Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra G are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus τ and the Calogero-Moser couplings m to infinity, while keeping fixed the combination M = m e^iδθτ for some exponent δ. Critical scaling limits arise when 1/δ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras G⁽¹⁾ and (G⁽¹⁾)^V. The limits of the untwisted or twisted Calogero-Moser system, for δ less than these critical values, but non-zero, consists of the ordinary Toda system, while for δ = 0, it consists of the trigonometric Calogero-Moser systems for the algebras G and G^V respectively.     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.     

Lax representation with a spectral parameter for the Kowalewski top and its generalizations

We construct a Lax pair with a spectral parameter for the Kowalewski top in two constant force fields combined with magnetic interaction, and for its higher-dimensional generalizations.     

Quantum affine algebras and universalR-matrix with spectral parameter

Using the previously obtained universalR-matrix for the quantized nontwisted affine Lie algebras U _q (A ₁ ⁽¹⁾ ) and U _q (A ₂ ⁽¹⁾ ), we determine the explicitly spectral dependent universalR-matrix for the corresponding quantum Lie algebras U _q (A ₁) and U _q (A ₂). As applications, we reproduce the well known results in the fundamental representations and we also derive an extremely explicit formula of the spectral-dependentR-matrix for the adjoint representation of U _q (A ₂), the simplest nontrivial case when the tensor product decomposition of the representation with itself has nontrivial multiplicity.     

Generalized Lax pairs,the modified classical Yang-Baxter equation,and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=–(M)L, where is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in gV. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV ^*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.     

Lax pairs for certain generalizations of the sine-Gordon equation

We derive the one-parameter family of isospectral linear eigenvalue problems which is the basic tool for treating certain generalized sine-Gordon equations by the inverse scattering method.     

Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}$. We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.We also consider Leibniz algebras with corresponding Lie algebra $n_{n,1}$ and such that the action $I\times n_{n,1}\to I$ gives rise to a minimal faithful representation of $n_{n,1}$. The classification up to isomorphism of such Leibniz algebras is given for the case of $n=4$.     

Iso-spectral deformations of general matrix and their reductions on Lie algebras

We study an iso-spectral deformation of the general matrix which is a natural generalization of the nonperiodic Toda lattice equation. This deformation is equivalent to the Cholesky flow, a continuous version of the Cholesky algorithm, introduced by Watkins. We prove the integrability of the deformation and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szegö. Using the formula, the solution to the LU matrix factorization can the constructed explicitly. Based on the root spaces for simple Lie algebras, we consider several reductions of the equation. This leads to generalized Toda equations related to other classical semi-simple Lie algebras which include the integrable systems studied by Bogoyavlensky and Kostant. We show these systems can be solved explicitly in a unified way. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.     

Standard complex for quantum Lie algebras

For a quantum Lie algebra Γ, let Γ^ be its exterior extension (the algebra Γ^ is canonically defined). We introduce a differential on the exterior extension algebra Γ^ which provides the structure of a complex on Γ^. In the situation when Γ is a usual Lie algebra, this complex coincides with the “standard complex.” The differential is realized as a commutator with a (BRST) operator Q in a larger algebra Γ^[Ω], with extra generators canonically conjugated to the exterior generators of Γ^. A recurrent relation which uniquely defines the operator Q is given.     

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA ₂,B ₂, andG ₂. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

Lax pairs, symmetries and recursion operators for Toda lattices

With the exception of some minor results and some conjectures, this paper is a survey of the finite nonperiodic Toda lattices and some of their generalizations. The areas investigated include Lax pairs, master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for such systems.     

Lie algebras associated with topological nilpotent groups

A Lie algebra structure is defined on the set of all continuous one-parameter groups of nilpotent topological groups. Extensions are given to some inductive and projective limits.     

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.     

Rational forms for twistings of enveloping algebras of simple Lie algebras

We give rational forms for twistings of classical enveloping algebras. We also remark a link with the generalized formalism of Gurevich, Manin, and Cartier.     

Lie point symmetry algebras and finite transformation groups of the general Broer--Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer--Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.     

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

r-Functions with quasi-dynamical spectral parameter

For certain 1 + 1-dimensional classical field theories, whose equations of motion can naturally be written in Lax form by introducing a quasi-dynamical spectral parameter (using sdiff2, the Lie algebra of symplectic diffeomorphisms of a two-dimensional manifold), the previously derived Poisson-commutativity of an infinite set of conserved charges also follows from the existence of an -function, whose functional form is given, and checked explicitly (as well as its Yang-Baxter equation) for an interaction that is exponential (respectively, inverse square) in the spatial derivative of the field.     

Quantum BRST charge for quadratically nonlinear Lie algebras

We consider the construction of a nilpotent BRST charge for extensions of the Virasoro algebra of the form {T _a ,T _b }=f _ab ^c T _c +V _ab ^cd T _c T _d , (classical algebras in terms of Poisson brackets) and [T _a ,T _b ]=h _ab I+f _ab ^c T _c +V _ab ^cd (T _c T _d )(quantum algebras in terms of commutator brackets; normal ordering of the product (T _c T _d ) is understood). In both cases we assume that the set of generators {T _a } splits into a set {H _i } generating an ordinary Lie algebra and remaining generators {S }, such that only theV _ij are nonvanishing. In the classical case a nilpotent BRST charge can always be constructed; for the quantum case we derive a condition which is necessary and sufficient for the existence of a nilpotent BRST charge. Non-trivial examples are the spin-3 algebra with central chargec=100 and theso(N)-extended superconformal algebras with levelS=–2(N–3).     

Quantum completely integrable systems connected with semi-simple Lie algebras

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators _k introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom.