An Introduction to Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras has become a very powerful tool and has found many applications in various branches of mathematics and physics. This book is intended as material for a graduate course on the subject. The author's intention is to keep the book as readable as possible and to avoid the technicalities needed in some proofs. Hence some proofs are only outlined and the details transferred to exercises, and others (e.g. the proofs of the Poincare-Birkhoff-Witt and Engels theorems) are omitted completely. The author instead gives some nice applications of the theory, e.g. a relation between the spectrum of the Laplace operator on the sphere S2 and representations of SO(3,R).
There are many exercises in the book. The reader is supposed to have some background in differential geometry, algebraic topology and some proofs even require some knowledge of homological algebra. The book starts with basics from the theory of Lie groups and Lie algebras. It contains a description of the usual basic parts of the theory (structure theory of Lie algebras, complex semisimple Lie algebras and their root systems, representations of semisimple Lie algebras, including highest weight representations, Verma modules, Harish-Chadra isomorphism and Bernstein-Gelfand-Gelfand resolution). The last appendix contains a useful detailed sample syllabus for a one-semester graduate course (two lectures a week).