Posts Tagged ‘Hopf algebroids’

Hopf algebroids and (quantum) groupoids (Part 2)

8 September 2014

Last time I defined weak Hopf algebras, and claimed that they have groupoid-like structure. Today I’ll make that claim more precise by defining the groupoid algebra of a finite groupoid and showing that it has a natural weak Hopf algebra structure. In fact, we’ll get a functor from finite groupoids to weak Hopf algebras.

First, recall how the group algebra works. If G is a group, its group algebra is simply the vector space spanned by elements of G, and with multiplication extended linearly from G to this vector space. It is an associative algebra and has a unit, the identity element of the group.

If G is a groupoid, we can similarly form the groupoid algebra. This may seem strange at first: you might expect to get only an algebroid of some sort. In particular, whereas for the group algebra we get the multiplication by linearly extending the group multiplication, a groupoid has only a partially defined “multiplication”—if the source of g differs from the target of h, then the composite gh is undefined.

However, upon “linearizing”, instead of saying gh is undefined, we can simply say it’s zero unless s(g)=t(h). This is essentially all there is to the groupoid algebra.  The groupoid algebra \mathbb{C}[G] of a groupoid G is the vector space with basis the morphisms of G, with multiplication given on this basis by composition whenever this is defined, zero when undefined, and extended linearly from there.

It’s easy to see that this gives an associative algebra: the multiplication is linear since we define it on a basis and extend linearly, and it’s associative since the group multiplication is. It is a unital algebra if and only if the groupoid has finitely many objects, and in this case the unit is the sum of all of the identity morphisms.

Mainly to avoid saying “groupoids with finitely many morphisms”, I’ll just stick to finite groupoids from now on, where the sets of objects and morphisms are both finite.

If we have a groupoid homomorphism, then we get an algebra homomorphism between the corresponding groupoid algebras, also by linear extension. So we get a functor

\mathbb{C}[\cdot]\colon\mathbf{FinGpd} \to \mathbf{Alg}

from the category of finite groupoids to the category of unital algebras.

But in fact, this extends canonically to a functor

\mathbb{C}[\cdot]\colon\mathbf{FinGpd} \to \mathbf{WHopf}

from the category of finite groupoids to the category of weak Hopf algebras.

To see how this works, notice first that there’s a canonical functor

\mathbb{C}[\cdot]\colon\mathbf{Set} \to \mathbf{Coalg}

from the category of sets to the category of coalgebras:  Every set is a comonoid in a unique way, so we just linearly extend that comonoid structure to a coalgebra.

In case that’s not clear to you, here’s what I mean in detail.  Given a set X, there is a unique map \Delta\colon X \to X \times X that is coassociative, namely the diagonal map \Delta(x) = (x,x). This is easy to prove, so do it if you never have.  Also, there is a unique map to the one-element set \epsilon\colon X \to \{0\}, up the choice of which one-element set to use.  Linearly extending \Delta and \epsilon, they become a coalgebra structure on the vector space with basis X. Moreover, any function between sets is a homomorphism of comonoids, and its linear extension to the free vector spaces on these sets is thus a homomorphism of coalgebras.  This gives us our functor from sets to coalgebras.

So, given a finite groupoid, the vector space spanned by its morphisms becomes both an algebra and a coalgebra.  An obvious question is: do the algebra and coalgebra structure obey some sort of compatibility relations?  The answer, as I already gave away at the beginning, is that they form a weak Hopf algebra.  The antipode is just the linear extension of the inversion map g \mapsto g^{-1}.

(More generally, for those who care, the category algebra \mathbb{C}[C] of a finite category C (or any category with finitely many objects) is a weak bialgebra, and we actually get a functor

\mathbb{C}[\cdot] \colon \mathbf{FinCat} \to \mathbf{WBialg}

from finite categories to weak bialgebras.  If C happens to be a groupoid, \mathbb{C}[C] is a weak Hopf algebra; if it happens to be a monoid, \mathbb{C}[C] is a bialgebra; and if it happens to be a group, \mathbb{C}[C] is a Hopf algebra. )

This is nice, but have we squashed out all of the lovely “oid”-iness from our groupoid when we form the groupoid algebra? In other words, having built a weak Hopf algebra on the vector space spanned by morphisms, is there any remnant of the original distinction between objects and morphisms? 

As I indicated last time, the key is in these two “loop” diagrams:


The left loop says to comultiply the identity, multiply the first part of this with an element g and apply the counit. Let’s do this for a groupoid algebra, where 1 = \sum_x 1_x, where the sum runs over all objects x.  Since comultiplication duplicates basis elements, we get

\Delta(1) = \sum_x 1_x \otimes 1_x

We then get:

g\mapsto \sum_x \epsilon(1_x\cdot g) \otimes 1_x = 1_{t(g)}

using the definition of multiplication and the counit in the groupoid algebra.  Similarly, the loop going in the other direction gives g \mapsto 1_{s(g)} , as anticipated last time. 

So, we can see that the image of either of the two “loop” diagrams is the subspace spanned by the identity morphisms.  This is a commutative subalgebra of the groupoid algebra, and these maps are both idempotent algebra homomorphisms.  So, they give “projections” onto the “algebra of objects”.   

In fact, something like this happens in the case of a more general weak Hopf algebra.  The maps described by the “loop” diagrams, are again idempotent homomorphisms and we can think of them as analogs of the source and target maps.  But there are some differences, too.  For instance, their images need not be the same in general, though they are isomorphic.  The images also don’t need to be commutative.  This starts hinting at what Hopf algebroids are like. 

But I’ll get into that later.


Hopf algebroids and (quantum) groupoids (Part 1)

1 September 2014

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:

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:

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:

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:

WHA-algand a coassociative coalgebra with counit:

wha-coalg(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:WHA-adj

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:

Screen shot 2014-08-22 at 4.29.47 PMsince 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 S which I’ll draw like this:WHA-antipode

and it satisfies these axioms:

WHA-s1-3Like 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 S 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:

g \mapsto 1_{t(g)}

where t is the map sending each morphism to its target, and 1_x denotes the identity morphism on the object x.  

Likewise, consider the dual of the previous axiom:


In the groupoid world, the left hand side gives the map

g \mapsto 1_{s(g)}

where s denotes the map sending each morphism to its source. 

So… if weak Hopf algebras really are like groupoids, then these two loop diagrams:

WHA-target-source 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 1_{s(1_{s(g)})}=1_{s(g)} (“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 1_{t(S(g))} = 1_{s(g)} does not hold,  if you draw it instead starting from the equation 1_{t(S(g))} = S(1_{s(g)}), then you get an equation that holds in any weak Hopf algebra.