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Regular version of the site
2024/2025

Infinite-Dimensional Lie Algebras

Type: Optional course (faculty)
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.
Learning Objectives

Learning Objectives

  • -
Expected Learning Outcomes

Expected Learning Outcomes

  • ---
Course Contents

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.
Assessment Elements

Assessment Elements

  • non-blocking Sheets
  • non-blocking Exam
Interim Assessment

Interim Assessment

  • 2024/2025 4th module
    0.5*(problem lists)+0.5*(final exam)
Bibliography

Bibliography

Recommended Core Bibliography

  • Infinite dimensional groups with applications, , 1985

Recommended Additional Bibliography

  • Солитоны: дифференциальные уравнения, симметрии и бесконечномерные алгебры, Мива, Т., 2005

Authors

  • Иконописцева Юлия Вахтаногвна
  • Uvarov Filipp Viktorovich