A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators {A, A * } ∈ A subject to q−deformed Dolan-Grady relations. Using the inverse scattering method, a new family of quantum integrable models is proposed. In the simplest case, the Hamiltonian is linear in the fundamental generators of A. For general values of q, the corresponding spectral problem is quasi-exactly solvable. Several examples of two-dimensional massive/massless (boundary) integrable models are reconsidered in light of this approach, for which the fundamental generators of A are constructed explicitly and exact results are obtained. In particular, we exhibit a dynamical Askey-Wilson symmetry algebra in the (boundary) sine-Gordon model and show that asymptotic (boundary) states can be expressed in terms of q−orthogonal polynomials.

A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and mutually commuting quantities -which ensure the integrability of the system -are written in terms of the fundamental generators of the new algebra. Relation with the deformed Dolan-Grady integrable structure recently discovered by one of the authors and Terwilliger's tridiagonal algebras is described. Remarkably, this (in)finite dimensional algebra is a "q−deformed" analogue of the original Onsager's algebra arising in the planar Ising model. Consequently, it provides a new and alternative algebraic framework for studying massive, as well as conformal, quantum integrable models.andwith fixed scalar ρ and k, l ∈ N. Finite dimensional representations are obtained, and examples of quantum integrable systems (XXZ spin chain, Sine-Gordon and Liouville field theories) which enjoy this symmetry are given. More generally, our framework opens the possibility of analyzing a large class of quantum integrable models from a new point of view, in the spirit of Onsager's approach [1]. The paper is organized as follows. In Section 2, the fundamental relations (4) are derived using its relation with a class of quadratic algebra, namely the reflection equation. Indeed, finite dimensional representations of its fundamental generators are shown to be generated from general solutions of the reflection equation. In particular, the closure of the algebra is ensured by the existence of a set of linear relations among the generators. Explicit expressions of mutually commuting quantities in terms of W −k , W k+1 , G k+1 ,G k+1 , generalizing (3) are obtained. Also, we argue that similarly to the undeformed case, the integrable structure generated from our "q−deformed" Onsager's algebra coincides with the deformed Dolan-Grady integrable structure recently discovered in [13,14]. In particular, we exhibit the correspondence in the simplest cases. In the last Section, we give some examples of quantum integrable systems which enjoy this (in)finite dimensional symmetry. Structure of the algebraIn the last thirty years, one of the most important progress in the approach of quantum integrable systems has been based on the star-triangle relations which originated in [1,15] and led to the Yang-Baxter equations, the theory of quantum groups as well as the quantum inverse scattering method. Although the star-triangle relations

The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on the tridiagonal algebraic structure associated with a deformation parameter $q$. Representations are shown to be generated from a class of quadratic algebras, namely the reflection equations. The spectral problem is briefly discussed. Finally, related massive quantum integrable models are shown to be superintegrable.Comment: 11 pages; LaTeX file with amssymb; ; v2: typos corrected, one reference added, to appear in Nucl. Phys.

The transfer matrix of the XXZ open spin-1 2 chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q−Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al. and Cao et al.. In the latter case the complete family of eigenvalues and eigenstates splits in two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.

The XXZ open spin chain with general integrable boundary conditions is shown to possess a q−deformed analogue of the Onsager's algebra as fundamental non-abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed 'Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime −1 < q < 0, level one infinite dimensional representation (q−vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo et al. are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using q−bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for q generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the q−Onsager algebra (generic non-diagonal case) or the augmented q−Onsager algebra (generic diagonal case). MSC: 81R50; 81R10; 81U15.Keywords: XXZ open spin chain; q−Onsager algebra; q−vertex operators; Thermodynamic limit Potts models [PeAMT, Dav, vGR] are explicit counterexamples of this idea (see also [Ar, AS]).

A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order q−difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.

We calculate the vacuum expectation values of local fields for the two-parameter family of integrable field theories introduced and studied in [11]. Using this result we propose an explicit expression for the vacuum expectation values of local operators in parafermionic sine-Gordon models and in integrable perturbed SU (2) coset conformal field theories.