Generalized B\"acklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups

TL;DR

This paper develops a unified cluster algebra framework for Coxeter-Toda systems on simple Lie groups, introducing generalized Bäcklund-Darboux transformations and combinatorial formulas, thereby advancing integrable systems theory.

Contribution

It generalizes cluster structures and Bäcklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups, linking them through cluster mutations and network formulations.

Findings

Constructed cluster structures on conjugation quotient Coxeter double Bruhat cells.

Developed generalized Bäcklund-Darboux transformations preserving Hamiltonian flows.

Provided combinatorial formulas for Coxeter-Toda Hamiltonians using network models.

Abstract

We derive the cluster structure on the conjugation quotient Coxeter double Bruhat cells of a simple Lie group from that on the double Bruhat cells of the corresponding adjoint Lie group given by Fock and Goncharov using the notion of amalgamation given by Fock and Goncharov, and Williams, thereby generalizing the construction developed by Gekhtman \emph{et al}. We will then use this cluster structure on the conjugation quotient Coxeter double Bruhat cells to construct generalized B\"{a}cklund-Darboux transformations between two Coxeter-Toda systems on simple Lie groups in terms of cluster mutations, thereby generalizing the construction developed by Gekhtman \emph{et al}. We show that these generalized B\"{a}cklund-Darboux transformations preserve Hamiltonian flows generated by the restriction of the trace function of any representation of the simple Lie group, from which we deduce that the family of Coxeter-Toda systems on a simple Lie group forms a single cluster integrable system. Finally, we also develop network formulations of the Coxeter-Toda Hamiltonians for the classical Lie groups, and use these network formulations to obtain combinatorial formulas for these Coxeter-Toda Hamiltonians.