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