Mini-course

(partially supported by a travel grant from LMS)

Quantum Groups and Yang-Baxter Equation for Probabilists

Abstract: In the field of "integrable probability," many models and results are motivated by the quantum Yang--Baxter Equation, which has its origins in quantum groups. These quantum groups, introduced by Drinfeld' and Jimbo in 1985, are q--deformations of classical Lie algebras. In these lectures, we provide an introduction to these quantum groups. The intuition and motivation will be tailored for an expert in probability theory, with minimal algebraic background necessary. An outline of each lecture is provided below

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Lecture 1. (Wdn. 30/10/2024, 1100-1200, Zeeman Building, Room B3.01)

Definition of Quantum Groups

We provide a definition for quantum groups, which are actually Hopf algebras which q-deform the classical Lie algebras. We will swiftly define the simple and affine Lie algebras, briefly stating the classification theorem, as well as the definition of Hopf algebras. These definitions will be given probabilistic and physical intuition and motivation, coming from interacting particle systems and spin chains.

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Lecture 2 (Thu. 31/10/2024, 1400-1500, Zeeman Building, Room MS.03)

R-matrices and Yang-Baxter Equation

The quantum groups in the previous lecture provide solutions to the "quantum Yang--Baxter equation" (qYBE). In the second lecture, we discuss various solutions to qYBE, and their relationship to interacting particle systems and stochastic vertex models. In particular, we emphasize how these R-matrices can be related to stochastic S-matrices.

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Lecture 3 (Fri. 01/11/2024, 1400-1500, Zeeman Building, Room B1.01)

Contemporary Research in Probability

In the final lecture, we review some key probabilistic results which use the Yang-Baxter equation. These are results in integrable probability (broadly defined), and use solutions to the Yang--Baxter equation from the previous lecture.

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Warwick Probability Seminar (Wdn. 30/10/2024, 16:00-17:00, Zeeman Building, Room B3.02)

Universality of dynamic processes using Drinfel'd twisters

Abstract: The concept of 'universality' motivates a wide variety of probability and mathematical physics problems, going back to the classical central limit theorem. Most recently, the Kardar--Parisi--Zhang universality class has been proven to have Tracy--Widom fluctuations in the long-time asymptotics. In this talk, I will present a new universality result about the long-time asymptotics of so--called ``dynamic'' processes. The asymptotic fluctuations are related to the Tracy--Widom distribution. The proof will utilize a duality of Markov processes, which is constructed using Drinfel'd twisters of the quantum group U_q(sl_2), viewed as a quasi--triangular quasi-Hopf algebra. The orthogonality of the duality functions allow for an asymptotic analysis.