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Ingo Runkel
Algebra II (Lie algebras) - Winter term 2011/12
Announcements:
- [30.3.2012] The results of exam 2 are available below. To have a look
at your exam script, please come into my office hour on Tuesday, 3.4 at 14:00.
Times and rooms:
Lecture Tuesday and Friday 12:15-13:45 in H4. Excercise class Friday 14:15-15:45 in 431.
Exercise sheets:
[1]
[2]
[C2.5]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[C12]
Hints and solutions:
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Overview of material covered:
[PDF],
and the proof of Weyl's theorem
[PDF].
Results of exam 1:
exam,
solutions,
marks,
distribution.
Results of exam 2:
exam,
solutions,
marks.
Exam:
In order to qualify for the exam, you need to obtain 40% of the points on the exercise sheets
(there are 11 sheets of 20 points each, so you will need 88 points or more).
The exam will be written, the dates are 7.2.12 (12-15h, H1) and 30.3.12 (12-15h, H1).
A good part of the exam will be based on homework problems.
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 arise chiefly as symmetry groups, namely when the symmetry
transformations of some (mathematical or physical) object depend on continuous
parameters. The group SO(n) of rotations in R^n is such an example.
Lie algebras in turn are "linearisations" of Lie
groups - 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.
This course is mainly aimed at Masters students in Mathematics and Mathematical Physics. We will study Lie algebras and their representations. A rough overview of the topics is: definition and basic properties of Lie algebras; universal enveloping algebra; nilpotent and solvable Lie algebras; complex simple Lie algebras; root systems and Dynkin diagrams; highest weight representations, example sl(n).
Prerequisites:
Basic notions from algebra (in particular groups, fields, linear algebra)
Literature:
Examples of 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
A script:
- A set of lecture notes which I will use for part of the course [PDF]
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