Lectures On Representation Theory

An expanded version of the lectures given at the Nankai Mathematical Summer School in 1997. It provides an introduction to Lie groups, Lie algebras and their representations and introduces some directions of current research for graduate students who have little specialized knowledge in representation theory. It only assumes that the reader has a good knowledge of linear algebra and some basic knowledge of abstract algebra. Parts One to Three of the book cover the relatively elementary material of representation theory of finite groups, simple Lie algebras and compact Lie groups. These theories are a natural continuation of linear algebra. The last chapter of Part Three includes some results on extension of Weyl’s construction to exceptional groups. Part Four covers some advanced material on infinite-dimensional representations of non-compact groups such as the orbit method, minimal representations and dual pair correspondence, which introduces some directions of research in representation theory.