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.

17 pages; LaTeX file with amssymb; v2: typos corrected, references added, minor changes;v3: other typos corrected, version to appear in Nucl.Phys.B

References in corpus (4)

• 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
• Leonard pairs and the Askey-Wilson relations

Cited by in corpus (56)

• Quantum symmetric Kac-Moody pairs
• Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
• The half-infinite XXZ chain in Onsager's approach
• 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
• Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
• The open XXZ and associated models at q root of unity
• The Askey-Wilson algebra and its avatars
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• Non-Abelian symmetries of the half-infinite XXZ spin chain
• Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• The alternating central extension of the $q$-Onsager algebra
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• FRT presentation of the Onsager algebras
• Generalized q-Onsager algebras and boundary affine Toda field theories
• Onsager algebra and algebraic generalization of Jordan-Wigner transformation
• The alternating PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• 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
• Form factors of the half-infinite XXZ spin chain with a triangular boundary
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A conjecture concerning the $q$-Onsager algebra
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Truncation of the reflection algebra and the Hahn algebra
• A classification of sharp tridiagonal pairs
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager 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
• Analogues of Lusztig's higher order relations for the q-Onsager algebra
• 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
• Braid group action and quasi-split affine $ı$quantum groups I
• Tridiagonal pairs of $q$-Racah type
• Central extension of the reflection equations and an analog of Miki's formula
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• The algebra $U_q^{+}$ and its alternating central extension ${\mathcal{U}}_q^{+}$
• Murphy elements from the double-row transfer matrix
• Circular Hessenberg Pairs
• Tensor K-matrices for quantum symmetric pairs
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Tridiagonal pairs, alternating elements, and distance-regular graphs
• On the shape of a tridiagonal pair
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Higher order relations for ADE-type generalized q-Onsager algebras
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Two non-nilpotent linear transformations that satisfy the cubic $q$-Serre relations
• The algebra ${\mathcal{A}}_q$, $q-$Onsager algebras and coideal subalgebras: two open problems
• A new PBW basis for the alternating central extension of the $q$-Onsager algebra
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$