I have used the following in courses on Lie groups. Introduction to Lie Groups

These notes assume some familiarity with topics from linear algebra and from advanced calculus, including variants of the inverse function theorem, the notion of a manifold, integration on manifolds, ODE concepts such as vector fields and flows. Some of this background is presented in Appendices A-B of these notes. There are other sources giving more complete introductions.

For one, background can be found in notes on differential geometry.

Alternatively, one can find such background, as well as a brief treatment of some basic results about Lie groups, in the following. Manifolds, vector bundles, and Lie groups

The following quick treatment of differential forms is also useful background for understanding integration on a Lie group. Integration of differential forms

Here is a more complete treatment of differential forms. Differential forms and applications

This is excerpted from an advanced calculus text, which also has basic material on manifolds, flows, and matrix groups, found on the Math 522 web page.

Here is a more advanced treatment of vector fields, manifolds, and differential forms. Basic Theory of ODE and Vector Fields

The Lie groups notes also make occasional use of some functional analysis. Background is sketched in Appendix C of these notes. A more complete treatment can be found in the early sections of the following:

Outline of functional analysis

We move on to more advanced material. The following gives a geometric approach to some evolution equations that are “integrable.”

Finite and Infinite Dimensional Lie Groups and Evolution Equations

The following are miscellaneous notes on various topics related to Lie groups.

- Musings on the discrete Heisenberg group
- Multiple eigenvalues of operators with noncommutative symmetry groups
- Notes on integration on Lie groups
- Averaging rotations
- Some matrix integrals related to random matrix theory
- Positive definite zonal functions
- Jacobi’s generalization of Cramer’s formula (This is really linear algebra.)
- DeRham cohomology of compact symmetric spaces
- The Octonions
- D(M)/D_X(M)
- The Euclidean algorithm and Sl(2,Z)

The following deals with the interface between representation theory and microlocal analysis. Smooth operators for principal series representations — microlocal properties