Quadratic algebras related to elliptic curves

DOI10.1007/s11232-008-0081-0zbMath1160.37023arXiv0710.1072OpenAlexW2593391005MaRDI QIDQ954337

B. E. Eshmatov

Publication date: 10 November 2008

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

Full work available at URL: https://arxiv.org/abs/0710.1072

zbMATH Keywords

Poisson structureintegrable system

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics (70H06) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures (37K30) Poisson algebras (17B63) Quadratic algebras (but not quadratic Jordan algebras) (17A45)

Related Items (13)

Difference operators of Sklyanin and van Diejen type. ⋮ Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems, and KZB equations ⋮ Classical integrable systems and soliton equations related to eleven-vertex \(R\)-matrix ⋮ Spectral duality between Heisenberg chain and Gaudin model ⋮ Modifications of bundles, elliptic integrable systems, and related problems ⋮ Trigonometric integrable tops from solutions of associative Yang-Baxter equation ⋮ Three-particle integrable systems with elliptic dependence on momenta and theta function identities ⋮ Quadratic algebras based on \(SL(NM)\) elliptic quantum \(R\)-matrices ⋮ Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices ⋮ Multiplicative Hitchin systems and supersymmetric gauge theory ⋮ Relativistic elliptic matrix tops and finite Fourier transformations ⋮ Field analogue of the Ruijsenaars-Schneider model ⋮ Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

Cites Work

• Sklyanin elliptic algebras
• Painlevé VI, rigid tops and reflection equation
• Some algebraic structures connected with the Yang-Baxter equation
• Discrete groups and the integrability of quantum systems
• Gaudin model, KZ equation and an isomonodromic problem on the torus
• Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus
• Elliptic Schlesinger system and Painlevé VI
• Classicalr-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions
• Solutions of the classical Yang-Baxter equation for simple Lie algebras

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