Flag Varieties and the Yang-Baxter Equation

We investigate certain bases of Hecke algebras defined by means of theYang–Baxter equation, which we call Yang–Baxter bases. These bases areessentially self-adjoint with respect to a canonical bilinear form. In thecase of the degenerate Hecke algebra, we identify the coefficients in theexpansion of the Yang–Baxter basis on the usual basis of the algebra withspecializations of double Schubert polynomials. We also describe theexpansions associated to other specializations of the generic Heckealgebra.     

Knitting Ansatz and Solutions to Yang-Baxter Equation

We suggest a new method,named knitting ansatz,to generate solutions to Yang-Baxter equation with lower dimensional representations of braid group.To support our ansatz,we work out an example of a new 16×16 R-matrix constructed along this idea,with two 4×4 braid group representations of familiar 6-vertex type with different q-parameters.     

Dynamical Yang-Baxter Equation and Quantum Vector Bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.The research is supported in part by the Israel Academy of Sciences grant no. 8007/99-03, the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by Russian Foundation for Basic Research grant no. 03-01-00593.Deceased January 2004Acknowledgement We are grateful to J. Bernstein, V. Ostapenko, and S. Shnider for stimulating discussions within the Quantum groups seminar at the Department of Mathematics, Bar Ilan University. We appreciate useful remarks by M. Gorelik, V. Hinich, and A. Joseph during a talk at the Weizmann Institute. Our special thanks to P. Etingof for his comments on various aspects of the subject.     

On Set-Theoretical Solutions to Quantum Yang-Baxter Equation

The problem on the set-theoretical solutions to the quantum Yang-Baxter equation was presented byDrinfel'd as a main unsolved problem in quantum group theory. The set-theoretical solutions are a natural extensionof the usual (linear) solutions. In this paper, we not only give a further study on some known set-theoretical solutions(the Venkov's solutions), but also find a new kind of set-theoretical solutions which have a geometric interpretation.Moreover, the new solutions lead to the metahomomorphisms in group theory.     

On Triple Product and Rational Solutions of Yang-Baxter Equation

The Yang-Baxter equation is reinvestigated in the framework of triple system. By requiring the rational R matrix of the Yang-Baxter equation satisfying the generalized Filippov condition, we derive a relation with respect to the rational R matrix. Moreover the case of the super Yang-Baxter equation is also investigated.     

Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure

The necessary and sufficient condition for the representations of Temperley-Lieb algebra arising from representations of Heck algebra is presented. Using the condition, the representations of braid group with a quadratic reduction relation are divided into two kinds. One has a representation of Temperley-Lieb algebra behind it and another has not. Some concrete examples are discussed with the condition and their Baxterizations are also mentioned.     

Wu Algorithm and Solutions of Yang-Baxter Equation for Eight-Vertex Model

In this note, we claim that we have obtained all nondegenerate solutions of Yang-Baxter equation with spectral parameter for square-lattice eight-vertex model without "zero-field" conditions by several solution transformations and six basic solutiofis with the help of Wu algorithm.     

Solutions to Braid Group,Yang-Baxter Equation and Integrable Fermion Systems

Braid group representations are found in ferrnion systems in one space dimension. Explicit baxterization is performed to find corresponding new trigonometric solu tions of Yang-Baxter equation. The quantum algebra structures implied in the new solutions are discovered. Algebraic Bethe ansatz method is applied to solving these systems. The relationships between these fermion systems and polaron model, spin-1/2 Heisenberg spin chain are discussed.     

A Systematic Method' for Constructing Solutions to the Yang-Baxter Equation

In this paper a systematic method, based on Jimbo's theorem, for constructing the spectrumdependent trigonometric solutions, as well aa the rational ones, to the Yang-Baxter equation is presented. Some solutions obtained by this method are discussed. The link polynomials constructed from the trigonometric solutions are obtained.     

A Free-Fermion Type Solution of Quantum Dynamical Yang-Baxter Equation

The symmetries of the quantum dynamical Yang-Baxter (QDYB) equation without spectral parameters for glz are discussed. The classification of the six-vertex type solu tjons to the QDYB equation without spectral parameters is given and a free-fermion type solution is obtained.     

A New Representation of Birman-Wenzl Algebra and Solution of Yang-Baxter Equation

Corresponding to a new braid group representation, a representation of Birman-Wenzl algebra is provided. The trigonometric type solutions of spectral-dependent Yang-Baxter equation are obtained in the framework of the Yang-Baxterization, and the related vertex models are also mentioned.     

Coherent State Realization of Exponential Solutions of Yang-Baxter Equation and Its Generalizations

In this letter, it is shown that the generator of coherent state can be used to realize exponential solutions of Yang-Baxter equation with a spectral parameter. Based on generalization of this observation, the non-unitary finite dimensional solutions of Yang-Baxter equation are constructed in terms of the Heisenberg-Weyl algebra. The unitary realization of the solutionis also obtained in connection with the Hamiltonian of the system of forced harmonic oscillators.     

Quantum Discord in Two-Qubit System Constructed from the Yang–Baxter Equation

Quantum correlations among parts of a composite quantum system are a fundamental resource for several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, we investigate the quantum discord of the two-qubit system constructed from the Yang-Baxter Equation. The density matrix of this system is generated through the unitary Yang-Baxter matrix R. The analytical expression and numerical result of quantum discord and geometric measure of quantum discord are obtained for the Yang-Baxter system. These results show that quantum discord and geometric measure of quantum discord are only connect with the parameter θ, which is the important spectral parameter in Yang Baxter equation.     

Pairs of Compatible Associative Algebras,Classical Yang-Baxter Equation and Quiver Representations

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, a linear deformation of the matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are described in terms of quiver representations.     

Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-Electric Condition

A local (or modified) Yang–Baxter equation (LYBE) gives a functional map from the parameters of the weights in the left-hand side to the parameters of the corresponding weights in the right-hand side of the LYBE. Such maps solve the functional tetrahedron equation. In this Letter, all the maps associated with LYBE's of the ferro-electric type with a single parameter in each weight matrix are classified.     

Solutions to the Yang-Baxter Equation for the Spinor Representations of q-De

In this paper, both trigonometric and rational solutions to tlre Yang-Baxter equation associated with the spinor representations of the quantum D_l univcrsal enveloping algebras are obtained. The quantum Clebsch-Gordan matrices, the quantum projectors and the solutions are the block matrices with the dimensions of tire submatrices to Le 1 and 2^n-3 , 1l', where _l' = _l/2 + 1, if _l' is even, _l' = (_l + 1)/2, if _l is odd. Tlre csplicit forms of the submatrices with the same dimensions are indcpendcrlt of _l. As esamples, we discuss the solutions for the spinor representations of the quantum D₄ to D₇, and prcscnt tlre explicit forms of those submatrices with the dimensions 1, 2, 8 and 32. Tlre corresponding representations of the braid group and the link polynomials are also computed tlrrougll a standard method.     

Action in Hamiltonian Models Constructed by Yang-Baxter Equation: Entanglement and Measures of Correlation

By using the quantum Yang-Baxterization approach, we investigate the dynamics of quantum entanglement under the actions of different Hamiltonians on the different two-qubit input states and analyze the effects of the Yang-Baxter operations on it. During any quantum process that takes place in a noisy environment, quantum correlations display behavior that does not increase. We point out that for two-qubit systems subject to actions of different Yang-Baxter operations the loss of correlations can be mitigated by the appropriate choice of the initial states and the Yang-Baxterization process. We show that in a noisy environment it possible to create the optimal conditions for performing any quantum information task.     

From Braid Group Representations with Four Eigenvalues to Solutions of Yang-Baxter Equation

A recursive formula of Jimbo-type trigonometric Yang-Baxterization is presented. The consistency conditions of the Yang-Baxterization for four eigenvalues are obtained. Several 4 × 4 solutions of Yang-Baxter equations are obtained from a braid group representation.     

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.