Etsuro Date

TRANSFORMATION GROUPS FOR SOLITON EQUATIONS

Abstract A new approach to soliton equations, based on τ functions (or Hirotas dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy). The infinite-dimensional orthogonal group acts on the space of BKP τ-functions. The Sawada-Kotera equation is obtained as a reduction of BKP. Its infinitesimal transformations constitute the Euclidean Lie Algebra A2(2).

Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–

The hierarchy of Kadomtsev-Petvisahvili (KP) equation is studied on the basis of free fermion operators. Particular emphasis is laid on relating the operator approach to the Grassmann formulation of M. and Y. Sato. A new bilinear identity for wave functions is derived, and is shown to generate the series of Hirota bilinear equations for the KP hierarchy. Extension to the multicomponent case is also discussed.

Method for Generating Discrete Soliton Equations. V

As a continuation of previous work, discretization of nonlinear Schrodinger equation and its analogues are discussed as the reduction of 2-component KP hierarchy.

Exactly Solvable Sos Models: Local Height Probabilities and Theta Function Identities

Abstract The local height probabilities (LHPs) of a series of solvable solid-on-solid (SOS) models are obtained. The models have been constructed by a fusion procedure from the eight-vertex SOS model. The LHP results are expressed in terms of modular functions (which we call the branching coefficients) appearing in appropriate theta function identities. The critical behavior is studied by using their automorphic properties. Some of these identities result from the representation theory of affine Lie algebras. The branching coefficients are (not necessarily irreducible) characters of the Virasoro algebra constructed from the pairs (A 1 (1) ⊗ A 1 (1) , A 1 (1) ) or (A 2 l −1 (1) , C l (1) ). As the critical exponents the present lattice models realize the anomalous dimensions of known 2D conformal field theories (the minimal unitarizable theory and its supersymmetric extensions, the Z N -invariant theory, etc.)

KP Hierarchies of Orthogonal and Symplectic Type : Transformation Groups for Soliton Equations VI

A series of new hierarchies of soliton equations are presented on the basis of the Kadomtsev-Petviashvili (KP) hierarchy. In contrast to the KP case, which admits GL( ∞) as its transformation group, these new hierarchies are shown to correspond to O( ∞) or Sp( ∞). Soliton, rational and quasi-periodic solutions are constructed.

Fusion of the eight vertex SOS model

A higher spin analogue is presented of the eight vertex-SOS correspondence in the sense of Andrews-Baxter-Forrester. The resulting hierarchy of solvable models contain the hard hexagon model and its recent multi-state generalizations.

One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras Xn=An, Bn, Cn, Dn. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of Xn(1) modules. We also present similar results for the restricted face models of types Bn(1), Cn(1), Dn(1).

The structure of quotients of the Onsager algebra by closed ideals

We study the Onsager algebra from the ideal theoretic point of view. A complete classification of closed ideals and the structure of quotient algebras are obtained. We also discuss the solvable algebra aspect of the Onsager algebra through the use of formal Lie algebras.

Crystal base and

Theq-deformed vertex operators of Frenkel and Reshetikhin are studied in the framework of Kashiwaras crystal base theory. It is shown that the vertex operators preserve the crystal structure, and are naturally labeled by the global crystal base. As an application the one point functions are calculated for the associated elliptic RSOS models, following the scheme of Kang et al. developed for the trigonometric vertex models.

q

The energy and momentum of the spin model related to the vector representation of the quantized affine algebra of type are computed in the framework of Davies et al. Commutation relations among creation and annihilation operators are also derived.