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.     

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.     

Regular solutions of quantum Yang-Baxter equation from weak hopf algebras

Generalization of Hopf algebraslq (2) by weakening the invertibility of the generatorK, i.e., exchanging its invertibilityKK ⁻¹=1 to the regularity K K=K is studied. Two weak Hopf algebras are introduced: a weak Hopf algebrawslq (2) and aJ-weak Hopf algebravslq (2) which are investigated in detail. The monoids of group-like elements ofwslq (2) andvslq (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. A quasi-braided weak Hopf algebraŪqw is constructed fromwslq (2). It is shown that the corresponding quasi-R-matrix is regular R^{w w}R^w=R^w. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001 Project (No. 19971074) supported by the National Natural Science Foundation of China.     

On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutionsX(u, v) of the classical Yang-Baxter equation (CYBE), whereX takes values in a simple complex Lie algebrag, and left the classification problem of the rational one open. In 1984 Drinfeld conjectured that if a rational solution is equivalent to a solution of the formX(u,v)=C ₂/(u–v)+r(u,v), whereC ₂ is the quadratic Casimir element andr is a polynomial inu,v, then deg _u r=deg _v r1. In another paper I proved this conjecture forg=sl (n) and reduced the problem of listing nontrivial (i.e. nonequivalent toC ₂/(u–v)) solutions of CYBE to classification of quasi-Frobenius subalgebras of g. They, in turn, are related with the so-called maximal orders in the loop algebra of g corresponding to the vertices of the extended Dynkin diagramD ^e (g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofD ^e (g) with coefficient 2 or 3. In particular I will find all solutions forg=o=(5) and some solutions forg=o(7),o(10),o(14) andg ₂.     

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.     

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.     

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.     

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.     

Linear superposition of Wronskian rational solutions to the KdV equation

A linear superposition is studied for Wronskian rational solutions to the Kd V equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear Kd V equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.     

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.     

New solutions of the shape equation

The general shape equation describing the forms of vesicles is a highly nonlinear partial differential equation for which only a few explicit solutions are known. These solvable cases are briefly reviewed and a new analytical solution which represents the class of the constant mean curvature surfaces is described. Pearling states of the tubular fluid membranes can be explained as a continuous deformation preserving membrane mean curvature. Received 2 February 2002 / Received in final form 4 February 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: mladenov@obzor.bio21.bas.bg     

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.     

Multikink solutions and deformed defects

In this work we consider an application of the deformation procedure that enables us to construct, systematically, scalar field models supporting multikinks. We introduce a new deformation function in order to realize this task. We exemplify the procedure with three different starting models already known in the literature, and the resulting deformed models have rich minima structures which are responsible for the appearance of multikink configurations. We have also considered an application to braneworld scenarios, where we have obtained interesting configurations corresponding to the multikink solutions.