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401-3236-25L 4 Credits DR , MSC D-MATH

Infinite-Dimensional Lie Algebras and Integrable Systems

Lecturers & Examiners: Dr. Denis Nesterov
VVZ CR n/a

Last Updated: 2026-06-01 11:33:19

Abstract

The course is an introduction to the representation theory of infinite-dimensional Lie algebras, focusing on the Lie algebra of infinite matrices. The aim is to understand the Kadomtsev–Petviashvili equation using representation-theoretic methods. This equation exemplifies Integrable systems, a class of differential equations with rich symmetries enabling exact solutions.

Objective

The objective of the course is to introduce the basics of (infinite-dimensional) Lie algebras, their relation to Integrable systems and related mathematical objects, like Schur polynomials. The topics presented in the course will include: - Lie algebras and their representations - Heisenberg algebra and infinite matrices - Schur polynomials - Boson-Fermion correspondence - Infinite-dimensional Grassmannian - Integrable systems, Solitons and Kadomtsev–Petviashvili equation Using the theory developed throughout the course, we will describe solutions of the Kadomtsev–Petviashvili (KP) equation in terms of an infinite-dimensional Grassmannian and construct explicit solutions called solitons. The KP equation is a non-linear differential equation that models waves in shallow waters, fundamental to the theory of Integrable systems. The Lie algebra of infinite matrices can be regarded as an algebra of its symmetries. The approach to the subject will be almost purely algebraic. Other aspects of the KP equation, and Integrable systems in general, will be mentioned but not emphasised. The theory has widespread applications in enumerative geometry, mathematical physics, theory of symmetric functions, etc. For example, the smaller sister of the KP equation - Korteweg–De Vries equation - features in the context of moduli spaces of Riemann surfaces, e.g. Witten-Kontsevich theorem.

Content

The course will be mainly based on the relevant chapters of the book by Kac, Raina and Rozhkovskaya together with supplementary introductory material. The part of the course concerning integrable systems will also use the book by Miwa, Jimbo and Date and the lecture notes by Beisert. The relation between infinite matrices and integrable systems is discovered by Sato and Date, Jimbo, Kashiwara and Miwa.

Resources

Lecture Notes

Lecture notes will be typed in the process.

Literature

1. V. Kac, A.K. Raina and N. Rozhkovskaya, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, 2nd Edition, Advanced Series in Mathematical Physics: Volume 29 2. T. Miwa, M. Jimbo and E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press 3. N. Beisert, Introduction to Integrability, lecture notes, ETH Zurich HS16 4. T. Gomez, The boson-fermion correspondence, Bachelor's thesis, Utrecht University 2014

Learning Materials (Links)

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes
The exam is only offered in the summer 2025 and winter 2026 examination sessions.

Course Components

Type Title Time & Place Hours
lecture Infinite-Dimensional Lie Algebras and Integrable Systems
  • Thu 10:15-12:00 (HG G 26.5)
2 h weekly

Offered In