W-INFINITY AND THE RACAH-WIGNER ALGEBRA

We examine the structure of a recently constructed W∞ algebra, an extension of the Virasoro algebra that describes an infinite number of fields with all conformal spins 2,3,..., with central terms for all spins. By examining its underlying SL(2, R) structure, we are able to exhibit its relation to the algebras of SL(2, R) tensor operators. Based upon this relationship, we generalise W∞ to a one-parameter family of inequivalent Lie algebras W∞(μ), which for general μ requires the introduction of formally negative spins. Furthermore, we display a realisation of the W∞(μ) commutation relations in terms of an underlying associative product, which we denote with a lone star. This product structure shares many formal features with the Racah-Wigner algebra in angular-momentum theory. We also discuss the relation between W∞ and the symplectic algebra on a cone, which can be viewed as a co-adjoint orbit of SL(2, R). © 1990.