Vertex-IRF transformations, dynamical quantum groups and harmonic analysis

Universal Vertex-IRF Transformation for Quantum Affine Algebras

We construct a universal Vertex-IRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical Yang-Baxter equation. This universal Vertex-IRF transformation satisfies the generalized coBoundary equation and is an extension of our previous work to the quantum affine $U_q(A^{(1)}_r)$ case. This solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that the evaluation of this universal solution in the evaluation representation of $U_q(A_1^{(1)})$ gives the stand…

Vertex-IRF transformations and quantization of dynamical r-matrices

Motivated by the correspondence between the vertex and IRF models in statistical mechanics, we define and study a notion of vertex-IRF transformation for dynamical twists that generalizes a usual gauge transformation. We use vertex-IRF transformations to quantize all completely degenerate dynamical r-matrices on finite-dimensional Lie algebras.

11顶点有理R矩阵的高秩推广：IRF-顶点关系与结合Yang-Baxter方程

Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation

本文将11顶点有理R矩阵推广到GL_N情形，通过IRF-顶点变换建立其与动力学及半动力学R矩阵的联系，并证明量子与结合Yang-Baxter方程及相关恒等式。

Vertex--IRF correspondence and factorized L-operators for an elliptic R-operator

As for an elliptic $R$-operator which satisfies the Yang--Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex--IRF correspondence for the elliptic $R$-operator is obtained. The vertex--IRF correspondence implies that the Boltzmann weights of the IRF model satisfy the star--triangle relation. By means of these intertwining vectors, the factorized L-operators for the elliptic $R$-operator are also constructed. The vertex--IRF correspondence and the factorized L-operators for Belavin's $R$-matrix are reproduced from those of the elliptic $R$-operator.

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

In this paper we propose versions of the associative Yang-Baxter equation and higher order $R$-matrix identities which can be applied to quantum dynamical $R$-matrices. As is known quantum non-dynamical $R$-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical $R$-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is descr…

Algebraic Bethe ansatz for the elliptic quantum group $E_{τ,η}(sl_2)$

To each representation of the elliptic quantum group $E_{τ,η}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a $q$-deformation of Hermite's solution of the Lamé equation.