Related papers: Dynamical differential equations compatible with r…
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the…
We study the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the "superperiodic" operators for which all solutions of the equation $(L+1)\psi=0$ are…
An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.
In this paper, we study intertwining operators between subregular Whittaker modules of $\gl_N$ generalizing, on the one hand, the classical exchange construction of dynamical quantum groups, on the other hand, earlier results for principal…
Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent…
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely…
We model generalized harmonic functions on rings of differential operators and complex function spaces. The differential operators in the second Weyl-algebra that commute with rotations are described and leads to a natural notion for such…
We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of KZ system is rational too. We give the method of…
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…
We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $\mathcal{L}$ on Lie groups $\mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is…
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…
We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations…
We employ Weyl's method and Vinogradov's method to analyze skew-shift dynamics on semi-algebraic sets. Consequently, we improve the quantum dynamical upper bounds of Jitomirskaya-Powell, Liu, and Shamis-Sodin for long-range operators with…
Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they…
Given a weighted $\ell^2$ space with weights associated to an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a…
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of…
Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $\kzz$ and…
In this note we propose a compatible set of equations which commutes with the Knizhnik-Zamolodchikov equations based on the $\mathfrak{{gl}}(n\vert m)$ symmetry algebra and establish the Matsuo-Cherednik correspondence in this context.
We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to…
We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials.…