Department of Mathematics

INDIAN INSTITUTE OF SCIENCE

MA 305: Lie Groups and Lie Algebras

Credits: 3:0

Lie groups, definition and examples, Invariant vector fields and the exponential map, The Lie algebra of a Lie group, Lie subgroups and Lie subalgebras, Correspondence between connected Lie subgroups and Lie subalgebras, Cartan’s theorem, Lie group and Lie algebra homomorphism and their correspondence, Covering space theory of Lie groups, Commutative Lie groups and classification of connected abelian Lie group, Adjoint representation, Normal subgroups and ideals, Lie Group action and Lie transformation Groups, Coset Spaces and homogeneous spaces, Complexification, Classical Lie groups and their examples (Linear groups, Orthogonal Groups, Unitary Groups, Compact symplectic groups, Non-compact symplectic group). Topological properties and fundamental groups of classical Lie groups, The Killing form, Nilpotent and Solvable Lie algebras, Semisimple Lie algebras, Compact Lie algebras

prerequisite: Basic knowledge of Differential Geometry and Algebraic topology

Suggested books and references:

1. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press.
2. C. Chevalley, Theory of Lie groups, Dover.
3. F. Warner, Foundations of differentiable manifolds and Lie groups, Springer.
4. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Trim.
5. A. Knapp, Lie groups beyond an Introduction, Birkhaeuser.
6. D. Bump, Lie groups, Springer.

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