When I was a grad student (not too long ago), my advisor would occasionally get excited about what seemed to me rather minor discoveries. They were often notational efficiencies, or an observation that some extra structure comes for free. Since he was the one with tenure, I figured there was probably a good reason to think that these were important ideas. Eventually, I decided that if you can represent math more efficiently in your head, then you can fit more math into active processing at a time, and you’re more likely to pull something interesting out. This is a big deal if you’re trying to formulate a highly structured argument in a proof, or if you just want to learn some math without wasting large chunks of your life.
Today, I’ll explain one of these ideas, which is that groups and Hopf algebras are really the same thing, even though a lot of people will tell you otherwise.
If you’ve taken a class in abstract algebra, you’ll recall that a group is a set equipped with a multiplication map , an inversion map , and an identity element , where is a one point set. These maps are required to satisfy some compatibilities, like associativity of multiplication. You should try to write them down.
Suppose you want to express the fact that inversion takes elements to their inverses, but you don’t want to refer to individual elements. This is difficult using only the data listed above, but there are two additional maps we get for free: and , where the second is the obvious diagonal map: . Using these, we can say: , i.e., multiplying an element with its inverse yields the unit. These extra maps allow us to write the definition of group using only abstract nonsense.
If you’ve seen Hopf algebras before, these structures should look really familiar. If you haven’t seen them, here’s a quick introduction. Given a commutative ring , a Hopf algebra over is a six-tuple , where is an -module, is a multiplication map, is a unit, is called the antipode, is a counit, and is called comultiplication. These are required to satisfy relations, basically asserting that multiplication and unit makes a ring, comultiplication and counit makes a coring (just turn all the previous arrows around), comultiplication and counit are a ring maps, and the antipode satisfies the identity in the previous paragraph.
Again, if you’re new to Hopf algebras, this large amount of structure may seem kind of unwieldy, but it is useful if you want to study representations (also known as modules, or “stuff that your thing acts on”). The counit gives you a trivial representation, the comultiplication gives you a way to combine representations, and the antipode provides duals. One then says that the category of representations has a monoidal structure with duals. There are also many examples of these objects in nature, such as group rings, cohomology of H-spaces, and differential operators on formal groups (these are universal enveloping algebras of Lie algebras in characteristic zero).
Now, suppose we want to formulate the notion of Hopf algebra, but with sets instead of -modules. Our ring is replaced by a singleton (one might say that we were working over the field with one element). We have a unit and an associative multiplication, and this gives us a monoid structure. The counit is uniquely defined, since singletons are terminal in sets. The comultiplication has to satisfy , so it has to take any element to (This was giving me trouble, until Noah pointed out that it was an axiom). Finally, the antipode has to satisfy the identity I gave before: , i.e., it takes any element to something that multiplies to identity, i.e., an inverse. This recovers the structure of a group.
Advanced paragraph: We can get a similar uniqueness result when instead of sets, we consider schemes or topological spaces. The distinguishing feature of these categories, as opposed to -modules, is that the monoidal structure is the categorical product, and this forces the counit to map to the terminal object, making it unique. The additional monoidal structure in -modules allows the coalgebra structure to be non-unique, and this is what makes the theory of Hopf algebras very rich (or a complete mess, depending on who you ask). Extra monoidal structures appear elsewhere in math – Beilinson and Drinfeld define about five different tensor products on D-modules over a curve in their chiral algebras book, although some of these are only pseudomonoidal.
Now that we’ve gone back and forth, we can say that groups are Hopf algebras in sets, or that Hopf algebras are groups in -modules, where group in this case means something with multiplication, and a way to get a monoidal structure with duals on representations. When people say that quantum groups aren’t groups, they are just being unnecessarily restrictive in their notion of group. In fact, the word “quantum” here is just a way to emphasize some noncommutativity. This might come in useful if you’re trying to understand Borcherds’ definition of quantum field theory, which seems to be “a group acting on something.” If you want to construct the standard model from this definition, keep in mind that the group and the something are going to be rather complicated.