One more piece of structure we need. We take a bialgebra , and we add an “antipode”, which behaves sort of like an inverse operation. Then what we have is a Hopf algebra.
An antipode will be a linear map on the underlying vector space. Here’s what we mean by saying that an antipode “behaves like an inverse”. In formulas, we write that:
On either side, first we comultiply an algebra element to split it into two parts. Then we use on one or the other part before multiplying them back together. In the center, this is the same as first taking the counit to get a field element, and then multiplying that by the unit of the algebra.
By now it shouldn’t be a surprise that the group algebra is also a Hopf algebra. Specifically, we set . Then we can check the “left inverse” law:
One thing that we should point out: this is not a group object in the category of vector spaces over . A group object needs the diagonal we get from the finite products on the target category. But in the category of vector spaces we pointedly do not use the categorical product as our monoidal structure. There is no “diagonal” for the tensor product.
Instead, we move to the category of coalgebras over . Now each coalgebra comes with its own comultiplication , which stands in for the diagonal. In the case of we’ve been considering, this comultiplication is clearly related to the diagonal on the underlying set of the group . In fact, it’s not going too far to say that “linearizing” a set naturally brings along a coalgebra structure on top of the vector space structure we usually consider. But many coalgebras, bialgebras, and Hopf algebras are not such linearized sets.
In the category of coalgebras over , a Hopf algebra is a group object, so long as we use the comultiplications and counits that come with the coalgebras instead of the ones that come from the categorical product structure. Dually, we can characterize a Hopf algebra as a cogroup object in the category of algebras over , subject to a similar caveat. It is this cogroup structure that will be important moving forwards.