Classification of the GL(3) Quantum Matrix Groups

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given.Comment: 28 pages, Late

Parametrization of semi-dynamical quantum reflection algebra

We construct sets of structure matrices for the semi-dynamical reflection algebra, solving the Yang-Baxter type consistency equations extended by the action of an automorphism of the auxiliary space. These solutions are parametrized by dynamical conjugation matrices, Drinfel'd twist representations and quantum non-dynamical $R$-matrices. They yield factorized forms for the monodromy matrices.Comment: LaTeX, 24 pages. Misprints corrected, comments added in Conclusion on construction of Hamiltonian

The XXZ model with anti-periodic twisted boundary conditions

We derive functional equations for the eigenvalues of the XXZ model subject to anti-diagonal twisted boundary conditions by means of fusion of transfer matrices and by Sklyanin's method of separation of variables. Our findings coincide with those obtained using Baxter's method and are compared to the recent solution of Galleas. As an application we study the finite size scaling of the ground state energy of the model in the critical regime.Comment: 22 pages and 3 figure

On the developments of Sklyanin's quantum separation of variables for integrable quantum field theories

We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (time dependent correlation functions) of integrable quantum field theories (IQFTs). This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.Comment: 8 pages; invited contribution to the Proceedings of the XVIIth INTERNATIONAL CONGRESS ON MATHEMATICAL PHYSICS, August 2012, Aalborg, Danemark; accepted for publication on the ICMP12 Proceedings by World Scientific. The material here presented is strictly connected to that introduced in arXiv:0910.3173 and arXiv:1204.630

Solutions to a system of Yang-Baxter matrix equations

In this article, we take a system, $XAX=BXB$, $XBX=AXA$, of Yang-Baxter type matrix equations that ``generalizes" the matrix Yang-Baxter equation. We completely characterize the case when $A,B$ are orthogonal idempotents

Six-vertex model and non-linear differential equations I. Spectral problem

In this work we relate the spectral problem of the toroidal six-vertex model's transfer matrix with the theory of integrable non-linear differential equations. More precisely, we establish an analogy between the Classical Inverse Scattering Method and previously proposed functional equations originating from the Yang-Baxter algebra. The latter equations are then regarded as an Auxiliary Linear Problem allowing us to show that the six-vertex model's spectrum solves Riccati-type non-linear differential equations. Generating functions of conserved quantities are expressed in terms of determinants and we also discuss a relation between our Riccati equations and a stationary Schr\"odinger equation.Comment: 42 pages, 3 figure

Classification of the solutions of constant rational semi-dynamical reflection equations

We propose a classification of the solutions K to the semi-dynamical reflection equation with constant rational structure matrices associated to rational scalar Ruijsenaars-Schneider model. Four sets of solutions are identified and simple analytic transformations generate all solutions from these sets.Comment: 12 pages, no figure. Dedicated to Daniel Arnaudo

Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups

The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization of the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and Felder. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang-Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang-Baxter equation, obtained in our previous paper q-alg/9703040. All solutions we found can be obtained from Felder's elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum Yang-Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang-Baxter equation. In this paper we propose a similar language, originating from Felder's ideas, which we found to be adequate for the dynamical Yang-Baxter equation. This is the language of dynamical quantum groups (or \h-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper q-alg/9703040.Comment: 55 pages, amste