61 research outputs found
Generalized Onsager algebras
Let be the Kac-Moody algebra with respect to a
symmetrizable generalized Cartan matrix . We give an explicit presentation
of the fix-point Lie subalgebra of with
respect to the Chevalley involution. It is a presentation of
involving inhomogeneous versions of the Serre relations, or, from a different
perspective, a presentation generalizing the Dolan-Grady presentation of the
Onsager algebra. In the finite and untwisted affine case we explicitly compute
the structure constants of in terms of a Chevalley type basis
of . For the symplectic Lie algebra and its untwisted affine
extension we explicitly describe the one-dimensional representations of
.Comment: 20 pages. v2: typos corrected and references adde
Hyperbolic beta integrals
Hyperbolic beta integrals are analogues of Euler's beta integral in which the
role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma
function. They may be viewed as -bibasic analogues of the
beta integral in which the two bases and are interrelated
by modular inversion, and they entail -analogues of the beta integral for
. The integrals under consideration are the hyperbolic analogues of the
Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman
integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal
limit case of Spiridonov's elliptic Nassrallah-Rahman integral.Comment: 35 pages. Remarks and references to recent new developments are
added. To appear in Adv. Mat
Connection problems for quantum affine KZ equations and integrable lattice models
Cherednik attached to an affine Hecke algebra module a compatible system of
difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ)
equations. In case of a principal series module we construct a basis of power
series solutions of the quantum affine KZ equations. Relating the bases for
different asymptotic sectors gives rise to a Weyl group cocycle, which we
compute explicitly in terms of theta functions. For the spin representation of
the affine Hecke algebra of type C the quantum affine KZ equations become the
boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show
that in this special case the results lead to an explicit 4-parameter family of
elliptic solutions of the dynamical reflection equation associated to Baxter's
8-vertex face dynamical R-matrix. We use these solutions to define an explicit
9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard
(KZB) equations.Comment: 47 pages. v2: small corrections; v.3: small corrections in the proof
of Thm. 3.1
An expansion formula for the Askey-Wilson function
The Askey-Wilson function transform is a q-analogue of the Jacobi function
transform with kernel given by an explicit non-polynomial eigenfunction of the
Askey-Wilson second order q-difference operator. The kernel is called the
Askey-Wilson function. In this paper an explicit expansion formula for the
Askey-Wilson function in terms of Askey-Wilson polynomials is proven. With this
expansion formula at hand, the image under the Askey-Wilson function transform
of an Askey-Wilson polynomial multiplied by an analogue of the Gaussian is
computed explicitly. As a special case of these formulas a q-analogue (in one
variable) of the Macdonald-Mehta integral is obtained, for which also two
alternative, direct proofs are presented.Comment: 24 pages. Some remarks added in section 6 on the connection with
moment problem
- β¦