Representation of General Lie Groups and General Lie Algebras

Abstract

Based on root systems, this chapter deals with representations of a Lie group via those of the corresponding Lie algebra. This chapter discusses a non-compact Lie group as well as a compact Lie group. To discuss both, we focus on the relation between a (real) Lie algebra and a complex Lie algebra. Since representations of a non-compact group are very complicated, this chapter treats only a part of its representations that are related to quantum theory. As such special representation has analogies with representations of a compact Lie group, they can be more easily understood than the general case. Since this chapter is composed of very advanced topics and such sections are labeled with *, the reader can omit this chapter in the first time.