A q-analogue of U(g[(N+1)), Hecke algebra,and the Yang-Baxter equation

We study for g=g[(N+1) the structure and representations of the algebra (g), a q-analogue of the universal enveloping algebra U(g). Applying the result, we construct trigonometric solutions of the Yang-Baxter equation associated with higher representations of g.     

Deformation of the exterior algebra Ω(M n) and the Yang-Baxter equation

We show that the deformation of the exterior algebra on a given manifold is related to the existence of the Yang-Baxter equation. We prove that this deformed algebra involves a differential operator generating the algebra. The obtained differential calculus is not commutative and we recover the classical one for the classical limit of the deformation parameters. The q-analogue of the Leibniz rule corresponding to the purposed differential operator is given.     

Extended conformal algebra and the Yang-Baxter equation

Working in the context of classical Toda field theory, we derive their extended conformal symmetry algebra from their integrability properties.     

Classical solutions of the quantum Yang-Baxter equation

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the flip map exchanging components and suitably implanted in each cartesian powerS ⁿ , generates a symplectic action of the braid groupB _n onS ⁿ . Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.Research partially supported by NSF Grant DMS-90-01089Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania     

On twisting solutions to the Yang-Baxter equation

Sufficient conditions for an invertible two-tensor F to relate two solutions to the Yang-Baxter equation via the transformation R F ₂₁ ^-1 RF are formulated. Those conditions are equivalent to the problem of twisting for certain quasitriangular bialgebras.     

The four-state solution of the Yang-Baxter equation

A new four-state solution of the Yang-Baxter equation is constructed with the help of the lowest-dimensional cyclic L-operator related to a three-state R-matrix. Some special choice of the parameters on which this solution depends leads to an exactly solvable spin model on a chain with Hermitian Hamiltonian.     

Spectra of quantum chains without the Yang-Baxter equation

We study one-dimensional reaction-diffusion models described by master equations and their associated two-state quantum hamiltonians. By choosing appropriate rates, the equations of motion decouple into certain subsets. We solve the first subset which has a close relation to the problem of lattice electrons in an electric field. In this way we obtain L(L − 1) + 1 energy levels of a quantum chain with L sites. The corresponding hamiltonian depends on seven parameters and does not look integrable using conventional methods. As an application, we compute the dynamical critical exponent of a new type of kinetic Ising model.     

Yang-Baxter equation and representation theory: I

The problem of constructing the GL(N,) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered. In caseN=2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2,) are obtained and their eigenvalues are calculated. Some results for the caseN>2 are also presented.     

Boundary Yang-Baxter equation in the RSOS/SOS representation

We construct and solve the boundary Yang-Baxter equation in the RSOS/SOS representation. We find two classes of trigonometric solutions; diagonal and nondiagonal. As a lattice model, these two classes of solutions correspond to RSOS/SOS models with fixed and free boundary spins, respectively. Applied to (1 + 1)-dimensional quantum field theory, these solutions give the boundary scattering amplitudes of the particles. For the diagonal solution, we propose an algebraic Bethe ansatz method to diagonalize the SOS-type transfer matrix with boundary and obtain the Bethe ansatz equations.     

How to find the Lax pair from the Yang-Baxter equation

We show explicitly how to construct the quantum Lax pair from the Yang-Baxter equation. We demonstrate the new method by applying it to the Heisenberg XYZ model.Supported by DOE contract DE-FG02-88ER25053     

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a new class of quantum groups, deformed Yangians, is used to obtain new matrix rational solutions of the Yang-Baxter equation (YBE). The deformed Yangians arise from rational solutions of the classical Yang-Baxter equation of the form c ₂/u + const. The image of the universal quantum R-matrix for the deformed Yangian in finite-dimensional representations gives these new matrix rational solutions of YBE.     

Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation

We construct Drinfel’d twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians. Presented by D. Arnaudon at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

Hyperbolic complex Yang-Baxter equation and hyperbolic complex multiparametric quantum groups

The hyperbolic complex Yang-Baxter equation is equivalent to a system consisting of two ordinary Yang-Baxter equations, and a hyperbolic complex quantum group is isomorphic to a direct product of two quantum groups. As a concrete example, the quantum groupGL _H(; _ij) with hyperbolic complex multiparameter is isomorphic to a direct product of two quantum groupsGL(X; q _ij ) andGL(Y; r _ij ) with ordinary multiparameter.     

Construction of a new quantum double and new solutions to the Yang-Baxter equation

A new quantum double is established from a new Hopf algebra and a new kind of quantum R-matrix is obtained.     

The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong ^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.