Lie Groups, Cluster Variables and Integrable Systems

Abstract

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to several alternative descriptions of relativistic Toda systems, but allows to formulate in general terms some new class of integrable models.

Based on talks given at Versatility of integrability, Columbia University, May 2011; Simons Summer Workshop on Geometry and Physics, Stony Brook, July-August 2011; Classical and Quantum Integrable Systems, Dubna, January 2012; Progress in Quantum Field Theory and String Theory, Osaka, April 2012; Workshop on Combinatorics of Moduli Spaces and Cluster Algebras, Moscow, May-June 2012

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• N=2 Quantum Field Theories and Their BPS Quivers
• Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
• Cluster X-varieties, amalgamation and Poisson-Lie groups
• Discrete evolution for the zero-modes of the Quantum Liouville Model
• Higher pentagram maps, weighted directed networks, and cluster dynamics

Cited by in corpus (16)

• Exact quantization conditions for the relativistic Toda lattice
• Relativistic Classical Integrable Tops and Quantum R-matrices
• Cluster integrable systems, q-Painleve equations and their quantization
• Operators and higher genus mirror curves
• Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
• Cluster Toda chains and Nekrasov functions
• Cluster integrable systems and spin chains
• Solution of tetrahedron equation and cluster algebras
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• On Lie Groups and Toda Lattices
• Toda chain from the kink-antikink lattice
• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations
• Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons
• The cluster variety face of quantum groups
• Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$