Double Bruhat Cells in Kac-Moody Groups and Integrable Systems

Abstract

We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter elements in the affine Weyl group. In the process we extend various results on double Bruhat cells in simple algebraic groups to the setting of Kac-Moody groups. We also generalize some fundamental results in Poisson-Lie theory to the setting of ind-algebraic groups, which is of interest beyond our immediate applications to integrable systems.

24 pages; to appear in Lett. Math. Phys

References in corpus (8)

• Five Dimensional Gauge Theories and Relativistic Integrable Systems
• Kac-Moody groups and cluster algebras
• Dimer Models and Integrable Systems
• Factorization dynamics and Coxeter-Toda lattices
• Lie Groups, Cluster Variables and Integrable Systems
• Dimers and cluster integrable systems
• On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups
• Connected components of real double Bruhat cells

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• Toda Systems, Cluster Characters, and Spectral Networks
• Symplectic groupoids for cluster manifolds
• Shifted Poisson geometry and meromorphic matrix algebras over an elliptic curve
• Q-Systems, Factorization Dynamics, and the Twist Automorphism
• Affine cluster monomials are generalized minors