Generalized Bäcklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups

In the first part of this thesis, we construct generalized Bäcklund-Darboux transformations between two Coxeter-Toda systems on simple Lie groups from a cluster algebraic prospective, using the cluster structure on Coxeter double Bruhat cells developed by Fock-Goncharov, Gekhtman et al. and Williams, thereby generalizing the construction developed by Gekhtman et al. We will then show that these generalized Backlund-Darboux transformations preserve Hamiltonian flows generated by the restriction of the trace function of any representation of the simple Lie group. In the second part of this thesis, we develop network formulations of the Coxeter-Toda Hamiltonians for the classical Lie groups, and use these network formulations to obtain combinatorial formulas for the conserved quantities of certain Q-systems.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Lin, Mingyan Simon

Contributors dc:contributor

• Kedem, Rinat
• Di Francesco, Philippe
• Yong, Alexander
• Gekhtman, Michael

• Cluster algebras
• Integrable systems

Statement dc:rights

• Copyright 2021 Mingyan Lin

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113186

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/113186