A new (in)finite-dimensional algebra for quantum integrable models

Abstract

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

A deformed analogue of Onsager’s symmetry in the XXZ open spin chain

“…In addition, later on it was argued [18] that this algebra is related with a q−deformed analogue of the Onsager's algebra 5 constructed in [18]. As briefly mentionned in [15,18] this remarkable tridiagonal algebraic structure survives for the most general (non-diagonal) solutions of the reflection equation.…”

“…5 For q = 1 and special value of ρ, the algebra introduced in [18] can be related with the Onsager's algebra which played a crucial role in the original solution of the planar Ising model (see [2] for details).…”

“…8 To understand the general case N , it is instructive to look at the special cases N = 1 and N = 2 first [15,18]. For N = 1 in where, for the special case k = 0 we identify…”

“…Expanding in the spectral parameter u, one obtains a pair of relations linear in terms of the elements W −k , W k+1 which ensures the closure of the algebra. For N = 1, they correspond to the Askey-Wilson relations [14,15] [18]. In general, given N a pair of relations of order 2N + 1 in W 0 , W 1 exists, which can be written as a pair of linear recurrence relation of length N + 1 in the fundamental elements W −k , W k+1 , k = 0, 1, ..., N where W −N , W N +1 are nonlinear combinations of lower elements W −k , W k+1 with k = 0, 1, ..., N − 1.…”

“…10 More general relations can be obtained as a consequence of these fundamental ones. For details, we refer the reader to [18], Appendix B. For q = 1, note that the relations (17), (18) as well as their generalizations [18] reduce to the finite recurrence relations for the generators A k , G l in Onsager's algebra [21].…”