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     

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.     

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.     

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SO _q (n+1,n–1) are defined as real forms of quantum orthogonal groups SO _q (n+1,n–1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.     

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.     

New approach in theory of Clebsch-Gordan coefficients foru(n) andU q(u(n))

A new method for calculation of Clebsch-Gordan coefficients (CGCs) of the Lie algebrau(n) and its quantum analogU _q(u(n)) is developed. The method is based on the projection operator method in combination with the Wigner-Racah calculus for the subalgebrau(n−1) (U _q(u(n−1))). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form of the projection operator ofu(n) andU _q(u(n)). It is shown that theU _q(u(n)) CGCs can be presented in terms of theU _q(u)(n−1)) q−9j-symbols. Presented at the 9th International Colloquium: “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163. Supported in part by the U.S. National Science Foundation under Grant PHY-9970769 and Cooperative Agreement EPS-9720652 that includes matching from the Louisiana Board of Regents Support Fund.     

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.     

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].     

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.

     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.     

Vector fields on complex quantum groups

Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA _n–1 ,B _n ,C _n , andD _n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA ₁ it is shown that this Hopf algebra coincides withU _q sl(2, ).     

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 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.     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

q-Fuzzy Spheres and Quantum Differentials on Bq[SU2] and Uq(su2)

We provide a new unified construction of the two-parameter Podleś two-spheres as characterised by a projector e with trace _q (e) = 1 + λ. In our formulation the limit in which q → 1 with λ fixed is the fuzzy sphere, while the limit λ → 0 with q fixed is the standard q-deformed sphere. We show further that the non-standard Podleś spheres arise geometrically as ‘constant time slices’ of the unit hyperboloid in q-Minkowski space viewed as the braided group B _q [SU ₂]. Their localisations are then isomorphic to quotients of U _q (su ₂) at fixed values of the q-Casimir precisely q-deforming the fuzzy case. We also use transmutation and twisting theory to introduce a C_q[G_\mathbb C]{C_q[G_\mathbb {C}]} -covariant differential calculus on general B _q [G] and U _q (g), with Ω(B _q [SU ₂]) and Ω(U _q (su ₂) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Ω(C _q [SU ₂]) prior to twisting.     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed. Special representations of the translation part are investigated.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

(Z N×) n−1 generalization of the chiral Potts model

We show that theR-matrix which intertwines twon-by-N ^n–1 state cyclicL-operators related with a generalization ofU _q(sl(n)) algebra can be considered as a Boltzmann weight of four-spin box for a lattice model with two-spin interaction just as theR-matrix of the checkerboard chiral Potts model. The rapidity variables lie on the algebraic curve of the genusg=N ^{2(n–1)((n–1)N-n)+1} defined by 2n–3 independent moduli. This curve is a natural generalization of the curve which appeared in the chiral Potts model. Factorization properties of theL-operator and its connection to the SOS models are also discussed.     

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.     

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).     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.