The Canonical Symmetry and Hamiltonian Formalism. II. Hamiltonian Operators

It is shown how the canonical symmetry is used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem.Comment: 13 pages, LaTeX file, IHEP 93-6

UV manifold and integrable systems in spaces of arbitrary dimension

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form

On Z-graded loop Lie algebras, loop groups, and Toda equations

Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.Comment: 24 pages, talk given at the Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007

The Canonical Symmetry for Integrable Systems

The properties of discrete nonlinear symmetries of integrable equations are investigated. These symmetries are shown to be canonical transformations. On the basis of the considered examples, it is concluded, that the densities of the conservation laws are changed under these transformations by spatial divergencies.Comment: 17 pages, LaTeX, IHEP 92-14

A possible combinatorial point for XYZ-spin chain

We formulate and discuss a number of conjectures on the ground state vectors of the XYZ-spin chains of odd length with periodic boundary conditions and a special choice of the Hamiltonian parameters. In particular, arguments for the validity of a sum rule for the components, which describes in a sense the degree of antiferromagneticity of the chain, are given.Comment: AMSLaTeX, 15 page

Multidimensional Toda type systems

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.Comment: 29 pages, LaTeX fil

Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2

Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov 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^{+- 2 pi i/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 Delta 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.Comment: v2: cosmetic changes, new section on refined TSSCPPs vs refined ASM

The Wave Functions for the Free-Fermion Part of the Spectrum of the SUq(N)SU_q(N) Quantum Spin Models

We conjecture that the free-fermion part of the eigenspectrum observed recently for the $SU_q(N)$ Perk-Schultz spin chain Hamiltonian in a finite lattice with $q=\exp (i\pi (N-1)/N)$ is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous $SU_q(N-1)$ vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SU(3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model ($q=\exp (2 i \pi/3)$), which permit us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SU(3) spin chain. Similarly as in the known case of the $SU_q(2)$ case at $q=\exp(i2\pi/3)$ our numerical and analytical studies induce some conjectures for special rates of correlation functions.Comment: 25 pages and no figure

Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde