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MC452 Lie Algebras

MC452 Lie Algebras

Credits: 20 Convenor: Dr Alexander Baranov Semester: 1
Prerequisites: essential: MC241, MC254 desirable: MC341
Assessment: Regular coursework: 10% Three hour exam: 90%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

MC241, Linear Algebra, is essential. In the classification of semisimple Lie algebras certain finite groups are involved, the Weyl groups. Therefore, MC254, Algebra I, or MC341, Abstract Algebra, is helpful.

Course Description

Symmetry is a basic principle in all parts of science. Mathematical tools for describing or using symmetries are groups (finite or infinite) and algebras. Lie algebras are a tool to handle symmetries by linear algebra methods; they are vector spaces with additional structure.

The most important Lie algebras, at least for applications (e.g. to particle physics), are the socalled semisimple ones. A major goal of this module is to develop a full classification of these objects, and along the way to provide combinatorial tools which allow structural insights and efficient computations. Moreover, the concept of representations will be introduced and shown to be a fundamental tool.

Aims

To describe Lie algebras as a tool for handling symmetries by linear algebra methods;

To develop a full classification of semisimple Lie algebras;

To provide combinatorial tools allowing structural insights and efficient computations.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To be able to solve some routine problems in low and high dimensional topology.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

The ability to apply taught principles and concepts to new situations.

Syllabus

Basic concepts
Solvable and nilpotent Lie algebras
Semisimplicity
Killing form
The example sl(2)
Root systems and Weyl groups
The classification
Serre's theorem
Universal enveloping algebras
Representations
Weyl's character formula

Reading list

Recommended:

Humphreys, Introduction to Lie algebras and representation theory,

Background:

Serre, Lie algebras and Lie groups,

Bourbaki, Lie algebras,

Knapp, Lie groups, Lie algebras and cohomology,

Details of Assessment

The coursework will consist of regularly assigned exercise sheets. The June examination will have 6 questions, it being (theoretically) possible to obtain full marks by answering 4 of them.

Next: MC449 Galois Theory Up: Year 4 Previous: MC442 Differential Geometry

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Last updated: 2001-09-20
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