We study a mixed tensor product $\three^{\tensor m} \tensor \bthree^{\tensor n}$ of the three-dimensional fundamental representations of the Hopf algebra $\qSL{2|1}$, whenever $\q$ is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective $\qSL{2|1}$-module with the generating modules $\three$ and $\bthree$ are obtained. The centralizer of $\qSL{2|1}$ on the mixed tensor product is calculated. It is shown to be the quotient $\Alg_{m,n}$ of the quantum walled Brauer algebra $\qwb_{m,n}$. The structure of projective modules over $\Alg_{m,n}$ is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over $\Alg_{m,n}$. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over $\Alg_{m,n}\boxtimes\qSL{2|1}$. We give an explicit bimodule structure for all $m,n$.

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R -matrices. Here we study the simplest case – the 11-vertex R -matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n -particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau–Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is described.

A Monte Carlo simulation study of the critical and off-critical behavior of the Baxter–Wu model, which belongs to the universality class of the 4-state Potts model, was performed. We estimate the critical temperature window using known analytical results for the specific heat and magnetization. This helps us to extract reliable values of universal combinations of critical amplitudes with reasonable accuracy. Comparisons with approximate analytical predictions and other numerical results are discussed.

We study integrable models solvable by the nested algebraic Bethe ansatz and described by gl(2|1) or gl(1|2) superalgebras. We obtain explicit determinant representations for form factors of the monodromy matrix entries. We show that all form factors are related to each other at special limits of the Bethe parameters. Our results allow one to obtain determinant formulas for form factors of local operators in the supersymmetric t–J model.

Given a spanning forest on a large square lattice, we consider by Kirchhoff theorem a correlation function of $k$ paths ($k$ is odd) along branches of trees or, equivalently, $k$ loop-erased random walks. Starting and ending points of the paths are grouped such that they form a $k$-leg watermelon. For large distance $r$ between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as $r^{-\nu} \log r$ with $\nu = (k^2-1)/2$. Considering the spanning forest stretched along the meridian of the watermelon, we show that the two-dimensional $k$-leg loop-erased watermelon exponent $\nu$, corresponding to c=-2 CFT is converting into the scaling exponent for the reunion probability (at a given point) of $k$ (1+1)-dimensional vicious walkers, $\tilde{\nu} = k^2/2$, described by RMT.

We introduce a Kazhdan–Lusztig-dual quantum group for (1,*p*) Virasoro logarithmic minimal models as the Lusztig limit of the quantum *sℓ*(2) at *p*th root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with the fusion of irreducible and logarithmic modules in the (1,*p*) Virasoro logarithmic minimal models.

We show that from the spectra of the Uq(sl(2)) symmetric XXZ spin-1/2 finite quantum chain at = −1/2 (q = ei/3) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for = 0 (q = ei/2). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(m|n)-invariant R-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system of Bethe equations obeys the same properties. In this way we prove a generalized Gaudin hypothesis for the norm of the Hamiltonian eigenstates.

Currently there are two proposed ansatze for NSR superstring measures: the Grushevsky ansatz and the OPSMY ansatz, which for genera g<=4 are known to coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a vanishing two point function in genus four, which can be constructed from the genus five expressions for the respective ansatze. This is inconsistent with the known properties of superstring amplitudes.

In the present paper we show that the Grushevsky and OPSMY ansatze do not coincide in genus five. Then, by combining these ansatze, we propose a new ansatz for genus five, which now leads to a vanishing two-point function in genus four. We also show that one cannot construct an ansatz from the currently known forms in genus 6 that satisfies all known requirements for superstring measures.

Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window.

The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG.

We emphasize the role of corrections *and* of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.

We discuss the Matsuo-Cherednik type correspondence between the quantum Knizhnik-Zamolodchikov equations associated with GL(N) and the n-particle quantum Ruijsenaars model, with n being not necessarily equal to N. The quasiclassical limit of this construction yields the quantum-classical correspondence between the quantum spin chains and the classical Ruijsenaars models

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a *N *-column box is viewed as a time-ordered product of (N×N)-matrices consisting of a single *sl*2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space HN=SL(N,R)/SO(N). In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in HN. The coordinates of HN are interpreted as the coordinates of particles of the one-dimensional Toda chain. The group-theoretic structure of the system and links to some random matrix models are also discussed

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by superalgebra. Using coproduct properties of the Bethe vectors we obtain a *sum formula* for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.