2024/2025
Infinite-Dimensional Lie Algebras
ID 906192
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Filipp Uvarov
Language:
English
ECTS credits:
3
Course Syllabus
Abstract
Infinite-dimensional Lie algebras naturally arise in quantum field theory and differential geometry (for example, as Lie algebras of vector fields). Unlike in finite-dimensional case, the understanding of infinite-dimensional Lie algebras and their representations is far from being complete. In this course, we will cover the most famous and studied examples of infinite-dimensional Lie algebras, whose structure theory and representations on the one hand, resemble those of finite-dimensional simple Lie algebras, on the other hand, posses new interesting features.
Course Contents
- Representations of 𝔤𝔩(∞). Semi-infinite wedge product. Japanese cocycle.
- Heisenberg and Virasoro algebras, Fock modules.
- Affine Lie algebras as central extensions of loop algebras. The Sugawara construction.
- Generalized Cartan matrices and Kac – Moody Lie algebras.
- Affine Lie algebras as Kac – Moody Lie algebras.
