Department Mathematik
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Department of Mathematics
Joachim Wehler
LMU München
Winter term 2022/23

Lie Algebras (4+2)

Winter term 2022/23

Description

Lecture notes

References

Problem session

Examination

 

1. Description

The course on Lie algebras makes a first step beyond the introductory course on Linear algebra. The Lie algebra course continues with the theory of finite dimensional vector spaces and linear maps. But we now add a second interior composition, the so-called Lie bracket. It is a bilinear map which captures the commutator of each pair of matrices or linear endomorphisms. The resulting abstract object is named a Lie algebra.

Lie algebras allow to linearize Lie groups. The latter are important examples of symmetry groups in physics. In this context Lie algebras are often called infinitesimal groups.

The lecture will also investigate some properties of classical matrix groups, but it will not cover the general theory of Lie groups.

It is planned to cover the following topics:

  • Power series of matrices
  • The Lie algebras of the classical groups
  • Nilpotent and solvable Lie algebras
  • Killing form and semisimple Lie algebras
  • Cartan decomposition of complex semisimple Lie algebras
  • Root systems
  • Classification of complex semisimple Lie algebras
  • Outlook

Target audience: Master mathematics or physics, TMP students; 9 ECTS (Module WP36). Also advanced undergraduates.

Prerequisites: Linear algebra, Power series. Helpful: Covering theory.

Scheduled time: Tuesday 10-12 am and Thursday 12-2 pm (lecture), Tuesday 12-2 pm (problem session).

The course will be held online via Zoom. The room B 039 is reserved for the participants during the time of the lecture and the problem session. Please register by emailing to me. Then I will send you the Zoom-link just before each lecture.

2. Lecture notes

Lecture notes

The lecture notes will be updated continuously during the course, always before the corresponding lesson.

3. References

  • Humphreys, James E.: Introduction to Lie Algebras and Representation Theory.
  • Hilgert, J.; Neeb K.-H.: Lie-Gruppen und Lie-Algebren.
  • Hall, Brian: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction.

For detailed references see the lecture notes.

4. Problem session

In addition to the lecture, each week a problem session will be held. The basis is a series of problems for homework. The students are encouraged to discuss their solutions during the problem session.

Problem sheets

5. Examination

To obtain the ECTS-credits the participants have to pass the examination ("Modulprüfung") at the end of the course.

Admitted to the examination are those students from LMU or TUM, including Erasmus-students, who registered per email for the course on Lie algebras not later than November 2022. The examination takes place as an oral examination (30-60 minutes) via Zoom during the time 22.2.- 23.2.2023. You may take the examination in English or in German language.

The examination will cover the topics from the lecture and from the problem sheets. Any notes in electronic or in printed form are not permitted during the examination.

If you want to attend the examination please email to me until Friday, 13.1.2023, attaching a signed copy of the following agreement.

Your email should contain the following data:

Name - prename - email address - LMU/TUM student number - major (Mathematics/physics bachelor or master) - number of your current semester - if student with major master mathematics LMU: module number to credit (default WP 36).

The examinations are scheduled as follows:

  • 22.2.2023. 9.00-10.00 am: Mats Hansen

  • 22.2.2023. 10.00-11.00 am: Lingrui Cheng

  • 22.2.2023. 11.00-12.00 am: Emilie Falourd

  • 22.2.2023. 13.00-14.00 pm: Sophia Fitz



  • 23.2.2023. 9.00-10.00 am: Johannes Bartenschlager

  • 23.2.2023. 11.00-12.00 am: Loic Honet

I will send a Zoom-Link to the candidate about 5 minutes before the start of the examination. Each candidate will be informed about the result of the examination by email after the last candidate has finished the examination.

The room B 251 is reserved during the time of the examination for those participants, who want to join their Zoom-session from within the premises of the university.