Exchange Dynamical Quantum Groups

For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U _q (?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of U _q (?), and is an algebraic structure standing behind these relations. Received: 24 March 1998 / Accepted: 14 February 1999     

Embedding of Dynamical Symmetry Groups U(1, 1) and U(2) of a Free Particle on AdS 2 and S 2 into Parasupersymmetry Algebra

Using two different types of the laddering equations realized simultaneously by the associated Gegenbauer functions, we show that all quantum states corresponding to the motion of a free particle on AdS ₂ and S ² are splitted into infinite direct sums of infinite-and finite-dimensional Hilbert subspaces which represent Lie algebras u(1, 1) and u(2) with infinite- and finite-fold degeneracies, respectively. In addition, it is shown that the representation bases of Lie algebras with rank 1, i.e., gl(2, C), realize the representation of nonunitary parasupersymmetry algebra of arbitrary order. The realization of the representation of parasupersymmetry algebra by the Hilbert subspaces which describe the motion of a free particle on AdS ₂ and S ² with the dynamical symmetry groups U(1, 1) and U(2) are concluded as well.     

Relating the approaches to quantised algebras and quantum groups

This paper constructs two representations of the quantum groupU _q g' by exploiting its quotient structure and the quantum double construction. Here the quantum group is taken as the dual to the quantised algebraU _q g, a one parameter deformation of the universal enveloping algebra of the Lie algebra g, as in Drinfel'd [6] and Jimbo [10]. From the two representations, the Hopf structure of the quantised algebraU _q g is reexpressed in a matrix format. This is the very structure given by Faddeev et al. [7], in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated Lie group to g.Supported by a SERC studentship     

Generalized hypergeometric functions on the torus and the adjoint representation ofU q (sl 2)

We study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures. We define an action of the quantum enveloping algebraU _q (sl ₂) on a space of relative cycles, extending results obtained previously for the sphere. Absolute cycles are identified with singular vectors. In the case of one puncture, we find that the resulting topological representation is essentially the adjoint representation.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

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.     

Embedding euclidean lie algebras into quantum structures

We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group,E(2), as simple irrational functions of certainq deformed expressions involving the generators of the quantum algebraU _q (so(2, 1)). We consider implications of these results for the representation theory of the Lie algebra ofE(2). We briefly discess analogous results forU _q (so(2, 2)). Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

Cyclic Representations of Quantum (Super) Algebras At qp=1

Cyclic representations of quantum (super) algebras are studied at q^p=1 using two methods:the quotient module method and the q-boson realization method.For the quantum algebras associated with any finite dimensional simple Lie algebra the general theory of two methods is given,and is generated to the quantum superalgebra U_qosp(1.2).By constructing the cyclic representation of q-Heisenberg-Wey1 superalgebras the q-boson realization method is generated to construction of cyclic representations of some high-rank quantum superalgebras.     

(p,q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

Abstract

A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra U_pq (su(1, 1)) is proposed. The representation of the algebra U_pq (su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two- and three-point correlation functions of quasi-primary fields are calculated.     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).     

The dual (p,q)-Alexander-Conway Hopf algebras and the associated universal ℐ-matrix

The dually conjugate Hopf algebrasFun _p,q (R) andU _p,q (R) associated with the two-parametric (p,q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebraU _p,q (R) is extracted. The universal ?-matrix forsFun _p,q (R) is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ?-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ?-matrix and the FRT matrix generators,L ^(±), forU _p,q (R) are derived from the ?-matrix.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000     

Models of Q algebra representations

This article gives a review of various straightforward models ofQ algebra representations. This is done using one and two variable function space models of theq-analogues of Lie enveloping algebras. The algebras considered are the quantum algebraU _q (su ₂ ) and aq analogue of the oscillator algebra. We present only the general framework and refer the reader to references of the joint work of the author and Willard Miller, Jr.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

Uq[sl(2)] Quantum Algebra in Quantum Hall Effect

For a two-dimensional system of electronsdescribed by a Hamiltonian involving two- and three-bodyinteractions and an external transverse magnetic field,we construct the U_q[sl(2)] quantum algebra,where the deformation parameter q is related to thefilling factor . We show that the Laughlin statesform a representation of this algebra.     

u q(2,1) noncompact quantum algebra: Intermediate discrete series of unitary irreducible representations

The unitary irreducible representations of the u _q(2,1) quantum algebra that belong to the intermediate discrete series are considered. The q analog of the Mickelsson-Zhelobenko algebra is developed. Use is made of the U basis corresponding to the reduction u _q(2,1) ? u _q(2). Explicit formulas for the matrix elements of the generators are obtained in this basis. The projection operator that projects an arbitrary vector onto the extremal vector of the intermediate-series representation is found.     

The nonstandardq-deformation of enveloping algebraU(so n ): results and problems

We describe properties of the nonstandardq-deformationU _q ^/′ (so _n ) of the universal enveloping algebraU(so _n ) of the Lie algebra so _n which does not coincide with the Drinfeld-Jimbo quantum algebraU _q(so _n ) and is important for quantum gravity. Many unsolved problems are formulated. Some of these problems are solved in special cases. The research of this paper was made possible in part by Award UP1-2115 of U.S. Civilian Research and Development Foundation. Presented at DI-CRM Workshop held in Prague, 18–21 June 2000.     

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