Integrable quantum spin chains, non-skew symmetric \(r\)-matrices and quasigraded Lie algebras

DOI10.1016/j.geomphys.2006.02.002zbMath1101.81064OpenAlexW1969640107MaRDI QIDQ856759

T. V. Skrypnik

Publication date: 7 December 2006

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

Full work available at URL: https://doi.org/10.1016/j.geomphys.2006.02.002

zbMATH Keywords

integrable systemsinfinite-dimensional Lie algebrasclassical \(r\)-matrices

Mathematics Subject Classification ID

Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, (W)-algebras and other current algebras and their representations (81R10) Applications of Lie groups to the sciences; explicit representations (22E70) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Exactly solvable models; Bethe ansatz (82B23) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20) Groups and algebras in quantum theory and relations with integrable systems (81R12)

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Cites Work

• Unnamed Item
• Definition of functions for generalized affine Lie algebras
• What is a classical r-matrix?
• Separation of variables in the elliptic Gaudin model.
• Compatible Lie brackets and integrable equations of the principal chiral model type
• Hitchin systems, higher Gaudin operators and \(R\)-matrices
• Quasigraded deformations of loop algebras and classical integrable systems.
• Quasigraded Lie algebras on hyperelliptic curves and classical integrable systems
• Classical Yang–Baxter equations and quantum integrable systems
• Integrability and exact spectrum of a pairing model for nucleons
• Quadratic algebras and integrable systems

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