MATH32112 - 2010/2011

 

Specification

Aims

To introduce students to some more sophisticated concepts and results of Lie theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.

Brief Description of the unit

The course unit will deal with finite-dimensional Lie algebras, that is, with anticommutative algebras satisfying the Jacobi identity. These algebras have various applications in representation theory, mathematical physics, geometry, engineering and computer graphics. Lie theory is currently a very active area of research and provides many interesting examples and patterns to other branches of mathematics.

Learning Outcomes

On successful completion of the course students will have acquired:

• A sound understanding of basic concepts of the theory Lie algebras.
• Knowledge of some fundamental results of the theory of Lie algebras.
• Knowledge of the Killing-Cartan classification of the finite dimensional simple Lie algebras.

Future topics requiring this course unit

Syllabus

• Definitions and first examples. Ideals and homomorphisms. [4]
• Nilpotent Lie algebras. Engel's theorem. [3]
• Solvable Lie algebras. Lie's theorem. Radical and semisimplicity. [3]
• The Killing form and Cartan's criterion. [2]
• The structure of semisimple Lie algebras. [3]
• Representation theory of the Lie algebra sl(2). [3]
• Toral subalgebras and root systems. Integrality properties. Simple Lie algebras and irreducible root systems. [4]

For MATH42112 the lectures will be enhanced by additional reading on related topics.

Textbook

• Karin Erdmann and  Mark J. Wildon Introduction to Lie Algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited, 2006.
• J.E. Humphreys Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer, 1972.
  

Teaching and learning methods

2 Lectures per week, 1 Tutorial per week, Private Study: 7 hours per week.

Assessment

Coursework: 20% (MATH32112), 15% (MATH42112)

Examination: 80% (MATH32112), 85% (MATH421122)

Examination is of 2 hours duration (MATH32112), 3 hours duration (MATH42112) at the end of the First Semester.

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Arrangements