Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions

For any classical Lie algebra $\mathfrak{g}$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems survive after quantization.     

Integrable Evolution Equations on Associative Algebras

This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painlevé transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found. Received: 12 March 1997 / Accepted: 27 August 1997     

Classical integrable systems and their field-theoretical generalizations

Investigations of the interrelations between classical integrable systems, construction of their field theory generalizations, and some problems of quantization of Poisson manifolds are reviewed.     

Angular quantization and form factors in massive integrable models

We discuss an application of the method of angular quantization to the reconstruction of form factors of local fields in massive integrable models. The general formalism is illustrated with examples of the Klein-Gordon, sinh-Gordon and Bullough-Dodd models. For the latter two models the angular quantization approach makes it possible to obtain free field representations for form factors of exponential operators. We discuss an intriguing relation between the free field representations and deformations of the Virasoro algebra. The deformation associated with the Bullough-Dodd models appears to be different from the known deformed Virasoro algebra.     

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Form factors,deformed Knizhnik-Zamolodchikov equations and finite-gap integration

We study the limit of asymptotically free massive integrable models in which the algebra of nonlocal charges turns into affine algebra. The form factors of fields in that limit are described by KZ equations on level 0. We show the limit to be connected with finite-gap integration of classical integrable equations.     

First Integrals,Liouville Theorem,and Dirac Brackets

In this paper, we discuss the conditions for the existence of first integrals of movement and the Liouville theorem on integrable systems. We revise the core results of the Hamilton-Jacobi theory and discuss the extension of the formalism to encompass constrained systems using Dirac brackets, originally developed in the context of the canonical quantization of constrained systems. As an application, we analyze a Hamiltonian that represents the classical limit of a Fermionic system of oscillators.     

Generalized Hitchin Systems and the Knizhnik–zamolodchikov–bernard Equation on Ellipic Curves

The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.     

Quantum Integrable Toda-Like Systems in Deformation Quantization

We define and discuss the notion of quantum integrability of a classically integrable system within the framework of deformation quantization, i.e. the question whether the classical conserved quantities (which are already in involution with respect to the Poisson bracket) commute with respect to some star product on the phase space after possible quantum corrections. As an example of this method, we show by means of suitable 2 by 2 quantum R-matrices that a list of Toda-like classical integrable systems given by Y. B. Suris is quantum integrable with respect to the usual star product of the Weyl type in flat 2n-dimensional space.     

Mechanical systems with Poincaré invariance

Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincaré algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural quantum version. We follow this early work finding and classifying mechanical systems with such an action. New solutions are found together with a new class of models exhibiting an action of the Galilean algebra. These are related to the functional identities underlying the various Hirzebruch genera. The quantum mechanics is also discussed.     

A new (in)finite-dimensional algebra for quantum integrable models

A new (in)finite-dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite-dimensional representations are constructed and mutually commuting quantities—which ensure the integrability of the system—are written in terms of the fundamental generators of the new algebra. Relation with the deformed Dolan–Grady integrable structure recently discovered by one of the authors and Terwilliger's tridiagonal algebras is described. Remarkably, this (in)finite-dimensional algebra is a “q-deformed” analogue of the original Onsager's algebra arising in the planar Ising model. Consequently, it provides a new and alternative algebraic framework for studying massive, as well as conformal, quantum integrable models.     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

A Lie theoretic galois theory for the spectral curves of an integrable system: I

In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. The specific curves depend upon the representation of the Lie algebra. In this paper a Galois theory of spectral curves is given that classifies the spectral curves from an integrable system. The spectral curves correspond to conjugacy classes of certain subgroups of the Weyl group for the Lie algebra. The theory is illustrated with the periodic Toda lattice.Partially supported by a Louisiana Education Quality Support Fund grant LEQSF (87-89)-RD-A-8     

Algebraic model of a classical non-relativistic electron

An algebraic structure is constructed which serves as an algebraic analog of a phase space for a model of a non-relativistic classical electron. The structure consists of a type of Poisson bracket defined on the tensor product of a commutative algebra and a Grassmann algebra. The equivalent of Hamiltonian dynamics is defined and applied to specific models of an electron. A quantization procedure is introduced which leads to the usual quantum equivalents of the classical models.     

Symmetry-preserving perturbations of the Bateman Lagrangian and dissipative systems

Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, ?) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2,?), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems.     

Automorphisms of sl(2) and classical integrable systems

Outer automorphisms of infinite-dimensional representations of the sl(2) algebra are applied to produce some classical integrable systems with continuous and discrete time. The associated Lax pairs and r-matrix algebras are constructed.     

Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional supersymmetric gauge theories and A-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.