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.
If you are taking this course for credit, click
here .
(Not all of this is actually covered in class.)
Comments are very welcome!
Pages 1-213
Last update: 12/10/09 (will be further revised if I find the time..)
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.