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

Mingyan Simon Lin

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 Bäcklund-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 Bäcklund-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.

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

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 Bäcklund-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 Bäcklund-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.

Paper Structure

This paper contains 24 sections, 46 theorems, 177 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline of the paper
  4. Preliminaries
  5. Basic Lie theoretic data
  6. Double Bruhat cells
  7. Coxeter-Toda systems
  8. Cluster algebras
  9. Q-systems
  10. Elementary chips and network representations of the simple classical Lie groups
  11. Cluster structures on Coxeter double Bruhat cells
  12. Cluster structures on double Bruhat cells
  13. Cluster structures on conjugation quotients of Coxeter double Bruhat cells
  14. Generalized Bäcklund-Darboux transformations
  15. The elementary generalized Bäcklund-Darboux transformations
  16. ...and 9 more sections

Key Result

Theorem 2.1

FZ99 Let $u,v\in W$, and $\mathbf{i}=(i_1,\ldots,i_m)$ be a double reduced word for $(u,v)$. Then the map $x_{\mathbf{i}}:H\times\mathbb{C}^m\to G$, given by restricts to a biregular isomorphism between $x_{\mathbf{i}}:H\times(\mathbb{C}^{\times})^m$ and a Zariski open subset of the double Bruhat cell $G^{u,v}$.

Figures (33)

  • Figure 1: The elementary chips of $E_i(a_i)$, $E_{-i}(b_i)$ and $D(t_1,\ldots,t_r)$ in type $A$.
  • Figure 2: The elementary chip of $E_i(a_i)$, $E_{-i}(b_i)$, $i=1,\ldots,r-1$ in type $B$.
  • Figure 3: The elementary chips of $E_r(a_r)$, $E_{-r}(b_r)$, and $D(t_1,\ldots,t_r)$ in type $B$.
  • Figure 4: The elementary chips of $E_i(a_i)$, $E_{-i}(b_i)$, $i=1,\ldots,r-1$ in type $C$.
  • Figure 5: The elementary chips of $E_r(a_r)$, $E_{-r}(b_r)$ and $D(t_1,\ldots,t_r)$ in type $C$.
  • ...and 28 more figures

Theorems & Definitions (97)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5: Seed
  • Definition 2.6: Cluster mutation
  • Definition 2.7: Cluster variables and $\mathcal{X}$-coordinates
  • Definition 2.8: Cluster transformations
  • Definition 2.9: $\sigma$-periods
  • Theorem 2.10
  • ...and 87 more