A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N)

We prove that the deformed oscillator superalgebra W _q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U _q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Discrete series of unitary irreducible representations of the Uq(u(3, 1)) and Uq(u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U _q (u(3, 1)) and U _q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) _q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U _q (u(n, 1)) ⊇ U _q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.     

An extension of the Borel-Weil construction to the quantum groupU q (n)

The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU _q(n). Thisq-BW construction uses a recursion procedure forU _q(n) in which the fiber of the bundle carries an irrep ofU _q(n–1)×U _q(1) with sections that are holomorphic functions in the homogeneous spaceU _q(n)/U _q(n–1)×U _q(1). Explicit results are obtained for theU _q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007     

Heisenberg realization for U q (sl n ) on the flag manifold

We give the Heisenberg realization for the quantum algebra U _q (sl _n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U _q (sl _n ) on theq-symmetric algebraA _q (Mat _n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U _q [2].     

Quantum deformations of singletons and of free zero-mass fields

We consider quantum deformations of the real symplectic (or anti-De Sitter) algebra sp(4), spin(3, 2) and of its singleton and (4-dimensional) zero-mass representations. For q a root of –1, these representations admit finite-dimensional unitary subrepresentations. It is pointed out that U_q (sp(4, )), unlike U_q (su(2, 2)), contains U_q (sl ₂ ) as a quantum subalgebra.To Asim Barut, with all our friendship.     

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras U_q,q(gl(n)) and U_q,q(sl(n)) which are in duality with the multiparameter quantum groups GL_qq(n), SL_qq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, q_ij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers r_i and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GL_qq(n), SL_qq(n). The case n = 3 is treated in more detail.

     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in 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.     

Irreducible highest weight representations of quantum groupsU q (gl(n,C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasU _q (gl(n, C)) are given. Using them, irreducible highest weight representations ofU _q (gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.     

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     

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 ^·.     

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).     

Vertex operators of quantum affine Lie algebrasU q (D n (1) )

We give an explicit formula for the vertex operators related to the level 1 representations of the quantum affine Lie algebrasU _q (D _n ⁽¹⁾ ) in terms of bosons. As an application, we derive an integral formula for the correlation functions of the vertex models withU _q (D _n ⁽¹⁾ )-symmetry.NJ was supported in part by NSA grant MDA904-93-H-3005 and University of Kansas General Research allocation.SJK was supported in part by Basic Science Research Institute Program, Ministry of Education of Korea, BSRI-94-1414 and GARC-KOSEF at Seoul National University, Korea.     

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     

Representation theory of osp(1,2) q

We investigate the representations of the osp(1, 2) _q algebra. We derive all the finite-dimensional irreducible representations, whenq is not a root of unity. We also discuss the connection between those of osp(1, 2) _q and sl(2) _q .     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

The gl (M|N) super Yangian and its finite-dimensional representations

Methods are developed for systematically constructing the finite-dimensional irreducible representations of the super Yangian Y (gl(M|N)) associated with the Lie superalgebra gl(M|N). It is also shown that every finite-dimensional irreducible representation of Y (gl(M|N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite-dimensional are given.     

Drinfeld Twists and Algebraic Bethe Ansatz of the Supersymmetric Model Associated with U q (gl(m|n))

We construct the Drinfeld twists (or factorizing F-matrices) of the super-symmetric model associated with quantum superalgebra U_q(gl(m|n)), and obtain the completely symmetric representations of the creation operators of the model in the F-basis provided by the F-matrix. As an application of our general results, we present the explicit expressions of the Bethe vectors in the F-basis for the U_q(gl(2|1))-model (the quantum t-J model).