George Hart’s “12-Card Star” and suit permutations.

When I was at the JMM in San Antonio in January, I was happy to catch a workshop by George Hart:

I went to some great talks at the JMM, but a hands-on, interactive workshop was a nice change of pace in the schedule. Having seen some of George’s artwork before, I couldn’t resist. In the workshop, he taught us to build his sculpture/puzzle which he calls the 12 Card Star. Here’s what mine looked like:

He supplied the necessary materials: 13 decks of cards, all pre-cut (presumably with a band saw), like this:

We each took 13 card from these decks—the 13th, he said, was “just in case something terrible happens.”

He showed us how to put three cards together:

Then he gave us a clue for assembling the star: the symmetry group is the same as that of a …

Wait! Let’s not give that away just yet. Better, let’s have some fun figuring out the symmetry group.

Let’s start by counting how many symmetries there are. There are twelve cards in the star, all identically situated in relation to their neighbors, so that’s already 12 symmetries: given any card, I can move it to the position of my favorite card, which is the one I’ll mark here with a blue line along its fold:

But my favorite card also has one symmetry: I can rotate it 180, flipping that blue line from end to end around its midpoint, and this gives a symmetry of the whole star. (Actually, this symmetry is slightly spoiled since I drew the five of hearts: that heart right in the middle of the card isn’t symmetric under a 180 rotation, but never mind that. This would be better if I had drawn a better card, say the two of hearts, or the five of diamonds.)

So there are symmetries in total, and we’re looking for a group of order 24. Since 24 = 4!, the most obvious guess is the group of permutations of a 4-element set. Is it that? If so, then it would be nice to have a concrete isomorphism.

By a concrete isomorphism, I mean a specific 4-element set such that a permutation of that set corresponds uniquely to a symmetry of the 12-card star. Where do we get such a 4-element set? Well, since there are conveniently four card suits, let’s get a specific isomorphism between the symmetry group of Hart’s star and the group of permutations of the set

At the workshop, each participant got all identical cards, as you can see in the picture of mine. But if I could choose, I’d use a deck with three 2’s of each suit:

From this deck, there is an essentially unique way to build a 12-Card Star so that the isomorphism between the symmetry group and the group of permutations of suits becomes obvious! The proof is `constructive,’ in that to really convince yourself you might need to construct such a 12-card star. You can cut cards using the template on George’s website. He’s also got some instructions there. But here I’ll tell you some stuff about the case with the deck of twelve 2’s. % and from these it will be clear that (if you succeed) your star will have the desired properties.

First notice that there are places where four cards come together, like this:

In fact, there are six such places—call these the six 4-fold rotation points—and it’s no coincidence that six is also the number of cyclic orderings of card suits:

Now, out of the deck of twelve 2’s, I claim you can build a 12-card star so that each of these cyclic orderings appears at one 4-fold rotation point, and that you can do it in an essentially unique way.

This should be enough information to build such a 12-card star. If you do, then you can check the isomorphism. Think up an permutation of the set of suits, like this one:

and check that you can rotate your 12-card star in such a way that all of the suit symbols on all of the cards in the 12-card star are permuted in that way.  The rest follows by counting.

Sometime I should get hold of the right cards to actually build one like this.

Of course, there are other ways to figure out the symmetry group. What George Hart actually told us during the workshop was not that the symmetry group was the permutation group on 4 elements, but rather that the symmetry group was the same as that of the cube. One way to see this is by figuring out what the `convex hull’ of the 12-card star is. The convex hull of an object in Euclidean space is just the smallest convex shape that the object can fit in. Here it is:

This convex polyhedron has eight hexagonal faces and six square faces. You might recognize as a truncated octahedron, which is so named because you can get it by starting with an octahedron and cutting off its corners:

The truncated octahedron has the same symmetry group as the octahedron, which is the same as the symmetry group of the cube, since the cube and octahedron are dual.

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Thanks to Chris Aguilar for the Vectorized Playing Cards, which I used in one of the pictures here.

Hopf algebroids and (quantum) groupoids (Part 2)

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 of a groupoid is the vector space with basis the morphisms of , 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

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

But in fact, this extends canonically to a functor

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

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 , there is a unique map that is coassociative, namely the diagonal map . This is easy to prove, so do it if you never have.  Also, there is a unique map to the one-element set , up the choice of which one-element set to use.  Linearly extending and , they become a coalgebra structure on the vector space with basis . 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 .

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

from finite categories to weak bialgebras.  If happens to be a groupoid, is a weak Hopf algebra; if it happens to be a monoid,  is a bialgebra; and if it happens to be a group, 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 and apply the counit. Let’s do this for a groupoid algebra, where , where the sum runs over all objects .  Since comultiplication duplicates basis elements, we get

We then get:

using the definition of multiplication and the counit in the groupoid algebra.  Similarly, the loop going in the other direction gives , 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)

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. 

Explaining Cartan geometry

I recently got back from a week’s visit with Julian Barbour, which we spent talking about geometric foundations of Machian physics in general, and shape dynamics in particular.

Much of what Julian and I are discussing isn’t to the level of blog-worthy material yet, but one of the key ideas is Cartan geometry, particularly in its various “conformal” incarnations. So, one thing I did during the visit was to explain a bit of Cartan geometry to him. Explaining Cartan geometry is always fun for me: it’s an incredibly beautiful subject, can be understood on an intuitive level, and yet still seems to be rather underappreciated. We thought it would be fun to record part of our discussion here for others to read.

Essentially, Cartan geometry is a way of studying geometry by rolling one object around on another: the geometry of the one serves as a “prototype” for the geometry of the other, and the nontriviality of the rolling—i.e. the failure to come back to the same configuration after rolling around a loop—measures the geometric deviation from that of the prototype. I’ll explain this in more detail in a moment.

Physically speaking, Cartan geometry is all about gauge theory of geometry. Gravity is a kind of gauge theory, but unlike the gauge theories of particle physics, it is a gauge theory that determines the geometry of the space it lives on. This is precisely what Cartan geometry is good for. I’ve explained this elsewhere, where I’ve used the idea of “rolling without slipping” to study spacetime geometry by rolling a copy of, say, de Sitter space along it.

Julian isn’t immediately interested in spacetime geometry. As I mentioned before, he’s interested in physics where “time” plays no primary role: there is only space, or more precisely, only configuration space. But, in broad terms, he is currently studying a kind of “geometric gauge theory.” Cartan geometry should still be the most natural language for it.

So, we recently spent a day struggling to form some sort of synthesis of my work and his, using Cartan geometry and rolling without slipping to understand physics in a world without time. Here is what we came up with:

Oh, OK, so we got a bit further than that…

Cartan versus Levi-Civita

We really did use the globe as a prop for discussing Cartan geometry. Unfortunately, Julian’s book is too slick and too bulky to effectively simulate rolling a plane on a sphere without slipping, but this drink coaster with cork backing worked beautifully:

Here, Julian is experimentally verifying my claims about Cartan geometry. By rolling the coaster around a cleverly chosen loop on the globe, carefully avoiding any slipping or twisting, he can get back to a configuration with the same point of tangency on the globe, but where the coaster has been both rotated and translated.

This “rolling without slipping” of the coaster along a path on the globe by is one of the simplest examples of a Cartan connection. It’s a rule for moving a homogeneous space like a plane, represented here by the coaster, around on a not necessarily homogeneous space, represented here by the globe. (While Julian’s globe looks pretty spherical at the scale of the picture, it’s actually a relief globe, which made it convenient to remember which space was supposed to be the homogeneous one.) The deviation from the homogeneous geometry is measured by the failure to come back to the same configuration after going around some loop using this rule.

Notice that this kind of “parallel transport” that is qualitatively quite distinct from that done by the more familiar Levi-Civita connection. In particular, while the Levi-Civita connection transports tangent vectors in a linear way, “rolling” the tangent plane gave us translations, which are not linear transformations.

To perform the Levi-Civita parallel transport using our drink coaster model, place one finger in the middle of the coaster, right at the point of tangency with the sphere:

Then move the coaster around with just that finger. To do this, you of course have to slide the coaster—a forbidden maneuver in the Cartan version—but you should still be careful not to twist the coaster relative to the globe.

Actually, you can think of the Levi-Civita transport via rolling without slipping, if you keep making corrections as you go. Suppose we fix the origin on our drink coaster so that we can think of it as a vector space. Then draw a vector on it. To transport the vector along a path, first break the path up into small steps. After rolling along the first bit, the coaster’s origin will no longer be at the point of tangency with the globe. This is unacceptable, since we’re supposed to be carrying our vector along by a linear transformation! So, make a correction: fortunately, there is a canonical way to slide the coaster without rotating it, maintaining the point of contact on the globe, so that the origin goes back where it should be.

Now make these little corrections after each little step along the total path. If your steps were sufficiently small, once you finally arrive at your destination, you’ll have a very good approximation to the Levi-Civita transport. In the limit of infinitely many infinitesimal steps, you get the Levi-Civita transport exactly.

So, the Cartan connection knows about the Levi-Civita connection, but it also knows more: in fact, the additional information we’ve suppressed in forcing the origin to remain in contact with the globe is enough to reconstruct the metric on the sphere, up to a constant global scale, or, if you prefer, up to a global choice of unit of length. For details, see Proposition 3.2 in Sharpe’s book on Cartan geometry for details. The equivalence of a Riemannian metric (up to global scale) with this type of Cartan geometry is one of the most basic applications of Cartan’s method of equivalence.

In any case, rolling a plane around on a sphere is just one kind of Cartan geometry—there’s really a different flavor of Cartan geometry for each kind of Klein geometry. For example, we could also talk about spherical Cartan geometry by rolling a ball on Julian’s globe:

Even though the globe and the ball are both spheres here, the “rolling distribution” is nontrivial (and would be even if the globe were perfectly spherical), because they are spheres of different diameter. Rolling around a loop on the globe, we can get any transformation of the ball we wish.

If, on the other hand, the globe were a perfect sphere and we had a ball that was an exact mirror image of it, parallel transport by rolling would would be completely trivial: starting out in a configuration where, say, Nairobi, Kenya touches its mirror image, and going around any loop from Nairobi back to itself, no matter how convoluted, the two Nairobis always come back in contact in the end.

The easy intuitive proof of this fact is to imagine rolling the globe on an actual mirror.

Hamster geometry

By now you may be wondering if I’ll get beyond the picture of rolling homogeneous spaces and tell you more precisely what a Cartan geometry is. I will—at least up to a few details that you can look up.

I’ll assume you already know some Klein geometry, or that you at least have vague impressions of it and can fake the rest. Briefly, a (smooth) Klein geometry is a manifold equipped with a Lie group of symmetries acting transitively: there is at least one taking me from any point in to any other point. Picking any point , we can identify with the coset space . So, abusing terminology a bit, we often refer to “A Klein geometry ,” forgetting the name of the original homogeneous space .

Now, if is a manifold of the same dimension as the Klein geometry , then a Cartan geometry on , “modeled on ,” has two basic ingredients:

1. a principal bundle ,
2. a -valued 1-form on (the Cartan connection)

satisfying some properties that I won’t bother writing down here. Instead, I just want to describe the geometric meaning of these ingredients. I’ve explained this in my papers using what I call “hamster geometry,” and it is perhaps worth reiterating that explanation here.

In the example of a ball rolling on a surface, the ball has symmetry group and point stabilizer . So, Cartan geometry for this model involves an -valued 1-form on a principal bundle over a 2d manifold, namely a surface. To understand the geometric meaning of these things, think of the ball as being controlled by a hamster inside of it. Here is a hamster in a hamster ball on a clearly non-homogeneous torus:

Forgetting about the ball itself for the moment, a hamster can be placed at any point on the surface, facing in any of an ‘s worth of directions. So, the configuration space of a hamster on a surface is a principal bundle over the surface. That is the geometric meaning of the bundle.

Now what about the “Cartan connection”? In this case, it should be an -valued 1-form on our hamster configuration space.

The key to understanding this one form is to realize that, so long as there is no slipping of the ball on the surface, the motion of the ball is completely determined by the motion of the hamster. The -valued 1-form just describes the rotation of the ball as the hamster moves: it takes tangent vectors to hamster configuration space—”infinitesimal changes” in hamster configuration—and gives elements of —”infinitesimal rotations” of the model sphere.

Most importantly, you can “integrate” these infinitesimals (using the path-ordered exponential) to get actual rotations of the sphere from actual paths through hamster configuration space. It’s all just a precise setup for describing how the hamster drives the ball around.

For more general Cartan geometries, I often imagine a “generalized hamster” running around on my base manifold, pushing a copy of around as he goes. You may have a hard time visualizing a generalized hamster, but with a little practice, you can do it! I’ll explain how in an upcoming post.

When I do that, I also want to discuss some more particular examples: mainly various versions of conformal Cartan geometry, which I’ve been thinking about more lately, partly because of the discussions with Julian. In fact, this post was supposed to be about conformal Cartan geometry until I got carried away. It will have to wait for another time.