As in many mathematics departments, graduate students in McGill's Department of Mathematics and Statistics have to take a comprehensive examination comprising of two parts: a written part (Part A) and an oral part (Part B). The Part B exam is based on two topics related to the student's field of research. One of my Part B topics is algebraic groups.
The algebraic groups section will consist of some classical material found in usual sources, for which I am mainly using Humphreys' book, and Brian Conrad's notes on reductive group schemes, which can be found on his website.
Since one of my favourite aspects of learning is writing about what I learn, naturally I've decided to write a series of posts on algebraic groups, ranging from the classical to the modern. The idea of these posts is to illustrate some of the more technical material in a fashion that hopefully won't be too dry.
My approach shall be focused on the idea of many different perspectives: I shall describe different functors and different definitions of the same object, rather than rushing definitions to get to applications quickly.
I shall start with a bit of classical material, but I shall emphasize the approach through representable functors. Thus an algebraic group over $ k$ is thought of as an affine algebraic group scheme; that is, one that is represented by a finitely generated $ k$-algebra, where $ k$ is any commutative ring. However, when we look at classical results, we will assume that $ k$ is a field. Later we will return to the general case.
Topics to be Covered
Here is an approximate list of topics:
- Lie algebra of an algebraic group
- Some differentiation formulas
- Quotients and invariant theory
- Applications of Lie algebras
- Classification of reductive groups
- Reductive group schemes and root data