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
http://arxiv.org/abs/math-ph/0005032
Bryant - Introduction to Lie groups and symplectic geometry (1993 lectures)
http://www.math.duke.edu/~bryant/ParkCityLectures.pdf
Howe - 1983 - Very basic Lie theory
http://www.jstor.org/stable/2323277
Meinrenken - Lie groups and Lie algebras
http://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf
Meyer - 1996 - Exercises and solutions for a course on Lie groups
https://www.uni-math.gwdg.de/rameyer/download/liegrp.ps.gz
Milicic - Lectures on Lie groups
http://www.math.utah.edu/~milicic/Eprints/lie.pdf
Paradan - Symmetric spaces of noncompact type. Lie groups
http://math.univ-lyon1.fr/~remy/smf_sec_18_02.pdf
Samelson - Notes on Lie algebras
http://www.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf
Varadarajan - Lie groups
http://www.math.ucla.edu/~vsv/liegroups2007/liegroups2007.html
Ziller - Lie groups, representation theory and symmetric spaces
http://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf
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