A bispectral q-hypergeometric basis for a class of quantum integrable models

Abstract

Abstract. 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 with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q = 1).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

Theq-Onsager algebra and multivariableq-special functions

“…One then note that these recurrence relations have a structure analogous to that of the bivariate Hahn polynomials and GasperRahman polynomials found in [GV14] and [BM15], respectively. This suggests that the coefficients C {ñ} {n} could be related to known multivariate q−polynomials.…”

“…Such expressions should find applications in the context of quantum integrable models associated with higher rank symmetries, and provide a q−hypergeometric formulation of these models. For the sl 2 case, an example of such description is given in [BM15].…”

“…While a construction of a higher rank Askey-Wilson algebra along those lines is still awaited, an alternative framework to extend the algebraic picture to the multivariate realm for q not a root of unity is provided by the q−Onsager algebra [T99, B04]. In this last context, infinite and finite dimensional modules of the q−Onsager algebra have been constructed in terms of the multivariate Gasper-Rahman polynomials, in [BM15] N −pairs of operators generating the q−Onsager algebra were related to Iliev's families of Date: August, 2017. 1 The defining relations of the Askey-Wilson algebra are in terms of a scalar parameter q.…”

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

“…As we have constructed a similar abelian subalgebra of A q n that diagonalizes our basis, it seems plausible that the action of the diagonal operators in [27] can be extended to an action of the full algebra A q n . This would moreover complement the realization of the q-Onsager algebra by Ilievs difference operators proposed in [6]. These highly technical issues will be discussed in our subsequent work [8].…”

“…However, the operators Γ q A for |A| ∈ {2, 3} do not coincide with the tensor product elements Γ q A defined in (13)- (14). This is because the coproduct (43) on U Q (sl 2 ) is not compatible with the one on osp q (1|2) given in (6) under the correspondence (47). The relations (50) hence establish a different embedding of the q-Bannai-Ito algebra inside osp q (1|2) ⊗3 .…”