Motivation: representations of Lie groups
Sophus Lie was a Norwegian mathematician who lived from 1842 to 1899. Essentially single-handedly he discovered two fundamental classes of objects in modern mathematics, which now bear his name: Lie groups and Lie algebras. More importantly, he built a bridge between them; this ivs remarkable, because Lie groups seem to be part of differential geometry (in today’s language) while Lie algebras seem to be purely algebraic. In this chapter we will discuss a small part of Lie’s discovery.
1.1 Homomorphisms of general linear groups
Typically, Lie groups are infinite groups whose elements are invertible matrices with real or complex entries. So they are subgroups of the general linear group