Hopf Algebras

Take the diagrams of associativity and unital and flip all the arrows. These are below.

This is the comultiplication and counit of a coalgebra.

Let's put both an algebra and a coalgebra structure at the same time. And I'm going to make them get along. I will also provide an antihomomorphism S. If S^2=Id, then I would have a conjugation operation, but I'm not going to require that.

And this diagram commutes of course. It is just evil to draw non-commuting diagrams.

Let's use the example of the group algebra KG for some K and some G. The coproduct will be



The antipode map will be inversion of all group elements and the counit will send all group elements to 1 in K.

Let's check if this coalgebra structure behaves. We check it on a basis of KG namely g for all g in the group.

For the top path



The bottom path



The middle way



Success!

Another example: This one will generalize in a very rich manner.

Universal enveloping algebra of some Lie algebra.

Δ(x) = x ⊗ 1 + 1 ⊗ x for all x in the Lie algebra.
ε(x) = 0
S(x) = -x


These all respect commutators so they respect the ideal we mod out by in defining the universal enveloping algebra. Therefore they extend throughout the algebra. 

We are now ready to mess this structure by looking at the specific example of SL(2,C)

Start with M(2,C) it has a comultiplication given by



First of all remember these a's are actually functions on M(2,C) that read off the right coefficient. They are not complex numbers. That is why later we will be able to mess them up. This notation is a mishmash of tensor product and matrix multiplication but just replace multiplication by tensoring, but otherwise do matrix multiplication. Then you can just read off the coproducts of the a's which generate the regular functions on 2x2 matrices.

The counit is



Does this coproduct descend to SL(2,C)? Does the ideal generated by ad-bc-1 go both ways?





It is fine with comultiplication, it's also ok with counits. It's a coideal too.

That means we can use these coproducts and counits on SL(2,C)'s regular function space. We find a antipode and it is a Hopf algebra.

The antipode is given by S(f)(x)=f(x^-1). So the antipode of the function that reads the ij component is the function that reads off the ij component of the inverse matrix. This is why we concentrate on SL(2,C). There is a nice way to express the ij component of the inverse in terms of the original matrix components easily.

Well now we could suppose the a_ij's don't commute. Instead of commutativity, we put some different relations. (I am going to change my mind and call them a,b,c and d now)

ba=qab, db=qbd, ca=qac, dc=qcd, bc=cb, ad-da=(q-q^-1)bc

But we'll put the same comultiplication and counit. Again just like ad-bc, you just check that comultiplication and counit respect all the relations.

To get deformed SL(2,C) instead of deformed M(2,C) we see that the analog to ad-bc is now da-qbc. Apply comultiplication and counit and you'll see it is a coideal. It's also in the center of the algebra. It's okay to localize this quantum determinant to 1 and get to SL.

So we can get a Hopf algebra on the functions on any matrix group which are generated by the entry functions. Of course, this group may actually start off as a arbitrary topological group that was turned into a matrix group by some representation.

See http://en.wikipedia.org/wiki/Compact_quantum_group