I often hear physicists talk about Lie algebras and their representation theory, but most of the time hardly understand them because my knowledge of physics is very limited.
Does anyone know of any resources which explain the physical applications of Lie algebras and representation theory, but are fun to read and have a mathematical audience in mind?
The following recommendations are from the perspective of a physics grad student with an undergrad degree in pure math. My personal opinion is that learning mathematics for physics purely from either one of mathematicians or physicists alone is a pretty bad idea, but together, you can learn some pretty magical stuff, so I'd recommend trying to look at both math books, and mathy physics books in general if you really want to know what the heck is going on.
A standard book covering a nice variety of topics that physicists use is
Group Theory and Physics by Sternberg
It's written by a mathematician, so it's at essentially the level of rigor demanded by modern mathematics, and it has some really good commentary on physical applications
Most of the interesting applications of Lie Algebras in physics arise in general relativity, quantum mechanics, quantum field theory, and string theory. In the context of general relativity, Lie Algebras appear through isometry groups of semi-Riemannian manifolds. My favorite discussion of this is in chapter 9 of
Semi-Riemannian Geometry by O'Neill
In quantum mechanics, the most important Lie algebra is $\mathfrak{su}(2)$ because its representations describe how angular momentum arises in quantum systems. If you really want to see how physicists use this stuff, you should just open a book on quantum mechanics and turn to a chapter on angular momentum. My favorite quantum book is the (2-volume)
Quantum Mechanics by Cohen-Tannoudji
See, especially, chapters 9 and 10. As for quantum field theory, the basic object physicists study is relativistic quantum field theories in which the representation theory (both finite- and infinite-dimensional) the Lie algebra of the Poincare group $\mathbb R^4\rtimes \mathrm O(3,1)$ becomes fundamental to describing the symmetries of these theories. I'm not particularly familiar with mathematicians' accounts of this stuff, but if you want to know how physicists do things in this realm, you should take a look at the classic bible (mostly chapter 2)
The Quantum Theory of Fields by Weinberg
A branch of quantum field theory is conformal field theory where Lie algebras and affine Lie algebras are all over the place. Physicists use
Conformal Field Theory by Francesco, Mathieu, and Senechal
but a more mathematical treatment that I really quite like is
A Mathematical Introduction to Conformal Field Theory by Schottenloher.
For string theory, you basically need to learn about supersymmetry algebras. My personal favorite introduction to this (although not particularly geared towards mathematicians, is freely available online;
Introducing Supersymmetry by Sohnius
who, by the way, is very famous for the fundamental reason why there is such a supersymmetry craze among theoretical physicists: the Haag-Lopuszanski-Sohnius theorem.
You will probably also find this physics.SE question helpful:
https://physics.stackexchange.com/questions/6108/comprehensive-book-on-group-theory-for-physicists
Hope that helps!