An integrable structure related with tridiagonal algebras

Abstract

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.

11 pages; LaTeX file with amssymb; ; v2: typos corrected, one reference added, to appear in Nucl. Phys. B

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Cited by in corpus (57)

• 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
• Deformed Dolan-Grady relations in quantum integrable models
• 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
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$
• 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 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
• 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 classical Askey-Wilson algebras
• Generalized q-Onsager algebras and boundary affine Toda field theories
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• FRT presentation of the Onsager algebras
• 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
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Scalar product for the XXZ spin chain with general integrable boundaries
• 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
• The $q-$Onsager algebra and multivariable $q-$special functions
• Drinfeld J Presentation of Twisted Yangians
• 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 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
• Mock Tridiagonal Systems
• Central extension of the reflection equations and an analog of Miki's formula
• Tridiagonal pairs of $q$-Racah type
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Murphy elements from the double-row transfer matrix
• Some $q$-exponential formulas for finite-dimensional ${\square }_q$-modules
• 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
• Time and band limiting operator and Bethe ansatz
• On the shape of a tridiagonal pair
• The $S_3$-symmetric tridiagonal algebra
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• Double Lowering Operators on Polynomial
• 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
• The algebra ${\mathcal{A}}_q$, $q-$Onsager algebras and coideal subalgebras: two open problems