Loop groups, clusters, dimers and integrable systems. (English) Zbl 1417.37248
Álvarez-Consul, Luis (ed.) et al., Geometry and quantization of moduli spaces. Based on 4 courses, Barcelona, Spain, March – June 2012. Basel: Birkhäuser/Springer. Adv. Courses in Math., CRM Barcelona, 1-65 (2016).
From the introduction: The main idea of this work is to demonstrate the equivalence of two a priori different methods of construction and description of a wide class of integrable models, and thus, to propose a unified approach for their investigation. In the first, well-known method [A. G. Reyman and M. A. Semenov-Tian-Shansky, J. Sov. Math. 47, No. 2, 2493–2502 (1989; Zbl 0689.35091); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 176–188 (1987)], the phase space is taken as a quotient of double Bruhat cells of a Kac-Moody Lie group, with the Poisson structure defined by a classical \( r\)-matrix, and the integrals of motion are just the \(\text{Ad}\)-invariant functions.
For the entire collection see [Zbl 1362.14002].
For the entire collection see [Zbl 1362.14002].
MSC:
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 13F60 | Cluster algebras |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 82B23 | Exactly solvable models; Bethe ansatz |
Citations:
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