Math 533, Suggested Reading


Here, in a bit more detail, are the topics that we’ll cover (I hope).  The last two parts will be covered only lightly, as time permits.

Lie groups, Lie algebras and the exponential map (matrix version)

Matrix groups and local matrix groups

Exponential map

Matrix Lie algebras

Derivative of the exponential map

Local matrix subgroups

Lie's second theorem

Campbell-Baker-Hausdorff formulas

Lie groups, Lie algebras and the exponential map (manifold version)

Smooth manifolds and smooth submanifolds

Frobenius theorem

Lie subgroups

Fundamental groups and covering spaces

Lie’s second theorem, global form

Continuous versus differentiable in Lie theory

Warmup: continuous homomorphisms between vector groups

Cartan's theorem

Solvable versus semisimple

Nilpotent and solvable Lie algebras

Engel's theorem and Lie's theorem

The solvable radical

Semisimple Lie algebras

Characterizations of semisimple Lie algebras

Levi's theorem

Overview of the classification of complex semisimple Lie algebras

Lie group actions

Symmetric spaces

Flag varieties

Lie group representations

Finite-dimensional representations of compact Lie groups

Representations from group actions

Unitary representations of the Heisenberg group

The oscillator representation

And here are some suggested sources.

Lecture Notes Online

Hall - An Elementary Introduction to Groups and Representations

Bryant - Introduction to Lie groups and symplectic geometry (1993 lectures)

Howe - 1983 - Very basic Lie theory

Meinrenken - Lie groups and Lie algebras

Meyer - 1996 - Exercises and solutions for a course on Lie groups

Milicic - Lectures on Lie groups

Paradan - Symmetric spaces of noncompact type. Lie groups

Samelson - Notes on Lie algebras

Varadarajan - Lie groups

Ziller - Lie groups, representation theory and symmetric spaces

Some Recommended Texts

Duistermaat, Kolk - Lie groups

Hall - Lie groups, Lie algebras and representations

Howe, Tan - Non-abelian harmonic analysis. Applications of SL(2, R)

Humphreys - Introduction to Lie algebras and representation theory

Procesi - Lie groups

Rossmann - Lie groups.  An introduction through linear groups

Spivak - A comprehensive introduction to differential geometry, vol 1

Varadarajan - Lie groups, Lie algebras and their representations

Warner - Foundations of differentiable manifolds and Lie groups

Some Other Texts

Adams - Lectures on exceptional Lie groups

Bourbaki - Lie groups and Lie algebras

Chevalley - Theory of Lie groups 1

Helgason - Differential geometry, Lie groups, and symmetric spaces

Knapp - Lie groups, beyond an introduction

Nigel Higson - 2012