Lie Groups and Lie Algebras

The book is based on a year-long graduate sequence on Lie groups and Lie algebras taught by the author at MIT. It is split into 51 sections, roughly corresponding to 1.5-hour lectures. The first half (Sections 1-26) gives an introduction into the basic theory of Lie groups and Lie algebras and then works out the structure, classification and representation theory of complex semisimple Lie algebras, ending with the Weyl character formula. The second half (Sections 27-51) treats more advanced topics, such as Schur-Weyl duality, representations of classical groups, Peter-Weyl theorem for compact topological groups, a representation-theoretic treatment of the hydrogen atom, classification of real semisimple Lie algebras, structure of real semisimple Lie groups, Cartan and Iwasawa decomposition, cohomology of Lie groups and homogeneous spaces, flag varieties, and others.