Compatible Lie brackets and the Yang-Baxter equation

DOI10.1007/s11232-006-0016-6zbMath1177.37067OpenAlexW1984238964MaRDI QIDQ1048006

Igor Z. Golubchik, Vladimir Sokolov

Publication date: 9 January 2010

Published in: Theoretical and Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11232-006-0016-6

zbMATH Keywords

Yang-Baxter equationLie bialgebraManin triple

Mathematics Subject Classification ID

Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Lie bialgebras; Lie coalgebras (17B62)

Related Items (15)

R-matrix, Lax pair, and multiparameter decompositions of Lie algebras ⋮ Noncommutative Poisson bialgebras ⋮ The category and operad of matching dialgebras ⋮ Compatible \(L_\infty \)-algebras ⋮ Cohomology and deformations of compatible Lie triple systems ⋮ Maurer-Cartan characterizations and cohomologies of compatible Lie algebras ⋮ Cohomology and deformations of compatible Hom-Lie algebras ⋮ (Co)homology of compatible associative algebras ⋮ Homotopy transfer theorem for linearly compatible di-algebras ⋮ Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras ⋮ Bi-Hamiltonian structures and singularities of integrable systems ⋮ Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations ⋮ Generalization of the concept of classical \(r\)-matrix to Lie algebroids ⋮ Algebraic properties of compatible Poisson brackets ⋮ Universal enveloping algebra of a pair of compatible Lie brackets

Cites Work

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• Compatible Poisson brackets on Lie algebras
• Compatible Lie brackets and integrable equations of the principal chiral model type
• A simple model of the integrable Hamiltonian equation
• Factorization of the Loop Algebras and Compatible Lie Brackets
• Quantum groups
• Solutions of the classical Yang-Baxter equation for simple Lie algebras

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