Maths > Algebraic geometry > Algebraic 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.
Motivation
Most books on algebraic groups tell you early on about Hopf algebras, and prove an anti-equivalence between categories of Hopf algebras and algebraic groups. I have never understood why I should care about this duality, since Hopf algebras are not a concept I ever encountered before, and I don't have much idea what "comultiplication" means (beyond formally). The fact that the books then go on to use Hopf algebras frequently in their proofs was annoying, because it looks like unnecessary complication.
In trying to write this post however, I realised that it is not just obfuscation: some things are easier to prove using Hopf algebras than without. And then it follows that we should care about the duality because we can use it to prove things. Hence I shall give up on caring about what a Hopf algebra "means" and just treat it as a technical tool to prove things about algebraic groups.
Note: I know that Hopf algebras have other uses, but they are not things that currently matter to me.
Hopf algebras, coordinate rings and polynomials
If applications to algebraic groups are the only way in which you care about Hopf algebras, then it is possible to give more explanation of what they mean, and at the same time give some idea of why they simplify proofs.
Think first of the coordinate ring of an affine variety. Coordinate rings are less scary than Hopf algebras - probably because I have lots of previous experience of rings. They are also very useful in proving things about affine varieties.
By using generators and relations, any argument using the coordinate rings of varieties can be turned into something explicitly about polynomials. The benefit of using coordinate rings instead of polynomials is that it doesn't require you to choose an embedding of the variety in affine space (equivalently, a set of generators of the coordinate ring). From a practical point of view, embedding independence tends to mean that there are fewer indices to keep track of; everyone knows that there are deeper reasons why embedding independent arguments are a good thing, but I can't quite explain why.
The Hopf algebra of an algebraic group is just the coordinate ring of the underlying variety, together with some extra structure that comes from the group structure. Just as for coordinate rings, any argument about algebraic groups using Hopf algebras encodes an argument expressed in terms of polynomials. But again, by using Hopf algebras, you can write the argument in an embedding-independent manner.
Definition of a Hopf algebra
After that long introduction, I should explain what the Hopf algebra of an algebraic group is.
Let be a linear algebraic group (over
), and
the ring of
-regular functions on
.
The multiplication operation on defines a morphism of varieties
.
By duality between affine varieties and
-algebras,
we get a
-algebra homomorphism
.
This is called comultiplication.
Similarly the inverse operation on defines a morphism of varieties
,
and so a
-algebra homomorphism
, called the antipode.
The identity "element" of is actually a (consistent) choice of element in
for every
-algebra
,
or in other words a morphism from the one-point variety
to
.
Dualising this gives a
-algebra homomorphism
, the counit.
The above morphisms of satisfy the group axioms; when we dualise, we get some relations on the
-algebra homomorphisms:
(In the third relation, is the
-algebra multiplication and
is the morphism which makes
into a
-algebra.)
We define a -Hopf algebra to be a
-algebra
together with
-algebra homomorphisms
,
,
satisfying the above axioms.
Now it is easy to check that the anti-equivalence between -algebras and
-varieties induces an anti-equivalence between algebraic groups over
and finitely-generated geometrically reduced
-Hopf algebras.