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Ingo Runkel
Lie algebras
Announcements:
- [26.5] The 8th exercise sheet is now online, see below. The 7th and 8th sheet will be discussed together on Thursday, 1st of June.
- [11.5] For the lectures on Tue, 16.5 and Thu, 18.5 Anssi Lahtinen will kindly replace me. There will be no exercise class on Thu, 18.5. Instead, there will be a replacement exercise class on Wed, 24.5, 18:00-19:30 in room 142.
Times and rooms:
Lecture Tuesday 10:15-11:45 in H5 and Thursday 10:15-11:45 in H6. Excercise class Thursday 12:15-13:45 in 142.
Lecture notes:
Mr Weber has been kind enough to make the notes he takes during the course available here
[Dropbox link]. I have not verified the
notes, please contact Mr Weber if you have comments.
Description:
The importance of Lie algebras derives from their relation to Lie groups.
A Lie group is a group and a manifold, such that the group operations are
smooth maps. They often arise as symmetry groups in physics and mathematics.
The group SO(n) of rotations in R^n is such an example.
Lie algebras, the topic of this course, in turn are "linearisations" of Lie groups, or "infinitesimal symmetry transformations".
They consist of a vector space together with a bilinear operation,
the Lie bracket. Surprisingly, many of the properties of Lie groups can derived
from these linearisations. We will study:
- definition and basic properties of Lie algebras,
- nilpotent and solvable Lie algebras,
- universal enveloping algebra and PBW-basis,
- semisimple Lie algebras: root systems, Dynkin diagrams, classification, and
- highest weight representations.
This course is mainly aimed at Masters students in Mathematics and Mathematical Physics.
Prerequisites:
Basic notions from algebra (in particular groups, fields, linear algebra).
Literature:
Books with an emphasis on Lie algebras and representations are:
- Humphreys, Introduction to Lie algebras and representation theory, Springer
- Serre, Lie algebras and Lie groups, Springer
Some books which may be useful but go beyond this course are:
- Fulton, Harris, Representation theory, Springer
- Knapp, Lie groups beyond an introduction, Birkhäuser
- Fuchs, Schweigert, Symmetries, Lie algebras and representations, Cambridge University Press
- Baker, Matrix Groups, Springer
Exam:
To qualify for the exam, you should least 40% of the homework problems
and participate actively in the problem classes.
At the beginning of each class, you can mark on a list which problems (or problem parts, for
longer problems) you solved
and can present on the board. To qualify for the exam, you should have marked 40% or more of the problems by the end
of the course.
If you marked a problem as solved but it becomes apparent that you did not prepare it sufficiently in case you get asked to the board, the all problems for that week will be marked as not solved.
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