I’ve been thinking a lot about weak Hopf algebras and Hopf algebroids, especially in relation to work I’m doing with Catherine Meusburger on applications of them to gauge theory. I don’t want to talk here yet about what we’re working on, but I do feel like explaining some basic ideas. This is all known material, but it’s fun stuff and deserves to be better known.
First of all, as you might guess, the “-oid” in “Hopf algebroid” is supposed to be like the “-oid” in “groupoid”. Groupoids are a modern way to study symmetry, and they do things that ordinary groups don’t do very well. If you’re not already convinced groupoids are cool—or if you don’t know what they are—then one good place to start is with Alan Weinstein’s explanation of them here:
- Alan Weinstein, Groupoids: Unifying Internal and External Symmetry.
There are two equivalent definitions of groupoid, an algebraic definition and a categorical definition. I’ll use mainly categorical language. So for me, a groupoid is a small category in which all morphisms are invertible. A group is then just a groupoid with exactly one object.
Once you’ve given up the attachment to the special case of groups and learned to love groupoids, it seems obvious you should also give up the attachment to Hopf algebras and learn to love Hopf algebroids. That’s one thing I’ve been doing lately.
My main goal in these posts will be to explain what Hopf algebroids are, and how they’re analogous to groupoids. I’ll build up to this slowly, though, without even giving the definition of Hopf algebroid at first. Of course, if you’re eager to see it, you can always cheat and look up the definition here:
- Gabriella Böhm, Hopf Algebroids.
but I’ll warn you that the relationship to groupoids might not be obvious at first. At least, it wasn’t to me. In fact, going from Hopf algebras to Hopf algebroids took considerable work, and some time to settle on the correct definition. But the definition of Hopf algebroid given here in Böhm’s paper seems to be the one left standing after the dust settled. This review article also includes a brief summary of the development of the subject.
To work up to Hopf algebroids, I’ll start with something simpler: weak Hopf algebras. These are a rather mild generalization of Hopf algebras, and the definition doesn’t look immediately “oid”-ish. But in fact, they are a nice compromise between between Hopf algebras and Hopf algebroids. In particular, as we’ll see, just as a group has a Hopf algebra structure on its groupoid algebra, a groupoid has a weak Hopf algebra structure on its groupoid algebra.
Better yet, any weak Hopf algebra can be turned into a Hopf algebroid, and Hopf algebroids built in this way are rich enough to see many of features of general Hopf algebroids. So, I think this gives a pretty good way to understand Hopf algebroids, which might otherwise seem obscure at first. The strategy will be to start with weak Hopf algebras and consider what “groupoid-like” structure is already present. In fact, to emphasize how well they parallel ordinary groupoids, weak Hopf algebras are sometimes called quantum groupoids:
- Dmitri Nikshych and Leonid Vainerman, Finite Quantum Groupoids and Their Applications
So, here we go…
What is a Weak Hopf algebra? This is quick to define using string diagrams. First, let’s define a weak bialgebra. Just like a bialgebra, a weak bialgebra is both an associative algebra with unit:
Compatibility of multiplication and comultiplication is also just like in an ordinary bialgebra or Hopf algebra:
So the only place where the axioms of a weak bialgebra are “weak” is in the compatibility between unit and comultiplication and between counit and multiplication. If we define these combinations:
then the remaining axioms of a weak bialgebra can be drawn like this:
The two middle pictures in these equations have not quite been defined yet, but I hope it’s clear what they mean. For example, the diagram in the middle on the top row means either of these:
Just as a Hopf algebra is a bialgebra with an antipode, a weak Hopf algebra is a weak bialgebra with an antipode. The antipode is a linear involution which I’ll draw like this:
and it satisfies these axioms:
Like in a Hopf algebra, having an antipode isn’t additional structure on a weak Hopf algebra, but just a property: a weak bialgebra either has an antipode or it doesn’t, and if it does, the antipode is unique. The antipode also has most of the properties you would expect from Hopf algebra theory.
One thing to notice is that the equations defining a weak Hopf algebra are completely self-dual. This is easy to see from the diagrammatic definition given here, where duality corresponds to a rotation of 180 degrees: rotate all of the defining diagrams and you get the same diagrams back. Luckily, even the letter is self-dual.
There’s plenty to say about about weak Hopf algebras themselves, but here I want to concentrate on how they are related to groupoids, and ultimately how they give examples of Hopf algebroids.
To see the “groupoidiness” of weak Hopf algebras, it helps to start at the bottom: the antipode axioms. In particular, look at this one:
The left side instructs us to duplicate an element, apply the antipode to the copy on the right, and then multiply the two copies together. If we do this to an element of a group, where the antipode is the inversion map, we get the identity. If we do it to a morphism in a groupoid, we get the identity on the target of that morphism. So, in the groupoid world, the left side of this equation is the same as applying the target map, and then turning this back into a morphism by using the map that sends any object to its identity morphism. That is:
where is the map sending each morphism to its target, and denotes the identity morphism on the object .
Likewise, consider the dual of the previous axiom:
In the groupoid world, the left hand side gives the map
where denotes the map sending each morphism to its source.
So… if weak Hopf algebras really are like groupoids, then these two loop diagrams:
must essentially be graphical representations of the target and source maps.
Of course, I only said if Hopf algebras are like groupoids, and I haven’t yet explained any precise sense in which they are. But we’re getting there. Next time, I’ll explain more, including how groupoid algebras give weak Hopf algebras.
Meanwhile, if you want some fun with string diagrams, think of other things that are true for groupoids, and see if you can show weak Hopf algebra analogs of them using just diagrams. For example, you can check that the diagrammatic analog of (“the source of the source is the source”) follows from the weak Hopf algebra axioms. Some others hold make a trivial rephrasing: while the obvious diagrammatic translation of does not hold, if you draw it instead starting from the equation , then you get an equation that holds in any weak Hopf algebra.