Related papers: Duality for Knizhnik-Zamolodchikov and Dynamical O…

A duality transform for the coalgebra of the free difference quotient derivation-multiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic…

Symmetric functions appear in many areas of mathematics and physics, including enumerative combinatorics, the representation theory of symmetric groups, statistical mechanics, and the quantum statistics of ideal gases. In the commutative…

We study some basic and interesting quantum mechanical systems in dynamical noncommutative spaces in which the space- space commutation relations are position dependent. It is observed that the fundamental objects in the dynamical…

We study the superposition operators (also called Nemytskii operators) between spaces of almost periodic (respectively almost automorphic) functions in the sense of Stepanov. We state new results on the superposition, notably we give a…

The T-duality transformations between open and closed superstrings in different D-manifolds are generalized to curved backgrounds with commuting isometries. We address some global aspects like the occurrence of orientifold boundaries in…

This paper is a survey of our recent work on operator algebras associated to dynamical systems that lead to classification results for the systems in terms of algebraic invariants of the operator algebras.

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This…

We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these result for general investigations of quasi-periodic functions and distributions. We also…

In Bergman and Dirichlet spaces, the shift operator is not an isometry, but it is a left invertible operator. In this paper we give conditions on the left invertible operators such that a operator version, in the sense of Rosenblum and…

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of…

We give a fermionic formula for $R$-matrices of exterior powers of the vector representations of $U_q(\widehat{ \mathfrak{gl}}_N)$ and relate it to the dynamical Weyl group of Tarasov--Varchenko and Etingof--Varchenko, via a Howe…

For arbitrary second-order differential operators, the existence conditions and the construction of intertwining transmutation operators are shown. In an explicit form found hyperbolic equations with two independent variables and their…

Let $\frak F_{\l}$ be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the space $\mathfrak{D}_{\lambda,\mu}^k$ of k-th order linear differential operators from $\frak F_{\l}$ to $\frak F_{\mu}$ as…

We consider second order differential operators $P$ with polynomial coefficients that preserve the vector space $V_k$ of polynomials of degrees not greater then $k$. We assume that the metric associated with the symbol of $P$ is flat and…

The paper deals with the configuration of subalgebras in generic $n$-dimensional $k$-argument anticommutative algebras and ``regular'' anticommutative algebras.

The main aim of this paper is to generalize the classical concept of positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The…

For an inner function u we discuss the dual operator for the well-known compressed shift. We establish conditions for two dual compressed shifts to be unitarily equivalent/similar and we describe the invariant subspace structure for the…

We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive.…

We consider 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. This assertion confirms…

Gelfand and Ponomarev [Functional Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous…