Highlights in Linear Algebraic Groups 14: Singular Tori

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

From 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 »