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

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, . . .) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gómez-Ullate et al.'s are the first members of these infinitely many potentials.

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

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable onedimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of type I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the 'virtual state wavefunctions' which are 'deleted' by the generalisation of Crum-Adler theorem. Each polynomial has another integer n which counts the nodes.

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [1], Felder [2]). Frønsdal [3,4] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U q (g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U q (g). We also prove the shifted cocycle condition for the twistors, thereby completing Frønsdal's findings.This construction entails that, for generic values of the deformation parameters, representation theory for U q (g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra A q,p ( sl 2 ).