Main text: Kirillov, Jr., An introduction to Lie Groups and Lie Algebras
Lecture 1: Definition of manifolds and (real and complex) Lie groups. Implicit function theorem. Examples. Further reading: Spivak, Calculus on Manifolds. In particular Chapter 5.
Lecture 2: Connected manifolds and Lie groups. Discrete Lie groups. Open submanifolds. The connected component at the identity element. G/G^0 is discrete.
Lecture 3: Fundamental group of a manifold. Simply connected manifolds and Lie groups. Universal cover of a manifold and Lie group. Further reading: Hatcher, Algebraic Topology. Chapter 1.1 and 1.3. In particular Theorem 1.38 and the paragraph that follows.
Lecture 4: Tangent spaces and vector fields on manifolds, derivative (differential) of a morphism, left-invariant vector fields, Lie algebra (as a vector space) of a Lie group.
Lecture 5-6: Classical groups
Lecture 7: Open, immersed, embedded submanifolds. Closed Lie subgroups.
Lecture 8: Quotient groups and homogenous spaces
Lecture 9: One-parameter subgroups, the exponential map.
Lecture 10: Classes of manifolds. The commutator (bracket).
Lecture 11-12: Computing differentials using curves. Differential of Ad. Jacobi identity. Definition of Lie algebra. Derivations. Action of SL(2,K) and sl(2,K) on polynomials.
Lecture 13: Abelian Lie algebras. Lie subalgebras and Lie ideals. Lie algebras of subgroups and quotients. Vect(M).
Lecture 14: Baker-Campbell-Hausdorff formula. Fundamental theorems 3.40,3.41,3.42.
Lecture 15: Equivalence of categories (connected simply-connected Lie groups)/(finite-dimensional Lie algebras). Complexification and real forms.
Homework 1, due Jan 26. Exercises 2.2, 2.7-2.10.
Homework 2, due Feb 16. Choose three to hand in: Exercise 2.5, 3.5, 3.9, 3.11, 3.14.