ISU MATH 620X Spring 2018

Lie algebras and their representations

Course Material

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.