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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.

- Michio Jimbo, Satoru Odake
- 1997

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… (More)

- Satoru ODAKE, Ryu SASAKI
- 2005

Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant “discrete quantum mechanical systems” are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are “discrete” counterparts of Calogero… (More)

- Michio Jimbo, Satoru Odake
- 1998

We investigate the structure of the elliptic algebra Uq,p(ŝl2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra Uq(ŝl2), which are elliptic analogs of the Drinfeld currents. They enable us to identify Uq,p(ŝl2) with the tensor product of Uq(ŝl2) and a Heisenberg algebra… (More)

- Satoru Odake, Ryu Sasaki
- 2009

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)

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 WN 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)

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 C-M systems. We determine the polynomials describing the equilibrium positions… (More)

- Satoru Odake, Ryu Sasaki
- 2006

The annihilation-creation operators a(±) are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the ‘sinusoidal coordinate’. Thus a(±) are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the… (More)

- Hidetoshi Awata, Masafumi Fukuma, Satoru Odake, Yas-Hiro Quano
- 1994

By using the free field realizations, we analyze the representation theory of the W 1+∞ algebra with c = 1. The eigenvectors for the Cartan subalgebra of W 1+∞ are parametrized by the Young diagrams, and explicitly written down by W 1+∞ generators. Moreover, their eigenvalues and full character formula are also obtained.

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)