As an undergraduate on physics seeking a solid education on mathematics, I have recently stumbled upon some theories that make use of the formalism of Lie groups and Lie algebras.
In light of this, I would like to ask where should I look for references in order to learn more about Lie theory and, of course, about its applications on physics, especially in Quantum Mechanics.
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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.
Well from a point of view of Lie groups theory to be applied to physics there is an elementary book called "Lie groups, physics and geometry" by Robert Gilmore which deals with matrix groups, Lie algebras, even some operator algebras and a small bits on structure of Lie groups. It also includes the standard applications (quantum mechanics, Maxwell eqn.s,..).
But for the more general study of Lie theory one needs quite a bit of topology,differential geometry and algebra (beyond linear algebra.. its good to know things from say commutative algebra,algebraic groups.., so one can study the representation theory of Lie groups..). In this case i would recommend the book by Chevally "Theory of Lie groups".