Posted by Jason Polak on 07. June 2013 · Write a comment · Categories: algebraic-geometry, group-theory · Tags: , ,

IMG861From Highlights 12 and Highlights 13, we have gained quite a bit of information on connected reductive groups $ G$ of semisimple rank 1. Recall, this means that $ G/R(G)$ has rank 1 where $ R(G)$ is the radical of $ G$, which is in turn the connected component of the unique maximal normal solvable subgroup of $ G$.

But wait, why have we been looking at groups of semisimple rank 1 at all? Let’s take a quick look at how we can get a good source of this groups inside a general group $ G$.

Tori

Let $ G$ be a connected algebraic group over $ k=\overline{k}$. We are not assuming that $ G$ is reductive or anything else besides this. We start by dividing up the tori of $ G$ into two kinds: the regular tori and the singular tori. Both of these species will be important in our study of algebraic groups.
More »