Introduction to Quantum Groups and Yang-Baxter Equation For Probabilists

TL;DR

This paper introduces quantum groups and the Yang-Baxter equation to probabilists, explaining their algebraic structures, applications to integrable stochastic models like ASEP, and related topics through lecture notes.

Contribution

It provides an accessible introduction to quantum groups and the Yang-Baxter equation tailored for probabilists, including algebraic background and applications to stochastic models.

Findings

Defines integrability of ASEP via Yang-Baxter equation

Connects quantum groups to stochastic vertex models

Includes algebraic tools like Hecke algebras and matrix product ansatz

Abstract

These are a set of lecture notes for a mini-course I gave at The University of Warwick from October 30th to November 1st, 2024. Recordings of the lectures are available on Oleg Zaboronski's webpage at https://warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/jeffrey_kuan_visit/ . The main body of the notes covers the content of the lectures, and provides an introduction to Drinfel'd-Jimbo quantum groups and the Yang-Baxter equation, with a probabilist as the target audience. The appendix contains several topics, requested by colleagues during my visit to the United Kingdom, which all depend on the main set of notes. The notes begin by defining what it means for the asymmetric simple exclusion process (ASEP) to be integrable, in the sense of satisfying the Yang-Baxter equation. It then provides the algebraic background for the Yang-Baxter equation, by defining Drinfel'd-Jimbo groups as a quasi-triangular Hopf algebra. The algebraic background motivates generalizations of ASEP to stochastic vertex models and "fused" models. Each section corresponds to approximately an hour of lecture time. The appendix covers the F.R.T. construction, Hecke algebras, the matrix product ansatz, and orthogonal polynomial vertex weights. The topics in the appendix can be read independently of each other. Accessibility Statement: This PDF meets the technical standards of WCAG2.1AA, which complies with Ohio Administrative Policy IT-09 , Texas Administrative Code 206.70 and Title II of the Americans with Disabilities Act (effective April 24, 2026) . A webpage version of this PDF, typeset in MathML, is also available at https://go.osu.edu/QuantumKuan . To block web crawlers, the webpage is password protected. The password is TaySwift13. As an additional benefit, the webpage will have space for public comments and a list of updated errata, without the need to update the arXiv version.