Generalized shift elements and classical \(r\)-matrices: construction and applications

DOI10.1016/j.geomphys.2013.12.011zbMath1303.37028OpenAlexW1972467023MaRDI QIDQ2442612

T. V. Skrypnik

Publication date: 4 April 2014

Published in: Journal of Geometry and Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.geomphys.2013.12.011

zbMATH Keywords

integrable systemsLax operatorsclassical \(r\)-matrices

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Applications of Lie algebras and superalgebras to integrable systems (17B80) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures (37K30) Yang-Baxter equations (16T25)

Related Items (16)

Elliptic BCS-Richardson model and the modified algebraic Bethe ansatz ⋮ Ground-state energies of the open and closed \(p + ip\)-pairing models from the Bethe Ansatz ⋮ Integrable spin-${\frac{1}{2}}$ Richardson–Gaudin XYZ models in an arbitrary magnetic field ⋮ Reduction in soliton hierarchies and special points of classical \(r\)-matrices ⋮ ``Generalized algebraic Bethe ansatz, Gaudin-type models and \(Z_{p}\)-graded classical \(r\)-matrices ⋮ Reductions in finite-dimensional integrable systems and special points of classical r-matrices ⋮ Classical $r$-matrices, ``elliptic BCS and Gaudin-type Hamiltonians and spectral problem ⋮ Separation of variables in anisotropic models: anisotropic Rabi and elliptic Gaudin model in an external magnetic field ⋮ Elliptic Gaudin-type model in an external magnetic field and modified algebraic Bethe ansatz ⋮ Non-skew-symmetric classical r-matrices and integrable ``\(p_x +ip_y\) proton-neutron BCS models ⋮ Elliptic three-boson system, ``two-level three-mode JCD-type models and non-skew-symmetric classical r-matrices ⋮ Solution of the classical Yang-Baxter equation with an exotic symmetry, and integrability of a multi-species boson tunnelling model ⋮ General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters ⋮ Separation of variables for linear Lax algebras and classical r-matrices ⋮ Anisotropic BCS-Richardson model and algebraic Bethe ansatz ⋮ The generalized Lipkin-Meshkov-Glick model and the modified algebraic Bethe ansatz

Cites Work

• Integrable quantum spin chains, non-skew symmetric \(r\)-matrices and quasigraded Lie algebras
• Spin chains in magnetic field, non-skew-symmetric classical r-matrices and BCS-type integrable systems
• The Lagrange rigid body motion
• The Kowalevski top: its \(r\)-matrix interpretation and bihamiltonian formulation
• Lie algebras with triangular decompositions, non-skew-symmetric classical \(r\)-matrices and Gaudin-type integrable systems
• Non-skew-symmetric classical r-matrices, algebraic Bethe ansatz, and Bardeen–Cooper–Schrieffer–type integrable systems
• Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Generalized Gaudin systems in an external magnetic field and reflection equation algebras
• Non-skew-symmetric classicalr-matrices and integrable cases of the reduced BCS model
• Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz
• Generalized Gaudin spin chains, nonskew symmetric r-matrices, and reflection equation algebras
• Generalized Gaudin systems in a magnetic field and non-skew-symmetricr-matrices
• Quadratic algebras and integrable systems
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