The representation theory of Lie groups and Lie algebras are very related. In fact, in the case of Simply-connected Lie groups, the irreducible representations of these Lie groups are in bijection with the irreducible representations of its corresponding Lie algebra. In the case of a connected Lie group, the irreducible representations of its corresponding Lie algebra are in bijection with the irreducible representations of its universal covering space ( which is a Lie group as well ). If your Lie group is not connected, one can still use this correspondence by considering the connected component of the identity ( your group modulo this component will only be a discrete/dimension 0 group ).
Off the top of my head, I can give 2 good books that illustrate this correspondence very well: Representation theory: a first course (Fulton and Harris) and
Introduction to Lie Groups and Lie algebras ( Krilliov Jr ). You can also learn a good deal by reading the first couple of chapters of Representations of Compact Lie groups by Bockner. However, the rest of the book develops the theory almost entirely without any reference to Lie algebras.
From my own experience, I will advise you against every book of mathematical methods written specifically for physicist. From my point of view, is better to learn about mathematics from mathematically written books (it sounds so obvious but is not). For example, many people like Schultz, Geometrical methods of mathematical physics, but I prefer to learn about the common topics in Singer, Thorpe, Lecture notes on elementary topology and geometry. (I don't say it is not a good textbook, I only say I find difficult learning things on books written in a pretty informal way.)
The most complete work about methods of mathematical physics is probably
- Reed, Simon, Methods of modern mathematical physics,
that covers functional analysis, Fourier analysis, scattering theory, operator theory.
Since you are interested in cosmology, the best review on Loop Quantum Gravity is that by Thomas Thiemann,
- Thomas Thiemann, Modern and canonical quantum general relativity,
a 900 pages review, equipped with about 300 pages of mathematical methods (mathematical appendices are not a textbook however, but a collection of necessary results, eventually explored in some depth). References therein are very useful also.
Many people like
- Deligne et al., Quantum fields and strings: a course for mathematicians,
that joins a good part of your requests. (I haven't read it, however, I know it since is "famous".)
A celebrated book on methods of classical mechanics, concerning manifolds too, is
- Abraham, Mardsen, Foundations of mechanics.
Another is
- Choquet, Bruhat, Analysis, manifolds and physics.
All Arnold's books are always a great choice. (he wrote about ergodic theory and geometrical methods for differential equations, among the other things.)
There are a lots of more specific books, e.g. dealing with mathematical structure of quantum mechanics, but many of those are more and more specialized and is better to have very clear the general theory before try to get more involved into dangerous subjects such as, to say, quantum field theory. Once one has a strong background, the best opera on the subject of field theory probably is
- Zeidler, Quantum field theory,
an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,
- Haag, Local quantum physics.
EDIT. I'd like to add some book I've discovered more recently and I think fit very well:
Streater, Wightman, "PCT, Spin and all that",
Teschl, "Mathematical methods in Quantum Mechanics",
Bogolioubov, Logunov, Todorov, "Axiomatic Quantum Field Theory",
Lansdman, "Mathematical concepts between classical and quantum mechanics".
Best Answer
I've been in your same situation, and I think this might be the book you're looking for: Physics from Symmetry, by Jakob Schwichtenberg. It explains the fundamental concepts of Lie/representation theory carefully, in a quite intuitive manner, motivated via applications in Physics. There is a chapter dedicated exclusively to Quantum Mechanics, which fundamental principles are derived using mathematical tools alone, but you'll also find discussions of QFT, Electromagnetism and Classical Mechanics.
For a more mathematically rigorous approach, I'd recommend Naive Lie Theory by John Stillwell, also an excellent read, which successfully conveys complex ideas in a simple fashion. You'll specially enjoyed the historical notes at the end of the book.
Finally, I've read some good reviews of Group Theory and Physics by Shlomo Sternberg, which appears to be a main reference work.