Double Bruhat Cells in Kac–Moody Groups and Integrable Systems

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.