Martin's Blog

Character groups of algebraic tori

Posted by Martin Orr on Sunday, 24 January 2010 at 18:10

In this post I will return to the subject of algebraic tori. Just as Pontryagin duality classifies locally compact abelian groups through their characters, so algebraic tori are also classified by their characters.

In order to account for the arithmetic phenomenon of non-split tori, we need to include a Galois action on the character group. The primary result of this post is that there is an anti-equivalence of categories between {k-tori} and {finitely generated free abelian groups with a continuous action of \mathop{\mathrm{Gal}}(k^s/k)}.

2 comments Tags alg-geom, alg-groups, maths, tori

Hopf algebras and algebraic groups

Posted by Martin Orr on Sunday, 17 January 2010 at 21:12

This post was supposed to be about character groups of algebraic tori. But while writing about that, I found that I wanted to use Hopf algebras, which were something that previously seemed alien to me. So instead I have written about Hopf algebras and why they are useful in the study of algebraic groups.

no comments Tags alg-geom, alg-groups, maths

Algebraic tori

Posted by Martin Orr on Friday, 08 January 2010 at 16:16

Algebraic tori are the simplest examples of algebraic groups. In this post I will define algebraic tori and give some examples. Later I will write about their character groups and representations, and after that I will be able to talk about Hodge structures.

I have been trying to write a post about algebraic tori for several days, mainly because I was trying to sort out the proof that tori over separably closed fields are split. This is complicated and not very important as in practice I only care about perfect fields, so I have left it out.

Note that the algebraic tori considered here have nothing to do with the complex tori in my last post; indeed the complex points of an algebraic torus are not compact in the usual topology! They are called tori because they play the same role in the theory of algebraic groups as real tori play in the theory of Lie groups.

1 comment Tags alg-geom, alg-groups, maths, tori