MATHEMATICS 617- FALL 2001

LIE GROUPS, LIE ALGEBRAS AND THEIR REPRESENTATIONS

SECTION 60578- Instructor: Alex Freire (Mathematics Dept.)Proposed time* (Tuesday + Thursday 9:40-10:55)

*Please send me an e-mail if you are interested in the course but have a time conflict

COURSE DESCRIPTION:Introduction to Lie groups and Lie algebras, emphasizing: classification of compact

Lie groups and complex semi-simple Lie algebras and of their representations.

The approach will be interdisciplinary: applications and the terminology of quantum

mechanics/field theory or of differential geometry will be introduced whenever appropriate.

Complete discussion of representative examples will often be given priority over lengthy

classification proofs. The intended audience includes graduate students in mathematics and

physics.

PREREQUISITES:linear algebra; some experience in graduate-level mathematics or physicsGRADING:based on problem sets

TOPICS:The classifications of complex semisimple Lie algebras, compact Lie groups and

their representations were worked out by E.Cartan and H.Weyl in the 1920s. This is a vast subject,

with many applications to differential geometry (symmetric spaces), differential equations (integrable systems)

and quantum mechanics/quantum field theory. Classifying the irreducible unitary representations ofnoncompactsemisimple Lie groups is an active research area, interfacing with Fourier analysis, number theory

and dynamics/ergodic theory of group actions. Although we will only deal with finite-dimensional Lie algebras,

the more recent theory of representations of infinite-dimensional Lie algebras starts from the same basic concepts.

I-Lie algebras and Lie groups

Lie algebras of the classical compact groups

Lie algebra to Lie group: exponential map, Baker-Campbell -Hausdorff formula

Solvable, nilpotent, semisimple Lie algebras and groups

Representations of compact groups: Peter-Weyl theorem

Irreducible representations of SU(2), SO(3), sl(2,C)

Simply-connected groups, covering groups , complexification

2-Complex semisimple Lie algebras: classification and representations

Cartan subalgebras, root space decompositions

Cartan matrices, Dynkin diagrams and classification

Theorem of the highest weight; Weyl group

3-The irreducible representations ofSU(n)

Representations on spaces of tensors: Weyl's theorem, duality with symmetric group

Complex homogeneous spaces and line bundles: theorem of Borel-Weil

4-Clifford algebras,Spin(n)and spinors

5-Unitary representations of non-compact groups: the Poincare' group, SL(2,C), SL(2,R)

structure of non-compact semisimple groupsDepending on student interest, we will continue in the spring with integrable systems, loop groups and

applications to exact solutions in field theory (sigma-models, harmonic maps, etc.)REFERENCES

I-Mathematics

* Carter, Segal, Mac Donald: Lectures on Lie groups and Lie algebras, London Mathematical Society

A.W.Knapp- Lie groups and Lie algebras beyond an introduction (Birkhauser)

Th.Brocker, T, tomDieck- Representations of compact Lie groups (Springer)

H.B.Lawson, M.L.Michelson- Spin Geometry (Princeton)2-Physics

*S.Sternberg, Group Theory and Physics (Cambridge)

H.Georgi, Lie algebras in particle physics (Addison-Wesley)*= main texts for the course (will be ordered by the UT bookstore)

IF INTERESTED, please enroll early - the decision to run the course or not will be based on the

number of students enrolled by the end of April

QUESTIONS? Please email me at: freire@math.utk.edu