Double Bruhat Cells in Kac-Moody Groups and Integrable Systems

TL;DR

This paper constructs a family of integrable Hamiltonian systems with phase spaces as double Bruhat cells in Kac-Moody groups, extending known results from simple algebraic groups and Poisson-Lie theory.

Contribution

It generalizes the construction of integrable systems to Kac-Moody groups and extends fundamental results in double Bruhat cells and Poisson-Lie theory to ind-algebraic groups.

Findings

Constructed integrable systems generalizing the relativistic periodic Toda lattice.

Extended results on double Bruhat cells to Kac-Moody groups.

Generalized Poisson-Lie theory to ind-algebraic groups.

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.