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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.
On the basis of the collective field method, we analyze the Calogero– Sutherland model (CSM) and the Selberg–Aomoto integral, which defines , in particular case, the partition function of the matrix models. Vertex operator realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Vira-soro constraint for(More)
Using the collective field method, we find a relation between the Jack symmetric polynomials, which describe the excited states of the Calogero-Sutherland model, and the singular vectors of the W N algebra. Based on this relation, we obtain their integral representations. We also give a direct algebraic method which leads to the same result, and integral(More)
Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Pöschl-Teller potentials in terms of their degree ℓ polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (ℓ = 1, 2,. .(More)
Integrability and supersymmetry of the supersymmetric extension of the sine-Gordon theory on a half-line are examined and the boundary potential which preserves both the integrability and supersymmetry on the bulk is derived. It appears that unlike the boundary bosonic sine-Gordon theory, integrability and supersymmetry strongly restrict the form and(More)
The Ruijsenaars-Schneider systems are 'discrete' version of the Calogero-Moser (CM) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e ±β ′ p (β ′ is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of CM systems. We determine the polynomials describing the equilibrium(More)
We investigate the structure of the elliptic algebra U q,p (sl 2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U q (sl 2), which are elliptic analogs of the Drinfeld currents. They enable us to identify U q,p (sl 2) with the tensor product of U q (sl 2) and a Heisenberg(More)