MATH42112 - 2007/2008

 

Specification

Aims

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

[[x,y],z]+[[y,z],x]+[[z,x],y]=0.

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.

Learning Outcomes

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

None.

Syllabus

1. 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]
2. Representations. Course units. Casimir element. Weyl's theorem. Representations of sl(2). [7]
3. Root space decomposition. Maximal toral subalgebras and roots. Orthogonality properties. Integrality properties. [4]
4. Classification. Simple Lie algebras and irreducible root systems. Isomorphism theorem. The Lie algebra of type G₂ and octonions. Representations. [9]

Textbooks

• 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.

Assessment

Mid-semester coursework: 15%

End of semester examination: two and a half hours weighting 85%

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Arrangements