<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.
Big History specifically and purposefully looks at history from a much broader perspective, knowing full well that it's going to miss the complexity of large portions of what is going on at the much smaller scales. But it does this because different structures and ideas emerge at these larger scales. We see different types of things going on that we don't see at the smaller time scales.
If you analogize it to a chess game, there are some very simple and straightforward rules which are responsible for a terribly complex potential number of moves and games to be played. Most history has been studied at the level of "let's take a look at the first three moves of a game of chess" or let's look at the last three moves. This type of close examination doesn't allow for the broader sweeping view of entire games. Both Boris Spassky and a typical four year old could play the same 4 opening chess moves, but the complexity of competing against them diverges rapidly as the game progresses. By looking at the bigger picture, one can better appreciate what is happening later in the game. This also gives one a better platform for analyzing smaller portions of the mid-game as well.
I would suggest that Christian's audio lectures in [book:Big History: The Big Bang, Life On Earth, And The Rise Of Humanity|7956320] do a far better job of showing this viewpoint than his text [book:Maps of Time: An Introduction to Big History|745703].
I would say that big history is anything but "new age-y pseudo-science", and because it is rooted more heavily in physics, cosmology, biology, chemistry, microbiology and various other hard science disciplines, it is actually on more solid grounding than most other history-based courses. Those who have read C.P. Snow will recognize that big history has a grounding in both of the "two cultures" rather than being solely entrenched in just one. It's somewhat akin to taking a complexity theory-based approach to economics compared to the older 18th and 19th century simplified approaches followed by Adam Smith, Keynes, et al well into the 1980's. With approaches to economics exemplified by Kahneman/Tversky or more modern schools like the Santa Fe Institute, there is more being uncovered by shifting paradigms and taking more nuanced approaches to the subject than using the simplifications of the past.
Taking new approaches doesn't necessarily invalidate prior work in these fields, it just provides a broader basis for how all the various parts interact to create a larger whole.