic Diffeomorphisms

This article is a further contribution to our research [1] into a class of infinite-dimensional Lie algebras L∞(N+, N−) generalizing the standard W∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a group-theoretic framework. Here we interpret L∞(N+, N−) either as a infinite continuation of pseudo-unitary symmetries U(N+, N−), or as a “higher-U(N+, N−)-spin extension” of the diffeomorphism algebra diff(N+, N−) of a N = N+ + N− dimensional manifold. We also provide a deeper mathematical interconnection between Poisson (and symplectic diffeomorphism) algebras of functions on coadjoint orbits of pseudo-unitary groups U(N+, N−) and the classical limit of the corresponding tensor operator (and group) algebras. As potential applications, we comment on the formulation of integrable higher-dimensional dynamical (field) systems and gauge theories of higher-extended objects. Some remarks on non-commutative geometry are also made. PACS: 02.20.Tw, 02.20.Sv, 11.25.Hf, 03.65.Fd, 02.40.Gh