On a class of nonstandard quantum algebras and their representations

The nonstandardU _z sl(2, IR) quantum algebra is considered together with other nonstandard algebras sharing the same universalR-matrix as well as a fixed Hopf subalgebra. Some boson realizations for these nonstandard algebras are obtained which are later used in order to compute in a simplified way their (finite and infinite dimensional) representations. In the limit when the deformation parameterz vanishes these realizations turn into the well known (one or two-boson) Gelfand-Dyson realizations for the corresponding classical Lie algebras.     

sl(2) variational schemes for solving one class of nonlinear quantum models

Hamiltonians of a wide-spread class of G_inv invariant nonlinear quantum models, including multiboson and frequency conversion ones, are expressed as nonlinear functions of sl(2) generators. This enables us to use variational schemes, based on sl(2) generalized coherent states as trial functions, to obtain new (sl(2) cluster) quasi-classical solutions of spectral and evolution tasks which smoothly approximate exact ones and catch explicitly quantum cooperative features of the models under study (in contrast with standard independent-mode quasi-classical approximations). In such a way new analytical expressions are found for the energy spectra which are beyond both equidistant and quasi-equidistant ones obtained earlier.     

Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl(2)⊗sl(2)-valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes-Cummings-Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes-Cummings-Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.     

Complex quantum groups and their quantum enveloping algebras

We construct complexified versions of the quantum groups associated with the Lie algebras of typeA _n?1 ,B _n ,C _n , andD _n . Following the ideas of Faddeev, Reshetikhin and Takhtajan we obtain the Hopf algebras of regular functionals U_? on these complexified quantum groups. In the special exampleA ₁ we derive theq-deformed enveloping algebraU _q (sl(2, ?)). In the limitq→1 the undeformedU _q (sl(2, ?)) is recovered.     

On the dressing method for the generalised Zakharov–Shabat system

The dressing procedure for the generalised Zakharov–Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ‘dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.     

Embedding Uq(sl(2)) and Sine Algebras in Generalized Clifford Algebras

We establish the connection between certain quantum algebras and generalizedClifford algebras (GCA). To be precise, we embed the quantum tori Lie algebraand U _q (sl(2)) in GCA.     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

q-deformed conformal and Poincaré algebras on quantum 4-spinors

We investigate quantum deformation of conformal algebras by constructing the quantum space forsl _q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformedsu(2,2) algebra from the deformedsl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector inso _q (4,2) is constructed as a tensor product of two sets of 4-spinors. We obtain theq-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincaré algebra is derived through a contraction procedure.Work partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#030083)     

New algebraic structures in the Cλ-extended Hamiltonian system

A realization of various algebraic structures in terms of the C_λ-extended oscillator algebras is introduced. In particular, the C_λ-extended oscillator algebras realization of the Fairlie–Fletcher–Zachos (FFZ) algebra is given. This latter lead easily to the realization of the quantum U_t(sl(2)) algebra. The new deformed Virasoro algebra is also presented.     

Integrable quantum systems and classical Lie algebras

We have obtained six new infinite series of trigonometric solutions to triangle equations (quantumR-matrices) associated with the nonexceptional simple Lie algebras:sl(N),sp(N),o(N). TheR-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). TheseR-matrices provide an exact integrability of anisotropic generalizations ofsl(N),sp(N),o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.     

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)     

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U _q (sl(n)), Y(sl(n)) (the Yangian) and U _q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo     

Reflection K-matrices for 19-vertex models

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A₁⁽¹⁾ model, Izergin-Korepin or A₂⁽²⁾ model, sl(2|1) model and the osp(2|1) model. We find that there is a general solution for A₁⁽¹⁰⁾ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A₂⁽²⁾ and os(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.     

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.     

Quasi-exact solvability beyond the <Emphasis Type="Italic">sl</Emphasis>(2) algebraization

We present evidence to suggest that the study of one-dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(2) Liealgebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic Hamiltonian cannot be expressed as a polynomial in the generators of sl(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie algebraic approach. The text was submitted by the authors in English.     

Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

There exist a number of typical and interesting systems and/or models, which possess three-generator Lie-algebraic structure, in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl (2, C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of both the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schr?dinger equations governing various three-generator Lie-algebraic quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, it is shown that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors and its adiabatic limit in time-dependent systems is briefly discussed. Received 6 July 2002 / Received in final form 21 October 2002 Published online 11 February 2003     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

The algebra ,η( ) and infinite Hopf family of algebras

New deformed affine algebras, _,η( ), are defined for any simply laced classical Lie algebra g, which are generalizations of the algebra, _,η( ₂), recently proposed by Khoroshkin-Lebedev-Pakuliak (KLP). Unlike the work of KLP, we associate with the new algebras the structure of an infinite Hopf family of algebras in contrast to the one containing only finite number of algebras, introduced by KLP. Bosonic representation for _,η( ) at level 1 is obtained, and it is shown that, by repeated application of Drinfeld-like comultiplications, a realization of _,η( ) at any positive integer level can be obtained. For the special case of g = sl_r+1, (r + 1)-dimensional evaluation representation is given. The corresponding interwining operations are also discussed.