We find the ℓ-weights and the corresponding ℓ-weight vectors for the finite and infinite dimensional representations of the quantum loop algebras Uq(L(sl2)) and Uq(L(sl3)) obtained from the Verma representations of the quantum groups Uq(gl2) and Uq(gl3) via the Jimbo’s homomorphism. Then we find the ℓ-weights and the ℓ-weight vectors for the q-oscillator representations of the positive Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate the q-oscillator and prefundamental representations.

We find the ℓ-weights and the ℓ-weight vectors for the highest ℓ-weight q-oscillator representations of the positive Borel subalgebra of the quantum loop algebra Uq(L(sll+1)) for arbitrary values of l. Having this, we establish the explicit relationship between the q-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring q-oscillator representations.

We study the Schrödinger equation on the interval (0, 1) with a very high and localized potential wall. We consider the process where the position and the height of the wall change as follows: First, the potential increases from zero to a very large value, and so a narrow potential wall is formed and almost splits the interval into two parts; then, the wall moves to a different position, after which the height of the wall decreases to zero again. We show that even though the rate of variation of the potential’s parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the instantaneous energy eigenstates. Furthermore, we consider potentials with several narrow walls and show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary time, ∂ tΨ=(1/2∂ 2 x +V (X,t))Ψ whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian H = 1/2p 2+V(x,t) for the corresponding Painlevé equation. In present paper, we present explicit constructions for the first five equations from the Painlevé list.

This paper is a continuation of our previous paper where the Painlevé-Calogero correspondence has been extended to auxiliary linear problems associated with Painlevé equations. We have proved, for the first five equations from the Painlevé list, that one of the linear problems can be recast in the form of the non-stationary Schrödinger equation whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian for the corresponding Painlevé equation. In the present paper we establish the quantum Painlevé-Calogero correspondence for the most general case, the Painlevé VI equation. We also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding Schlesinger system with rational spectral parameter.

We define the quasi-compact Higgs G -bundles over singular curves introduced in our previous paper for the Lie group SL(*N*). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G at marked points of the curves. We demonstrate that in particular cases, this construction leads to the classical integrable systems of the Hitchin type. The examples of the systems are analogs of the classical Calogero-Sutherland systems related to a simple complex Lie group G with two types of interacting spin variables. These type models were introduced previously by Feher and Pusztai. We construct the Lax operators of the systems as the Higgs fields defined over a singular rational curve. We also construct the hierarchy of independent integrals of motion. Then we pass to a fixed point set of real involution related to one of the complex structures on the moduli space of the Higgs bundles. We prove that the number of independent integrals of motion is equal to the half of dimension of the fixed point set. The latter is a phase space of a real completely integrable system. We construct the classical *r*-matrix depending on the spectral parameter on a real singular curve, and in this way we prove the complete integrability of the system. We present three equivalent descriptions of the system and establish their equivalence.

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth and satisfy some other conditions then there exists a solution-giving C0-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e. as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.

In physics and in mathematics Zn 2 -gradings, n = 2, appear in various fields. The corresponding sign rule is determined by the "scalar product" of the involved Zn 2 - degrees. The Zn 2-supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define Zn 2-supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z·2-supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical "superization" to a Zn 2 - supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Zn 2-context.

We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles ofrational solutions to the matrix Kadomtsev-Petviashvili hierarchy with discrete time. The dynamics of poles is derived using the auxiliarylinear problem for the discrete flow

We discuss the quantization of the ̂ sl 2 coset vertex operator algebra W D(2,1;α) using the bosonization technique. We show that after quantization, there exist three families of commuting integrals of motion coming from three copies of the quantum toroidal algebra associated with gl 2 .

We take a peek at a general program that associates vertex (or chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.