Ideals of Kac-Moody algebras and realisations of W∞

We have recently constructed two new higher-spin extensions of the Virasoro algebra, denoted W1+∞ and W∞, with generators of all conformal spins s≥1 and s≥2 respectively, which admit central terms for all spins. In this paper, we show how these algebras may, respectively, be realised as enveloping algebras of the U(1) Kac-Moody algebra and the Virasoro algebra, factored by certain ideals. The algebra W1+∞ may be viewed as the algebra of all smooth differential operators on the circle. (orig.)