MATH42112 - 2007/2008
- Title: Lie Algebras
- Unit code: MATH42112
- Credits: 15
- Prerequisites: MATH10202 or MATH10212 Linear Algebra.
- Co-requisite units: None
- School responsible: Mathematics
- Members of staff responsible:
The course will provide basic knowledge of finite-dimensional Lie algebras for those students interested in Lie groups and representation theory.
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 group theory, mathematical physics, geometry and other branches of mathematics. Lie theory is also a very active area of independent interest in modern algebra.
On successful completion of the course unit students will have gained understanding of basic notions of Lie Theory up to the classification of finite-dimensional simple Lie algebras of characteristic zero.
Future topics requiring this course unit
- General theory. Definitions and first examples. Ideals and homomorphisms. Nilpotent Lie algebras. Engel's theorem. Solvable Lie algebras. Lie's theorem. Radical. Semisimplicity. Killing form. Cartan's criterion. Jordan-Chevalley decomposition. [10 lectures]
- Representations. Course units. Casimir element. Weyl's theorem. Representations of sl(2). 
- Root space decomposition. Maximal toral subalgebras and roots. Orthogonality properties. Integrality properties. 
- Classification. Simple Lie algebras and irreducible root systems. Isomorphism theorem. The Lie algebra of type G2 and octonions. Representations. 
- J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer 1972.
Teaching and learning methods
Three lectures and one half hour examples class each week. In addition students should expect to spend at least six hours each week on private study for this course unit.