Satoru Odake

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

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.

Infinitely many shape invariant potentials and new orthogonal polynomials

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 Poschl-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 Gomez-Ullate et al.s are the first members of these infinitely many potentials.

Quasi-Hopf twistors for elliptic quantum groups

AbstractThe 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. [FIJKMY1], Felder [Fe]). Frønsdal [Fr1, Fr2] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebraUq(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universalR matrix ofUq(g). We also prove the shifted cocycle condition for the twistors, thereby completing Frønsdals findings.This construction entails that, for generic values of the deformation parameters, the representation theory forUq(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 algebraAq,p(

{\scr W}_N

\widehat{\mathfrak{s}\mathfrak{l}}_2

Collective Field Theory, Calogero-Sutherland Model and Generalized Matrix Models

).

Another set of infinitely many exceptional (Xℓ) Laguerre polynomials

We derive a quantum deformation of theWN algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.

Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials

Abstract 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 opertor realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Virasoro constraint for the generalized matrix models and indicate relations with the CSM operators. Similar results are presented for the q -deformed case (the Macdonald operator and polynomials), which gives the generating functional of infinitely many conserved charges in the CSM.

Excited states of the Calogero-Sutherland model and singular vectors of the WN algebra

Abstract We present a new set of infinitely many shape invariant potentials and the corresponding exceptional ( X l ) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one-dimensional quantum mechanics and the corresponding X l Laguerre and Jacobi polynomials [S. Odake, R. Sasaki, Phys. Lett. B 679 (2009) 414]. The new X l Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X l Jacobi polynomials and the potentials, whereas the known X l Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known X l Jacobi polynomials and the potentials.