School of Mathematics School of Mathematics
MA3415 - Introduction to Lie Algebras 2011-12 (JS & SS Mathematics
SS Two-Subject Moderatorship )
Lecturer: Prof. R. Tange

Requirements/prerequisites: MA2215, MA2322 desirable, concurrent registration for MA3411 also desirable.

Duration: 11 weeks (Michaelmas Term)

Number of lectures per week: 3 including tutorials

Assessment:

ECTS credits: 5

End-of-year Examination: 2 hour exam in Trinity Term (April/May)

Description:

Lie algebras should be thought of as the infinitesimal analogue of groups. Lie theory is an important and very active branch of mathematics with many links to other areas: geometry, representation theory, mathematical physics amongst others.

Topics:

• Definitions, small examples, classical Lie algebras, subalgebras, ideals and homomorphisms.

• Nilpotent Lie algebras, Engel's theorem.

• Solvable Lie algebras, Radical and semisimplicity, Lie's theorem, the Killing form and Cartan's criterion.

• Representations of Lie algebras, Representation theory of the Lie algebra sl(2).

• The structure of semisimple Lie algebras, Jordan-Chevalley decomposition.

• The classification of semisimple Lie algebras, possibly with some details: toral subalgebras, root systems ...

Refer to http://www.maths.tcd.ie/~rtange/teaching/lie_algebras/Lie_algebras.html for further details.

Learning Outcomes: On sucessful completion of this module, students will be able to:

• Give the definitions of: Lie algebra, homomorphism of Lie algebras, subalgebra, ideal, derivation, centre, representation of a Lie algebra, submodule, irreducible module, homomorphism of g-modules, the Killing form of a Lie algebra and the trace form of a classical Lie algebra, the derived and descending central series of a Lie algebra, nilpotent Lie algebra, solvable Lie algebra, solvable radical, semisimple and simple Lie algebra, maximal toral subalgebra, root system, irreducible root system.

• Give the definitions of and calculate with the classical Lie algebras.
• Describe the construction of the irreducible representations of sl2.
• State Engel's Theorem, Lie's Theorem and Cartan's Criterion.
• Describe the direct sum decomposition into simple ideals and the Jordan-Chevalley decomposition for semisimple Lie algebras.
• Indicate how root systems correspond to semisimple Lie algebras and give the root space decomposition and root system of sln.

Nov 2, 2011

File translated from TEX by TTH, version 2.70.
On 2 Nov 2011, 09:48.