Lax Matrices & Clusters for Type A & C Q-Deformed Open Toda Chain

TL;DR

This paper reveals that a family of quantum Toda chains associated with Lie groups can be unified as a single cluster integrable system, which becomes unitary equivalent under positive representations, connecting different Coxeter word parameterizations.

Contribution

The authors demonstrate that various quantum Toda chains parameterized by Coxeter words are actually different clusters of a single integrable system, extending previous constructions.

Findings

Unified quantum Toda chains as a single cluster integrable system

Different Coxeter word parameterizations are cluster charts

Systems become unitarily equivalent in positive representations

Abstract

At the turn of the century, Etingof and Sevostyanov independently constructed a family of quantum integrable systems, quantizing the open Toda chain associated to a simple Lie group $G$. The elements of this family are parameterized by Coxeter words of the corresponding Weyl group. Twenty years later, in the works of Finkelberg, Gonin, and Tsymbaliuk, this was generalized to a family of quantum Toda chains parameterized by pairs of Coxeter words. In this paper, we show that this family is actually a single cluster integrable system written in different clusters associated to cyclic double Coxeter words. Furthermore, if we restrict the action of Hamiltonians to its positive representation, these systems become unitary equivalent.