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 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.     

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.     

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.     

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.     

A New Current Algebra and the Reflection Equation

We establish an explicit algebra isomorphism between the quantum reflection algebra for the U_q([^(sl₂)]) R{U_q(\widehat{sl_2}) R}-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).     

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.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.     

Topological representations of the quantum groupU q(sl 2)

We define a topological action of the quantum groupU _q(sl ₂) on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc. This action commutes with the monodromy action of the braid groupoid, which is given by theR-matrix ofU _q(sl ₂).Currently visiting the Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Supported in part by the NSF under Grant No. PHY89-04035, supplemented by funds from the NASA     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

Integral Presentations for the Universal R-Matrix

We present an integral formula for the universal R-matrix of quantum affine algebra U _q with Drinfeld comultiplication. We show that the properties of the universal R-matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For U _q we describe precisely the cycles and present a new simple expression for the universal R-matrix as a result of calculation of corresponding integrals.     

The multi-dimensional <Emphasis Type="Italic">q</Emphasis>-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

We obtain the inhomogeneous invariance quantum group for the multi-dimensional q-deformed bosonic Newton oscillator algebra. The homogenous part of this quantum group is given by the multiparameter quantum group $ GL_{X;q_{ij} } $ GL_{X;q_{ij} } of Schirrmacher where q _ij’s take some special values. We find the R-matrix which gives the non-commuting structure of the quantum group for the two dimensional case.     

Quasi-1 structure on q-Poincaré algebras

We use braided groups to introduce a theory of 1-structures on general inhomogeneous quantum groups, which we formulate as quasi-1 Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix from appropriate to the previous q-Euclidean and q-Minkowski space-time algebras R₂₁x₁x₂ = x₂x₁R and R₂₁u₁Ru₂ = u₂R₂₁u₁R. We obtain unitarity of the fundamental differential representations. We further show that the Euclidean and Minkowski-Poincaré quantum groups are twisting equivalent by another quantum cocycle.     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

Intertwining relations,asymmetric face model,and algebraic Bethe ansatz forSU p,q (2) invariant spin chain

Intertwining relations for the quantumR-matrix of theSU _p,q (2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parametersp andq, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.     

Some remarks on Yang-Baxter algebras

We point out the existence of an alternative algebraic structure in Yang-Baxter algebra with trigonometric R-matrix, which appears to be the generalization of the Yangian in Yang-Baxter algebras with rational R-matrix and which is described most naturally by q-commutators. Some properties are presented, in particular in the case of the well-known symmetric six-vertex model. Received: 13 February 1998 / Revised: 16 March 1998 / Accepted: 17 April 1998     

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     

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⁾ ).