A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

Abstract

A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.q-alg/9507034

Drinfeld-Sokolov reduction for quantum groups and deformations of W-algebras

“…The deformed screening operator S h 1 coincides with the screening operator for the q-Virasoro algebra introduced in [60,6,7]. This algebra may be defined as follows.…”

“…The definition of the deformed Virasoro algebra given in [60] was motivated by the theory of symmetric functions. Later a deformed analog for the algebra W k (sl N ) was introduced by the same authors in [6,7] and independently by Feigin and E. Frenkel in [22].…”

“…Later a deformed analog for the algebra W k (sl N ) was introduced by the same authors in [6,7] and independently by Feigin and E. Frenkel in [22]. In [34] E. Frenkel and Reshetikhin introduced a deformation of the algebra W k (g) for an arbitrary complex semisimple Lie algebra g. In this paper we call the deformed W-algebras introduced in [60,6,7,34] the q-W-algebras since, in fact, they are not deformations of the W-algebras W k (g) but quantizations of Poisson algebras of functions on certain reduced Poisson manifolds (see below).…”

Algebra of Screening Operators for the Deformed W n Algebra

“…The operator X (a) 12 and the signs ± in the commutation relations will be given later. The relations in the other cases σ = s 2 , s 1 s 2 s 1 are obtained by permuting the upper indices (a 0 , a 1 , a 2 ) → (a 2 , a 0 , a 1 ) successively.…”

“…In fact, the analogy with conformal field theory goes further. Each Fock space has the structure of a module over the deformed Virasoro algebra (DVA) discovered in [12], where the deformation parameter x (0 < x < 1) is the one which enters the Boltzmann weights of the models. As was shown in [10,13], the above complex is actually that of DVA modules, i.e., the operator d commutes with the action of DVA.…”