<i>Manny wrote: "I see Jacobson was an expert on Lie algebras... yes, that might certainly be useful! Thanks..."</i>
Many/most graduate math texts on Lie Groups/Algebras utilize manifold theory as their basis, which can make the field more daunting for physicists. As a result, I recommend looking at books that take a more linear algebraic bent to the subject, which can make some of the quantum mechanics related areas more transparent. I've used [book:Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|1051566], [book:Matrix Groups for Undergraduates.|1162374], and [book:Lie Groups, Lie Algebras, and Some of Their Applications|424103] for these types of applications and viewpoints.
About a year ago, I took a two quarter sequence at UCLA on Lie Groups from an ("easier") matrix group perspective which I imagine you may find somewhat useful in your reading on quantum mechanics. Several engineers, programmers, amateur mathematicians, physicists, and quantum mechanics enthusiasts had spent several years coaxing the professor into teaching it from this perspective. It was geared toward the advanced undergraduate level, and based on your comments here and your reading of Shankar, should be relatively easily followable.
We loosely followed Hall's textbook and portions of [book:Matrix Analysis|647523], which will give you some of the advanced linear algebra you could possibly be missing depending on your background. If it helps, here's a link to a downloadable pdf copy of the notes for the first class with the audio of the lecture embedded (using Livescribe.com digital pen technology which should let you click on the notes and jump to the audio portion related to where you've clicked): http:/
Having this background may make Weyl and Woit's developing textbook more easily manageable. My guess is that Woit is doing a more thorough job of developing the math than typical physics-oriented texts like [book:Geometry, Topology and Physics|439357] which do a lot of hand-waving at the math in an effort to get to the physics more quickly, but at the detriment of understanding what is happening mathematically.
I've dipped into some of Weyl's work in the past, but also keep in mind that some of his notation and definitions can be dated in relation to more modern presentations.