**Lie Algebras and Representation Theory**

**SS 2021**

**Dietrich Burde**

**Lectures:** Monday 15:00-16:30, Thursday 15:00-16:30,
Online Moodle.

This page provides informations and lectures notes.
Find here
the dates and a syllabus, including a bibliography.
The lecture gives an introduction to the structure theory and representation theory of Lie algebras.
The main focus here lies on the classification of finite-dimensional complex semisimple
Lie algebras and their representations.
The aim of this lecture is to provide the basic theory and knowledge on Lie algebras and
representation theory, as it is necessary for further directions of Differential Geometry,
Number Theory and many other areas.
After introducing basic notions of Lie algebra theory we discuss the theorems of Engel
and Lie, the Jordan-Chevalley decomposition, the Cartan criteria, Weyl's theorem, the theorems
of Levi and Malcev, the classification of complex semisimple Lie algebras and Serre's theorem.
In the chapter on representations of semisimple Lie algebras we present the classification
by highest weight, introducing also the universal enveloping algebra. We give several
applications such as Weyl's character formula and Weyl's dimension formula.

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pdf-files:

**Exam topics**:

- Lie algebras, derivations and homomorphisms
- Ideals and Semidirect sums
- Classification of low-dimensional Lie algebras
- Lie Algebra Representations
- Semisimple and reductive Lie Algebras
- Nilpotent and solvable Lie algebras, Engel's Theorem and Lie's Theorem.
- Weyl's Theorem, Levi's Theorem and Mal'cev's Theorem
- The classification of complex semisimple Lie algebras
- Classification of heighest wight modules
- Weyl's character formula and Weyl's dimension formula

Exam 2021:

Monday, 28th of January 2021, 15:00 - 16:15 Digital.

Dietrich Burde
Last modified: Mi Jun 23 10:43:51 CEST 2021