相关论文: The Racah algebra as a commutant and Howe duality
The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations…
Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and…
We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra…
We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We…
Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these…
Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic…
A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an…
We study various aspects of the metaplectic Howe duality realized by Fischer decomposition for the metaplectic representation space of polynomials on $\mathbb{R}^{2n}$ valued in the Segal-Shale-Weil representation. As a consequence, we…
The analysis of the $\mathbb{Z}_2^{3}$ Laplace-Dunkl equation on the $2$-sphere is cast in the framework of the Racah problem for the Hopf algebra $sl_{-1}(2)$. The related Dunkl-Laplace operator is shown to correspond to a quadratic…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist…
The Lie algebra of the classical group SU(2) is constructed from two quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder…
Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains…
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential…
Assume that ${\mathbb F}$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is the unital associative ${\mathbb F}$-algebra defined by generators and relations in the following way. The generators are $A$,…
We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a…
The Askey-Wilson algebra $AW(3)$ with three generators is shown to serve as a hidden symmetry algebra underlying the Racah and (new) generalized Clebsch-Gordan problems for the quantum algebra $sl_q(2)$. On the base of this hidden symmetry…
Equipping a non-equivariant topological $E_\infty$-operad with the trivial $G$-action gives an operad in $G$-spaces. For a $G$-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups.…
New bivariate Griffiths polynomials of $q$-Racah type are introduced and characterized. They generalize the polynomials orthogonal on the multinomial distribution introduced by R. Griffiths fifty years ago. They also correspond to a…