MAT 1120HF Lie groups and Clifford algebras
Classes:
M 12-1 at BA 2195, WF 11-12 at BA 2175.
The plan of this course is to present the basic theory of Clifford algebras, with applications to Lie groups and Lie algebras. Detailed lecture notes will be provided. Topic include:
- Quadratic forms
- Clifford algebras
- Spinor modules
- Dirac operators
- Lie groups
- Applications:
Poincare-Birkhoff-Witt theorem, Duflo theorem, Weyl character
formula, Borel-Weil theorem, multiplets of
representations,...
This is the second edition of
a course that I had taught in Fall 2005. The lecture notes for the
first edition are posted below, and will give an idea of what this
is all about. However, the notes will be substantially revised,
and in any case my choice of topics will be a bit different this
time.
If you are taking this course for credit, click
here .
(Not all of this is actually covered in class.)
Comments are very welcome!
Pages 1-142
Last update: 11/16/09
Pages 1-96
Last updated in 12/2005.
S. Sternberg, lecture notes on Lie algebras , available from
his website.
M. F. Atiyah, R. Bott, A. Shapiro:
Clifford modules .
Topology 3 (1964), 3--38.
B. Kostant:
A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups,
Duke Math. J. 100 (1999), no. 3, 447--501.
C. Chevalley:
The algebraic theory of spinors and Clifford algebras,
(reprinted version), Springer 1997.
B. Lawson, L. Michelson:
Spin geometry, Princeton University Press (1989).
A. Alekseev, E. Meinrenken:
The non-commutative Weil algebra , Inventiones Mathematicae 139 (2000), 135--172.
A. Alekseev, E. Meinrenken:
Clifford algebras and the classical dynamical Yang-Baxter equation,
Mathematical Research Letters 10 (2003), 253--268.