Quantum 401-1401-1401-1algebras and Macdonald polynomials

Abstract

We derive a quantum deformation of the W N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.q-alg/9508011

New q,p(sl(2)) algebras from the elliptic algebra q,p((2)c)

“…Moreover the closed algebraic relation (2.1) may, in such a frame, acquire a crucial importance as a q-deformation of the Virasoro-current commutations relations. This would then provide us with the full q-deformed structure required to construct possible generalizations of quantum Ruijsenaars-Schneider models, following the original derivation of q-deformed Virasoro algebras [8,9].…”

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“…To our knowledge, this is the only known example in the vertex operator approach 1) where the type II VO's do not involve integrals of screening operators. We note also the existence of a deformation of the W k currents, which we believe to be equivalent to the one in [16,17] with specialization r = k + 2. This conforms with the known equivalence between the Z k parafermionic CFT and the first member in the W k minimal series.…”

“…The level-rank duality also suggests the existence of the deformed W k currents [16,17] in the parafermionic description of the ABF models.…”

“…(8) in [16]), but have not verified such relations as W k (u) = 1 which are expected to hold only at the level of the cohomology.…”

“…We expect that these W j (u) (j ≥ 1) generate the same deformed W algebra for sl N in [16], under the following identification…”