On universalR-matrix for quantized nontwisted rank 3 affine KM algebras

Explicit formulas of the universalR-matrix are given for all quantized nontwisted rank 3 affine KM algebras U _q (A ₂ ⁽¹⁾ ), U _q (C ₂ ⁽¹⁾ ) and U _q (G ₂ ⁽¹⁾ ).     

The uniqueness theorem for the universal R-matrix

Up to now, the universal R-matrix for quantized Kac-Moody algebras is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras U _q (A _inf1 ^sup(1) ) and U _q (A _inf2 ^sup(2) ) are given.

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The exponential map for representations ofU p,q (gl(2))

For the quantum groupGL _p,q (2) and the corresponding quantum algebraU _p,q (gl(2)) Fronsdal and Galindo [Lett. Math. Phys.27 (1993) 59] explicitly constructed the so-called universalT-matrix. In a previous paper [J. Phys. A28 (1995) 2819] we showed how this universalT-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universalT-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universalR-matrix is discussed.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

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 group invariants and link polynomials

A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsU _q (E ₈),U _q (so(2m+1) andU _q (gl(m)) are considered as examples, and corresponding link polynomials are obtained.     

On the universalR-matrix for Uq(241-1241-1241-1)

A compact form for the universalR-matrix of U _q (sl _n ) is derived and illustrated by simple applications.     

UniversalR-matrix for quantized (super)algebras

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.     

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.     

q-Weyl group and a multiplicative formula for universalR-matrices

We define theq-version of the Weyl group for quantized universal enveloping algebras of simple Lie group and we find explicit multiplicative formulas for the universalR-matrix.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

On automorphisms and universal ℛ-matrices at roots of unity

Invertible universal ?-matrices of quantum Lie algebras do not exist at roots of unity. However, quotients exist for which intertwiners of tensor products of representations always exist, i.e. ?-matrices exist in the representations. One of these quotients, which is finite-dimensional, has a universal ?-matrix. In this Letter we answer the following question: under which condition are the different quotients of U _q (sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal ?-matrix of the one can be transformed into a universal ?-matrix of the other. We prove that this happens only whenq ⁴ = 1, and we explicitly give the expressions for the automorphisms and for the transformed universal ?-matrices in this case.     

Dynamical R-matrices for Calogero models

We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F, defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p)×U(q)), allows non-trivial solutions when |p−q|⩽1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p−q|=1 and for the elliptic models when |p−q|=1 or p=q.     

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.     

QuantumR-matrix and intertwiners for the Kashiwara algebra

We study the algebraB _q (g) presented by Kashiwara and introduce intertwiners similar toq-vertex operators. We show that a matrix determined by 2-point functions of the intertwiners coincides with a quantumR-matrix (up to a diagonal matrix) and give the commutation relations of the intertwiners. We also introduce an analogue of the universalR-matrix for the Kashiwara algebra.     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

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.     

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 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.     

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.