In this letter we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in paper [1]. As it turned out, for a special value of the asymmetry parameter ∆ = −1/2 the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites E = −3N/2. Using Mathematica we have found explicitly the corresponding eigenvectors for N ≤ 17. The obtained results support the conjecture of paper [1] that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wave function components are noticed. It is turned out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sing matrices.

Hanging about a hypothetical connections between the ground state vector for some special spin systems and the alternating-sign matrices, we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model considered by Batchelor, de Gier and Nienhuis. Our conjecture generalizes in a sense the conjecture of Bosley and Fidkowski, refined by Cohn and Propp, and proved by Wieland. I would say that imagination is a form of memory.(V. Nabokov)In paper [1] we made some conjectures related to combinatorial properties of the ground state vector of the XXZ spin chain for the asymmetry parameter ∆ = −1/2 and an odd number of cites. In the subsequent paper [2] Batchelor, de Gier and Nienhuis considered two variations of this model along with the corresponding O(n) loop model at n = 1 and notably increased the number of models and related combinatorial objects (see also [3]).During last months we have accumulated a lot of data on relation between the spin systems and alternating sign matrices (ASMs). In this note we limit ourselves to one qualitative conjecture which relates some classes of the states of the fully packed loop (FPL) model with the ground state vector of the dense O(n) loop model (see [4] and references therein). The states the FPL model is in bijective correspondence with the ASMs. Therefore if our conjecture is true then one can investigate the FPL model, and the ASMs, using the methods elaborated in mathematical physics for integrable model (see, for example, [5]).The background information on the ASMs and their different combinatorial forms can be found in the recent review by Propp [6] and in references therein. For more details on enumerative problems related to the states of the FPL model see the article by Wieland [7].Following the review paper by Propp [6] we define the "generalized tic-tac-toe" graph as the graph formed by n horizontal lines and n vertical lines meeting n 2 intersections of degree 4, with 4n vertices of degree 1 at the boundary. Then we number the vertices of degree 1. We start with the left top vertex and number clockwise every other vertex. Now 1

ABSTRACT. Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of L-operators for the quantum groups associated with the generalized Cartan matrices A

We consider an integrable conformally invariant two dimensional model associated to the affine Kac-Moody algebra sl 3 (C). It possesses four scalar fields and six Dirac spinors. The theory does not possesses a local Lagrangian since the spinor equations of motion present interaction terms which are bilinear in the spinors. There exists a submodel presenting an equivalence between a U (1) vector current and a topological current, which leads to a confinement of the spinors inside the solitons. We calculate the one-soliton and two-soliton solutions using a procedure which is a hybrid of the dressing and Hirota methods. The soliton masses and time delays due to the soliton interactions are also calculated. We give a computer program to calculate the soliton solutions.

ABSTRACT. We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group U q (L(sl 2 )) related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

The finite XXZ Heisenberg spin chain with twisted boundary conditions was considered. For the case of even number of sites N , anisotropy parameter −1/2 and twisting angle 2π/3 the Hamiltonian of the system possesses an eigenvalue −3N/2. The explicit form of the corresponding eigenvector was found for N ≤ 12.Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, the squared norm of the ground state vector is probably coincides with the number of half-turn symmetric alternating sign N × N matrices.

On the base of Lie algebraic and differential geometry methods,. Nirov, KS, "W-algebras for non-abelian Toda systems", Journal of Geometry and solution in terms of an Iwasawa type factorization of a large Lie group. (This has For the non-periodic Toda systems this large Lie group is a split case the theorem has the following geometric interpretation: If H e ?+ and ? is in. Classification of solutions to general Toda systems with .-UBC Math

Abstract. Integral formulae for polynomial solutions of the quantum KnizhnikZamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e ±2πi/3 and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter ∆ equal to −1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new statement relating the ground state components of XXZ spin chains and Temperley-Lieb loop models is formulated and proved.