Lie groups, Lie algebras and representations

摘要

The study of Lie groups raises pedagogical as well as mathematical issues. On the one hand, there seems to be general agreement that Lie groups play a central role in mathematics, and therefore undergraduate mathematics majors should be exposed to them; on the other hand, the very definition of a Lie group involves terms and ideas (most obviously the concept of a differentiable manifold) that most undergraduates don't ever get to see. Even at the graduate level, it seems unfortunate to require students to take a lot of differential geometry in order to introduce them to basic Lie theory.