Related papers: Koornwinder polynomials and the XXZ spin chain
We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The…
Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…
In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part of the paper the kernels associated with rank one root systems are used to define…
We study the question of when geometric extension algebras are polynomial quasihereditary. Our main theorem is that under certain assumptions, a geometric extension algebra is polynomial quasihereditary if and only if it arises from an even…
Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links.…
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article…
The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$,…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that…
The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy…
Using algebraic Bethe ansatz and the solution of the quantum inverse scattering problem, we compute compact representations of the spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field. At lattice distance m,…
By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \begin{equation*} \begin{aligned} B_n&=\frac{n!}{2\pi i}\oint \frac{z}{e^z-1}\frac{d…
For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials…
An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index $k$ was introduced in Journ Phys A 42 (2009) 242001 and was…
The paper is mainly devoted to the irreducibility of the polynomial representation of the double affine Hecke algebra for an arbitrary reduced root systems and generic "central charge" q. The technique of intertwiners in the non-semisimple…
We discuss in this paper combinatorial aspects of boundary loop models, that is models of self-avoiding loops on a strip where loops get different weights depending on whether they touch the left, the right, both or no boundary. These…
Making use of the real sl(2,R) Lie group algebra generating a spin 1/2 Lie group allows to create an explicitly given Lorentz invariant fermion wave. As the generators are real valued they can be interpreted as a deformation tensor in…
We study a Laurent polynomial representation $V$ of the double affine Hecke algebra of type $GL_n$ for specialized parameters $t^{k+1}q^{r-1}=1$. We define a series of subrepresentations of $V$ by using a vanishing condition. For some…
Using the natural notion of {\em Hasse--Schmidt derivations on an exterior algebra}, we relate two classical and seemingly unrelated subjects. The first is the celebrated Cayley--Hamilton theorem of linear algebra, "{\em each endomorphism…
In this letter we apply the methods of our previous paper hep-th/0108045 to noncommutative fermions. We show that the fermions form a spin-1/2 representation of the Lorentz algebra. The covariant splitting of the conformal transformations…