What are the mathematical prerequisites for studying Lie groups and Lie algebras?

http://www.amazon.com/Introduction-Algebras-Cambridge-Advanced-Mathematics/dp/0521889693 is excellent if you've covered all the topics mentioned and are mathematically inclined. Among other things for a math book, it covers the approximate symmetry group of the hydrogen atom and the famous Campbell–Hausdorff formula. Skimming through the table of contents will give you a sense of the requisites, but I'll point to a few resources here, both conceptually and in the form of textbooks. In particular, this book is an excellent follow up to http://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Edition/dp/0914098705, aka the pirate geometer's book, and some work on symmetry such as http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582 and http://www.amazon.com/Algebra-Edition-Featured-Titles-Abstract/dp/0132413779/ref=dp_ob_title_bk. You also get the sense that the authors of these books are deeply inspired by the beauty and utility of the concepts they introduce to the reader, which comes through in the writing. I'll just add Lie Groups and Lie Algebras are really the heart of modern theoretical physics, in both general relativity and quantum mechanics. For example, the symmetry group of general relativity is the http://en.wikipedia.org/wiki/Diffeomorphism group, an isometry of smooth manifolds, which is in a crude sense the mathematical manifestation of general covariance. In fact, the size of this group is a large reason that general relativity cannot be fully quantized using standard quantum field theory methods that are equipped to handle compact Lie groups like SU(N), in particular U(1), SU(2) and SU(3). Lie theory, pioneered by the Norwegian mathematician http://en.wikipedia.org/wiki/Sophus_Lie, connects seemingly disparate concepts that allows intuitive observations about physical symmetry and manifold structure from the atomic scale up to the cosmological scale to be identified with tractable algebraic calculations that yield highly accurate quantitative results. Note, for example, that U(1) and SU(2) are identified as manifolds with the circle and the 3-sphere, respectively, and that you can construct a 3-sphere with circles fibered over a 2-sphere with a http://en.wikipedia.org/wiki/Hopf_fibration. You can start to see in this example how different Lie groups as manifolds can be embedded inside each other in a strikingly beautiful and visual manner; and then, you can express that in the form of algebra and matrices using the connection Lie made between the information carried by the groups and the algebras. It's a magical thing to see this mathematical machinery go to work in physics, and then walk over to a laboratory full of electronic equipment analyzing some scale system and watch as the the numbers match step by step. I'd even say that Lie theory takes much of the mystery out of quantum physics, if not the magic! If you want to add topology and complex analysis as requisites, I'd add in http://www.amazon.com/Topology-Differentiable-Viewpoint-Willard-Milnor/dp/0691048339 and http://www.amazon.com/Complex-Analysis-Princeton-Lectures-No/dp/0691113858.