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:

and a coassociative coalgebra with counit:

(If the meaning of these diagrams isn’t clear, you can learn about string diagrams in several places on the web, like here or here.)

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:

since these are the same by associativity.

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.

Tags: expository, groupoids, Hopf algebroids, weak Hopf algebras

3 September 2014 at 06:58 |

Hi! Nice post!

One thing that’s confusing me is those two “twist” diagrams you draw, which are “essentially graphical representations of the target and source maps”. Clearly you want these to be different, but I’m confused about why.

At first I forgot how you defined the “cap” and “cup” from which you build these twists (together with the braiding). Now I remember, but I still don’t know if they obey the zigzag identities. If they don’t, you should warn us that not all topologically plausible moves are allowed. If they do, and your braiding is symmetric, it seems those two twists should be inverses of each other.

I’m fairly confused, but I hope you see my point. In a naive topological interpretation of your diagrams, you should be able to “pull straight” either twist and see that it’s the identity. In a less naive interpretation, the strings in your diagrams have a framing that can keep track of a 360 twist. Then one of your twist diagrams corresponds to a clockwise twist and the other corresponds to a counterclockwise twist.

Maybe you know the “Whitney trick” shown on page 15 of Kauffman’s book:

http://homepages.math.uic.edu/~kauffman/KFI.pdf

That’s the sort of thing that’s on my mind.

3 September 2014 at 10:25 |

Thanks, John!

Indeed, I should have said that the “twists” or “loops” are built, in order, from:

where the braiding is symmetric here, since I’m working in the symmetric monoidal category of vector spaces. (Of course, the diagrams also tell you how to do Hopf algebras is a braided monoidal category, but that’s another story.)

I should also have pointed out that the “cap” and “cup” here are not the ones people might expect from string diagrams for compact closed categories. Indeed, these other kinds of caps and cups might not exist in general, since a weak Hopf algebra needn’t be finite-dimensional. In a situation where you need both kinds of caps and cups (for example, if the

dualof a finite dimensional weak Hopf algebra is also heavily involved) my notation has to be modified. Adding arrows to all the strands suffices for that purpose, where all of the diagrams in my post have downward arrows.The zigzag laws don’t hold for my caps and cups! I guess adding arrows to the diagrams would also make it less tempting to pull zigzags straight without looking at the rules.

On the other hand, the

zigzagzigandzagzigzaglaws bothdohold. One of these says that a “zigzagzig” can be reduced to a “zig”, and the other says a “zagzigzag” can be reduced to a “zag”. I hope you know what I mean. These follow from sticking another unit or counit onto the middle strand in the equation relating the comultiplication to the unit and in the equation relating the multiplication to the counit.As you mentioned, it’s also nice to use a framing in these diagrams. Unfortunately, this still doesn’t make all of the allowed moves topologically obvious.

4 September 2014 at 04:42 |

Thanks! Now it’s more clear, and I even understand what you mean by “zigzagzig law”.

(But I’ve never quite decided which sort of wiggle is a “zig” and which is a “zag”. Once when I was falling asleep I thought I’d figured it out: a ziggurat should have lots of zigs. But when I woke up that no longer made sense.)

On another silly terminological note, I urge you to use “twist” instead of “loop” since “loop” means something else, while “twist” or “framing twist” is widely used to mean what you’ve drawn above.

More importantly, I’m eager to hear where this story goes next!

8 September 2014 at 10:45

If I weren’t dealing with self-dual things like weak Hopf algebras, I might start worrying about the difference between zigs and zags.

As for “twist” vs. “loop”, I don’t really like “twist” in this context. As with Hopf algebras, it’s sometimes nice to draw the

antipodeas a twist, and you can see from the weak Hopf algebra axioms that this isn’t the same as what I called a “loop”.The next bit of the story is now up, by the way.