The Algebraic Approach to Duality: An Introduction

TL;DR

This survey introduces the algebraic approach to Markov process duality, emphasizing the use of Lie algebra representations to simplify the analysis of duality functions, contrasting with the pathwise approach.

Contribution

It provides an elementary overview of the algebraic method for duality, highlighting recent developments involving Lie algebra representations and connecting older work on duality functions.

Findings

Algebraic approach offers a systematic way to analyze duality via Lie algebra representations.

Recent suggestions improve the choice of operators for duality, enhancing analytical tools.

Connections between intertwining and duality are clarified.

Abstract

This survey article gives an elementary introduction to the algebraic approach to Markov process duality, as opposed to the pathwise approach. In the algebraic approach, a Markov generator is written as the sum of products of simpler operators, which each have a dual with respect to some duality function. We discuss at length the recent suggestion by Giardin\`a, Redig, and others, that it may be a good idea to choose these simpler operators in such a way that they form an irreducible representation of some known Lie algebra. In particular, we collect the necessary background on representations of Lie algebras that is crucial for this approach. We also discuss older work by Lloyd and Sudbury on duality functions of product form and the relation between intertwining and duality.