Related papers: Integrability, intertwiners and non-linear algebra…
One introduces the notion of C*-algebra with polarization which could be considered as the quantum Kahler structure. The connection of these algebras with Kostant-Souriou geometric quantization is shown. The theory of polarized C*-algebra…
We discuss integrable many-body systems in one dimension of Calogero-Moser-Sutherland type, both classical and quantum as well as nonrelativistic and relativistic. In particular, we consider fundamental properties such as integrability, the…
We define an integral intertwining operator among modules for a vertex operator algebra to be an intertwining operator which respects integral forms in the modules, and we show that an intertwining operator is integral if it is integral…
Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown…
Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at…
We describe the $R$-matrix structure associated with integrable extensions, containing both one-body and two-body potentials, of the $A_N$ Calogero-Moser $N$-body systems. We construct non-linear, finite dimensional Poisson algebras of…
In the present paper the algebras of functions on quantum homogeneous spaces are studied. The author introduces the algebras of kernels of intertwining integral operators and constructs quantum analogues of the Poisson and Radon transforms…
We present a new class of 2d integrable models obtained as perturbations of minimal CFT with W-symmetry by fundamental weight primaries. These models are generalisations of well known $(1,2)$-perturbed Virasoro minimal models. In the large…
Recently, based on a supersymmetric approach, new classes of conditionally exactly solvable problems have been found, which exhibit a symmetry structure characterized by non-linear algebras. In this paper the associated ``non-linear''…
Some integration formulas which either occur or are implicit in Ha's recent exact calculation of some correlations in the Calogero-Sutherland model are discussed. These integration formulas include the calculation of the inner product…
We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are…
We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on…
We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra.…
We propose generalizations of Calogero models that exhibit invariance with respect to the infinite Weyl groups of affine, hyperbolic, and Lorentzian types. Our approach involves deriving closed analytic formulas for the action of the…
The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general…
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's…
We study the connection between periodic finite-difference Intermediate Long Wave hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum K-theory of the…
The Lax pair representation in Fourier space is used to obtain a list of integrable scalar evolutionary equations with quadratic nonlinearity. The famous systems of this type such as KdV, intermediate long-wave equation (ILW), Camassa-Holm…
Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Li\'enard type oscillators exhibit this interesting property. We show that a…
We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are…