Lie groups and Lie algebras
General data
Course ID: | 1000-135AGL |
Erasmus code / ISCED: | (unknown) / (unknown) |
Course title: | Lie groups and Lie algebras |
Name in Polish: | Algebry i grupy Liego |
Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
Course groups: |
(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics |
ECTS credit allocation (and other scores): |
6.00
|
Language: | English |
Main fields of studies for MISMaP: | mathematics |
Type of course: | elective courses |
Short description: |
Classical linear groups, abstract Lie groups, compact groups. Classical Lie's theory: correspondence between Lie groups and Lie algebras. The exponential map. Abstract approach to Lie algebras. Classification of simple Lie algebras. Representations of classical Lie groups and Lie algebras by the highest weight. Homogeneous spaces. |
Full description: |
1. Examples of classical Lie groups. Quaternions and the symplectic group. 2. Abstract Lie groups. Left-invariant vector fields, the exponential map, the adjoint representation. 3. Tori and their representations. The maximal tori in a compact Lie group. 4. Lie algebras associated to Lie groups. Classical matrix examples. 5. Classical Lie's theory: correspondence between Lie groups and Lie algebras. 6. Abstract approach to Lie algebras. Ideals, quotient Lie algebras and the corresponding group constructions. Relations between properties of Lie groups and Lie algebras. 7. Solvable, nilpotent and semisimple Lie algebras. Killing's form. Cartan's criteria fo solvability and semisimplicity. 8. Properties of Lie algebras associated to compact Lie groups. Invariant bilinear forms. Complex reductive Lie groups (as complexifications of compact Lie groups). 9. Classification of simple Lie algebras by root systems. 10. Representations of compact Lie groups. Characters of representations. 11. Representations of classical Lie groups and Lie algebras. Highest weight representations. 12. Representations of GL (n;C). Young diagrams (information about Pieri formula and Weyl's character formula). 13. Homogeneous spaces of classical groups. Torus action on G/P, fixed points, cell decompostion (using examples of Grassmannian and the flag variety). |
Bibliography: |
1. Adams, J.F. Lectures on Lie groups. 1969 2. Brocker, Theodor; tom Dieck, Tammo. Representations of compact Lie groups. GTM 98, 1985 3. Erdmann K., Wildon M. J. Introduction to Lie Algebras. 2006 4. Fulton, William, Harris, Joe. Representation theory. A rst course. 1991 5. Jacobson, Nathan. Lie algebras. 1962 (1979). 6. Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examplesconstitutes 1986 (2001). 7. Kirillov, Alexander, Jr. An introduction to Lie groups and Lie algebras. Cambridge Studies in Advanced Mathematics, 113. (2008) |
Learning outcomes: |
Student knows basic notions of Lie group and Lie algebras theory and related representation theory. In particular the student is fluent in the theory presented in the description of the lecture. This constitutes a star of further development and independent research. |
Assessment methods and assessment criteria: |
The lecture ends with a written and an oral exam. 20 % of the final grade consists of homework and active participation in the exercise sessions. |
Classes in period "Summer semester 2023/24" (in progress)
Time span: | 2024-02-19 - 2024-06-16 |
Navigate to timetable
MO TU W TH WYK
CW
FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Andrzej Weber | |
Group instructors: | Andrzej Weber | |
Students list: | (inaccessible to you) | |
Examination: | Examination |
Classes in period "Summer semester 2024/25" (future)
Time span: | 2025-02-17 - 2025-06-08 |
Navigate to timetable
MO TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Andrzej Weber | |
Group instructors: | Andrzej Weber | |
Students list: | (inaccessible to you) | |
Examination: | Examination |
Copyright by University of Warsaw.