2025/2026
Пуассоновы группы Ли
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Маршалл Йен Донен
Язык:
английский
Контактные часы:
30
Course Syllabus
Abstract
The first significant motivation for Poisson Lie Groups comes from the dressing transformation, when it was applied (in the mid 1970s) by Zakharov and Shabat in the context of inverse scattering theory, for generating soliton solutions to nonlinear systems.
With the emergence of an understanding that solitons and inverse scattering may naturally be interpreted as infinite-dimensional versions of integrable systems of classical mechanics, a natural question arose of how to cast the Zakharov – Shabat dressing as a symmetry of an appropriate Hamiltonian system. This conundrum was resolved (in the early 1980s) by Semenov – Tian-Shansky, with the essential observation that — as well as the system on which it acts — the group involved in the symmetry transformation might itself carry a Poisson bracket.
The notion of Poisson Lie Group was already not a new one. It had been observed by Drinfeld as being implicit in the classical Yang-Baxter equation, and as representing a classical version of Quantum Group; first proposed as far as back as in work of Wigner.
This idea of implementing Poisson Lie Groups as a useful tool in Classical Mechanics opened up a whole new area of mathematics.
In the proposed course we hope to uncover the simplest and most accessible parts of this theory. We shall study especially how it can be used in the context of Hamiltonian reduction to describe integrable Hamiltonian systems.