Related papers: Solutions of Trigonometric KZ Equations satisfy Dy…
Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the…
A dynamical $r$-matrix is associated with every self-dual Lie algebra $\A$ which is graded by finite-dimensional subspaces as $\A=\oplus_{n \in \cZ} \A_n$, where $\A_n$ is dual to $\A_{-n}$ with respect to the invariant scalar product on…
An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group S_N is presented for integer values of the parameter. The corresponding integrals…
Supersymmetric t-J Gaudin models with both periodic and open boundary conditions are constructed and diagonalized by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the Gaudin systems are derived, and used to…
It is shown that the rich algebraic structure of the standard $d$-dimensional Coulomb problem can be extended to its Dunkl counterpart. Replacing standard derivatives by Dunkl ones in the so($d+1$,2) dynamical algebra generators of the…
The spatially inhomogeneous large $N$ solutions to Kazakov--Migdal model are analyzed. The set of nonlinear differential equations is derived in the continuum limit. In one dimensional case these equations has a natural interpretation in…
We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schr\"odinger equation with stochastic forcing. The kynetic equation is obtained as a result of a double limiting…
We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are…
We discuss relations between different integral formulae for solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero in the $U_q(sl_2)$ case for $|q|<1$. Smirnov type formulae of M.Jimbo et al. are derived from the…
We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…
We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szego and Baxter. We also derive a new…
We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal…
In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of…
We prove new integral formulas for generalized hypergeometric functions and their confuent variants. We apply them, via stationary phase formula, to study WKB expansions of solutions: for large argument in the confuent case and for large…
The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by…
We show that the KZ system has a purely topological interpretation in the sense that it may be understood as a variation of complex mixed Hodge structure whose successive pure weight quotients are polarized. This in a sense completes and…
We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in…
We describe \,$q$-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^*F_\lambda$ of a partial flag variety \,$F_\lambda$\,. These…
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by…
We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By…