Related papers: The Weyl Algebra, Spherical Harmonics, and Hahn Po…
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding…
A systematic study of the representation theory of double affine Hecke algebras and related harmonic analysis is started in this paper. Continuing the previous papers we use the technique of intertwining operators to create Macdonald…
We categorify various Fock space representations on the algebra of symmetric functions via the category of polynomial functors. In a prequel, we used polynomial functors to categorify the Fock space representations of type A affine Lie…
Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right…
For the creation operator $\adag $ and the annihilation operator $a$ of a harmonic oscillator, we consider Weyl ordering expression of $(\adag a)^n$ and obtain a new symmetric expression of Weyl ordering w.r.t. $\adag a \equiv N$ and…
We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we…
We classify up to isomorphism the quantum generalized Weyl algebras and determine their automorphism groups in all cases in a uniform way, including those where the parameter q is a root of unity, thereby completing the results obtained by…
One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the most natural choice and that it has the best properties (e.g. symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there…
In this paper we give another characterization of the strictly nilpotent elements in the Weyl algebra, which (apart from the polynomials) turn out to be all bispectral operators with polynomial coefficients. This also allows to reformulate…
Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction…
For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…
The first part of the paper is devoted to the $G$-type spaces i.e. the spaces $G^\alpha_\alpha (\mathbb R^d_+)$, $\alpha\geq 1$ and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with…
The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the symmetry algebra of the harmonic oscillator on the…
We propose two complementary design principles for engineering three-dimensional (3D) Weyl semimetals and superconductors in a layer-by-layer setup which includes even and odd parity orbitals in alternating layers - dubbed orbital selective…
We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. Furthermore, the connection…
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…
We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are…
The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old'…
In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TC-algebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we…
By means of the notions of cross product algebras of the theory of quantum groups, in the context of classical Hopf algebra structures, we deduce some known structures of Weyl algebras type (as the Drinfeld quantum double, the restricted…