# Lie groups and bundles

# Lie groups and bundles

Lecturers

Marcel Nicolau (UAB)

Joan Porti (UAB)

##### Contents

**Lie Groups**. Matrix groups and Lie groups. The Lie algebra of a Lie group. The exponential map. Lie subgroups and homogeneous spaces. Symmetric spaces. Linear representations of Lie groups. An overview of the classification of Lie groups and of Lie algebras.**Fiber Bundles**. Vector bundles. The de Rham cohomology of a differentiable manifold. Linear connections and curvature. Principal bundles. The Weil homomorphism. Characterístic classes. The Euler class. Chern and Pontrjagin classes.**Sheafs theory**. Sheaves and presheaves. Axiomatic sheaf cohomology. Cech cohomology. The de Rham theorem. Cohomology of vector bundles.**Hodge theory**. Elliptic differential operators and elliptic complexes. The Hodge decomposition theorem on compact Riemannian manifolds.

##### Bibliography

- Warner, Frank W. Foundations of differentiable manifolds and Lie groups. Springer-Verlag, 1983.
- Greub, Werner; Halperin, Stephen; Vanstone, Ray. Connections, curvature, and cohomology. Vol. I and II. Academic Press, 1972
- Helgason, Sigurdur. Differential geometry, Lie groups, and symmetric spaces. American Mathematical Society, 2001.
- Bröcker, Theodor; tom Dieck, Tammo. Representations of compact Lie groups. Springer-Verlag, 1995.
- Milnor, John W.; Stasheff, James D. Characteristic classes. Princeton University Press, 1974.