Representations of Complex Semi-simple Lie Groups and Lie Algebras

Abstract

Lie groups and Lie algebras occupy a prominent and central place in mathematics, connecting differential geometry, representation theory, algebraic geometry, number theory, and theoretical physics. In some sense, the heart of (classical) representation theory is in the study of the semisimple Lie groups. Their study is simultaneously simple in its beauty, as well as complex in its richness. From Killing, Cartan, and Weyl, to Dynkin, Harish-Chandra, Bruhat, Kostant, and Serre, many mathematicians in the twentieth century have worked on building up the theory of semisimple Lie algebras and their universal enveloping algebras. Books by Borel, Bourbaki, Bump, Chevalley, Humphreys, Jacobson, Varadarajan, Vogan, and others form the texts for (introductory) graduate courses on the subject.