506 research outputs found
A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials
We define a one-parameter family of two-sided coideals in U_q(gl(n)) and
study the corresponding algebras of infinitesimally right invariant functions
on the quantum unitary group U_q(n). The Plancherel decomposition of these
algebras with respect to the natural transitive U_q(n)-action is shown to be
the same as in the case of a complex projective space. By computing the radial
part of a suitable Casimir operator, we identify the zonal spherical functions
(i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional
irreducible representations) as Askey-Wilson polynomials containing two
continuous and one discrete parameter. In certain limit cases, the zonal
spherical functions are expressed as big and little q-Jacobi polynomials
depending on one discrete parameter.Comment: 31 pages, AMS-TeX 2.1, no figure
A study on the fourth q-Painlev\'e equation
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its
symmetry structure and some particular solutions are investigated.Comment: 18 page
Similarity reduction of the modified Yajima-Oikawa equation
We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The
hierarchy is associated with a non-standard Heisenberg subalgebra in the affine
Lie algebra of type A_2^{(1)}. The system of equations for self-similar
solutions is presented as a Hamiltonian system of degree of freedom two, and
admits a group of B\"acklund transformations isomorphic to the affine Weyl
group of type A_2^{(1)}. We show that the system is equivalent to a
two-parameter family of the fifth Painlev\'e equation.Comment: latex2e file, 18 pages, no figures; (v2)Introduction is modified.
Some typos are correcte
The sixth Painleve equation arising from D_4^{(1)} hierarchy
The sixth Painleve equation arises from a Drinfel'd-Sokolov hierarchy
associated with the affine Lie algebra of type D_4 by similarity reduction.Comment: 14 page
Bott - Borel - Weil Construction For Quantum Supergroup Uq(gl(m|n))
The finite dimensional irreducible representations of the quantum supergroup
are constructed geometrically using techniques from the Bott -
Borel - Weil theory and vector coherent states.Comment: Latex, 22 page
Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent
polynomials. We write them equivalently as 2-vector-valued symmetric Laurent
polynomials. Then the Dunkl-Cherednik operator of which they are
eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result
made possible by this approach we obtain positive definiteness of the inner
product in the orthogonality relations, under certain constraints on the
parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also
becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as
limits both of the Askey-Wilson and of the little q-Jacobi case.Comment: 16 pages. Dedicated to Paul Butzer on the occasion of his 80th
birthday. v4: minor correction in (4.14
- …