Deformed Dolan-Grady relations in quantum integrable models

Abstract

A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators $\{\text{A},{\text{A}}^{\ast }\}\in \mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathcal{A}$ 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.

24 pages, LaTeX file with amssymb; v2: Clarifications to the text, references added; v3: Minor changes, misprints corrected, one reference added; to appear in Nucl.Phys.B

References in corpus (3)

• An integrable structure related with tridiagonal algebras
• Some algebra related to $P$-and $Q$-polynomial association schemes
• Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations

Cited by in corpus (62)

• Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
• The Bond-Algebraic Approach to Dualities
• The half-infinite XXZ chain in Onsager's approach
• An integrable structure related with tridiagonal algebras
• A new (in)finite dimensional algebra for quantum integrable models
• The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach
• The Universal Askey-Wilson Algebra
• A deformed analogue of Onsager's symmetry in the XXZ open spin chain
• A new current algebra and the reflection equation
• From affine Hecke algebras to boundary symmetries
• A family of tridiagonal pairs and related symmetric functions
• On $A_{n-1}^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)},A_{2n}^{(2)},A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$ Reflection $K$-Matrices
• The open XXZ and associated models at q root of unity
• Coideal algebras from twisted Manin triples
• The Bethe Ansatz Equations for Reflecting Magnons
• The Askey-Wilson algebra and its avatars
• Boundary non-local charges from the open spin chain
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• Non-Abelian symmetries of the half-infinite XXZ spin chain
• Braided module categories via quantum symmetric pairs
• The alternating central extension of the $q$-Onsager algebra
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• Generalized q-Onsager algebras and boundary affine Toda field theories
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• The alternating PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A conjecture concerning the $q$-Onsager algebra
• A classification of sharp tridiagonal pairs
• Truncation of the reflection algebra and the Hahn algebra
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The $q$-Onsager algebra and its alternating central extension
• Drinfeld J Presentation of Twisted Yangians
• From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Analogues of Lusztig's higher order relations for the q-Onsager algebra
• Central extension of the reflection equations and an analog of Miki's formula
• Mock Tridiagonal Systems
• Tridiagonal pairs of $q$-Racah type
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras
• Integrable boundaries for the q-Hahn process
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Generic triangular solutions of the reflection equation: $U_q(\widehat{s{l}_2})$ case
• Murphy elements from the double-row transfer matrix
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Higher order relations for ADE-type generalized q-Onsager algebras
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Time and band limiting operator and Bethe ansatz
• Higher Rank Askey-Wilson Algebras as Skein Algebras
• Tridiagonal pairs, alternating elements, and distance-regular graphs
• Integrability of the N=2 boundary sine-Gordon model
• On the shape of a tridiagonal pair
• The $S_3$-symmetric tridiagonal algebra
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Freidel-Maillet type presentations of $U_q(s{l}_2)$
• Two non-nilpotent linear transformations that satisfy the cubic $q$-Serre relations
• A new PBW basis for the alternating central extension of the $q$-Onsager algebra
• Finite size effects in integrable quantum field theory: the sine-Gordon model with boundaries