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 physics

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 of
noncompact semisimple 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 of SU(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 groups

Depending 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