Symmetry, Integrable Chain Models and Stochastic Processes

TL;DR

This paper presents a method to construct symmetric integrable chain models based on Lie algebraic structures, linking them to exactly solvable Markov chains with shared spectral properties.

Contribution

It introduces a general construction for symmetric integrable chain models and establishes their connection to solvable Markov chains with identical spectra.

Findings

Constructed chain models with Lie algebraic symmetries.

Linked integrable models to exactly solvable Markov chains.

Demonstrated spectral equivalence between models and chains.

Abstract

A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the corresponding integrable models.