The Connection Between Hopf Algebras and Groups

Hopf algebras are known to have a copious ammounts of structure which makes them useful in studying representations while, on the other hand, groups come equipt with little structure. In the following we will “realize” Hopf algebras as groups and comment on why we call quasitriangular Hopf algebras quantum groups. Hopefully, this will help demistify the confusion that comes along when studying Hopf algebras. For those of you who don’t know Hopf algebras, don’t worry! We will review them as well.

Groups

Lets review what a group is first before we dig any deeper. We eventually want to redefine the notation of a group by altering one of the axioms. Recall that a group G is a set with the following:

  • An assoicative binary map m:G \otimes G \to G.
  • An inverse map S:G \to G.
  • An identity element \eta:\ast \to G where \ast is the singleton set.

We also have two other natural maps which we get for free. The first is the obvious map \epsilon:G \to \ast, and the second is the diagonal map

\Delta:G \to G \otimes G \qquad g \mapsto g \otimes g.

Having recalled the defintion of a group, we now want to define the notion of a Hopf algebra. We will build up the defintion by defining an algebra, a coalgebra, a bialgebra, and then finally a Hopf algbera. All of our algebras and coalgebras will be assumed to be associative and unital (we will explain what this means in the following).

Algebras

Let k be a field and A be a k-vector space. We say that A is an algebra (remember assocaitive and unital) if there exists a map m:A \otimes A \to A called the multiplication and a map \eta:k \to A called the unit such that the following diagrams commute:

Observing these diagrams for a moment makes it clear why they are necessary and natural for the defintion of an algebra.

Coalgebras

Coalgebras, in a categorical sense, are the dual structures to algebras; we just need to reverse the directions of all the arrows. Let C be a k-vector space. Then C is a coalgebra (remember associative and unital) if there exists a map \Delta:C \to C \otimes C called the comultiplication, and a map \epsilon:C \to k called the counit such that the following diagrmas commute:

Observe that these diagrams are just those for an algebra but with the arrows reversed. Another moments thought shows that they are the natural maps one would want the comultiplication and counit to satisfy.

Bialgebras

Let B be a k-vector space posessing both an algebra and coalgebra structure over k. We would hope that the algebra and coalgebra structures interact in a “nice way”. They do not always do, but when they do we have the notion of a bialgebra. In particular, we say that B is a bialgebra if the following diagrams commute:

In the diagrams above

\tau:B \otimes B \to B \otimes B \qquad v \otimes w \mapsto w \otimes v

is called the twist map (it twists the factors in the tensor product by interchanging them). Yet another moments thought shows that all these maps really say is that the algebra and coalgebra structures behave nicely with each other.

Hopf Algebras

Hopf algebras are bialgebras with additional structure. Let H be a bialgebra over k. We say that H is a Hopf algebra if it posses a map S:H \to H called the antipode such that

m \circ (S \otimes id) \circ \Delta = \eta \circ \epsilon = m \circ (id \otimes S) \circ \Delta.

The double identity about is equivlaent to the following diagram being commutative:

The antipode is best thought of as a generalized inverse with respect to multiplication as can be realized by thinking about the diagram above for a moment.

Redefining Groups

We would like to redefine groups. As explained previously, a group G has maps m, \eta, \Delta, latex \epsilon, and S. We might of noticed by now that we have used the the same notation for the maps for algebras, coalgebras, and Hopf algebras. This is by no accident. If we replace the field k with the singleton set \ast, then it can be easily checked that G satisfies all of the commutative diagrams for an algebra, coalgebra, and bialgebra. In particular, we can think of G as a bialgebra over the field with one element \ast. Notice that we never need to use the inverse map S in verifying that G satisfies these commutative diagrams. So, we (re)define a group G as a bialgebra over \ast such that there exists a map S:G \to G such that

m \circ (S \otimes id) \circ \Delta = \eta \circ \epsilon = m \circ (id \otimes S) \circ \Delta.

In other words, it satisfies the axiom for a Hopf algebra! In terms of elements g \in G, this double identity becomes

S(g) = 1 = gS(g)

which is just another way of saying that S(g) is the inverse element of g so that S:G \to G is the inverse map! So, we see that groups are just Hopf algebra over \ast or Hopf algebras are just groups over an arbitrary field k. If we adopt \ast as a field, then we can say groups instead of Hopf algebras!

Quantum Groups

I thought I’d mention an aside about quantum groups for those who are familar with them (if you’re not familar but want to be, or just want learn more about the theory, see my notes here). For those who are familar with quantum groups, this should shed some light on why we call a quasitriangular Hopf algebra a quantum group. The “quantum” in quantum group expresses the relaxation of the cocommutativity of the Hopf algebra (which we will not explain why here), but more importantly the “group” in quantum groups really just means Hopf algbera.

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