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Department of Mathematics

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Lie-Algebras

Winter Semester 2021/22

This is the website for the lecture course on Lie algebras and its associated exercise classes.

Content

This course will give an introduction to Lie algebras and their representation theory, with a focus on the complex semisimple case. A Lie algebra is a vector space equipped with a (typically) non-associative multiplication called Lie bracket. Lie algebras arise naturally in many contexts in mathematics and physics, for example as tangent spaces at the identity of Lie groups, and they can be thought of as spaces of “infinitesimal” or “linearised symmetries”.

The content of the course includes:

  • the definition and basic properties of Lie algebras;
  • the classical Lie algebras, including a thorough discussion of sl(2,\C) early in the semester;
  • the classification problem, nilpotent and solvable Lie algebras;
  • semisimple Lie algebras, root systems;
  • representation theory of semisimple Lie algebras;
  • universal enveloping algebra and PBW theorem;
  • an outlook on applications.

This course is mainly aimed at Masters and advanced Bachelor students in Mathematics and Mathematical Physics.

Prerequisites: basic notions from algebra (groups, fields, linear algebra).

Coordinates

Lectures will be in person (please get in contact if you cannot participate in person):

  • Monday, 12:15–1:45 pm, Geomatikum H6, 11th Oct 2021–24th Jan 2022
  • Thursday, 12:15–1:45 pm, Geomatikum H5, 14th Oct 2021–27th Jan 2022

Exercise classes in two groups. The preparatory meeting was on 11th October.

Initially everyone was signed up for the online group. Between Monday 11th and Thursday 21st Oct 2021 it was possible to change registrations in STiNE.

Ressources

Lecture notes were provided during the semester (available now upon request). Additional notes (not all checked):

Additional references:

  • James E. Humphreys: Introduction to Lie Algebras and Representation Theory, Vol. 9. Graduate Texts in Mathematics. Third printing, revised. Springer-Verlag, New York-Berlin, 1980.
  • Jean-Pierre Serre: Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University, Vol. 1500 Lecture Notes in Mathematics. Firth printing, 2nd edition Springer-Verlag, New York-Berlin, 2006.

Exam

There will be an oral exam. To qualify for the exam, you should solve at least 40 % of the homework problems and participate actively in the exercise classes.

Contact