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     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

The classical su(2) invariance of the su(2) q -invariant XXZ spin chain

The XXZ spin-chain Hamiltonian has been constructed to be su(2) _q -invariant, but naively does not appear to be su(2)-invariant. However, using recently discovered deforming maps between representations of su(2) _q and corresponding representations of su(2), we prove a theorem which states that if a Hamiltonian is su(2) _q -invariant, it is also su(2)-invariant. The theorem generalizes to any quantized Lie algebra.     

On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U _q (so _n,1) are found.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Representations of the coordinate ring of GL q (3)

It is shown that finite-dimensional irreducible representations of the quantum matrix algebraM _q (3) (the coordinate ring of GL _q (3)) exist only whenq is a root of unity (q ^p = 1). The dimensions of these representations can only be one of the following values:p ³,p ³/2,p ³/4, orp ³/8. The topology of the space of states ranges between two extremes, from a three-dimensional torusS ¹ ×S ¹ ×S ¹ (which may be thought of as a generalization of the cyclic representation) to a three-dimensional cube [0, 1] × [0, 1] × [0, 1].     

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Braid group action on theq-Weyl algebra

We provide a braid group action on theq-deformed Weyl algebraW _q (n). The restriction of this action to the representations ofU _q (A _n–1 ) andU _q (C _n ) inW _q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

Nonstandard q-deformation of the Euclidean Lie algebra and its representations

A nonstandard q-deformed Euclidean algebra U _q(iso _n ), based on the definition of the twisted q-deformed algebra U _qso_n) (different from the Drinfeld–Jimbo algebra U _q(so _n )), is defined. Infinite dimensional representations R of U _q(iso _n ) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators R(T _n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra iso _n , these spectra are discrete and spectral points have one point of accumulation.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Two variable link polynomials from quantum supergroups

New two variable link polynomials are constructed corresponding to a one-parameter family of representations of the quantum supergroup U _q [gl(2 | 1)]. Their connection with the Kauffman polynomials is also investigated.     

Diagonalization of the braid generator on unitary irreps of quantum supergroups

It is shown that the braid generator associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U _q (gl(2|1)) and a family of eight-dimensional typical representations.     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

Irreducible representations of the quantum analogue of SU(2)

We give the complete set of irreducible representations of U(SU(2))_q when q is a mth root of unity. In particular, we show that their dimensions are less or equal to m. Some of them are not highest-weight representations.     

Nonstandard U q(so3) and U q(so4): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U _q(so₃) and U _q(so₄), which can be embedded into U _q(sl₃) and U _q(sl₄) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U _q(so₃) when q is positive. Homomorphisms from U _q(so₃) and U _q(so₄) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U _q(so₃) and U _q(so₄) for q equal to a root of unity are obtained.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997