If you are familiar with group rings, you might think that the title of this post is false. If G is a nonabelian group, multiplying the basis elements g and h in can yield , so we have a problem. In general, if you have a problem that you can’t solve, you should cheat and change it to a solvable one (According to my advisor, this strategy is due to Alexander the Great). Today, we will change the definition of commutative to make things work.
Given a group G, the group ring is defined to be the free abelian group whose generating basis is given by the elements of G, and with “convolution” multiplication. In coordinates, this means
These sums make sense, because they have finitely many nonzero summands. In general, we can ask for the coefficients to be elements in other rings, like , and these are still called group rings.
What does it mean for a ring to be commutative? Normally, we would say that for any two elements a and b, we have ab = ba. I’d like to phrase this more categorically, without referring to individual elements. Multiplication in a ring A is given by a map . This is a bilinear operation, in the sense that na*b = a*nb for all integers n, so we can view multiplcation as a linear map (Chern apparently liked to say that tensor products replace bilinear maps by linear maps). To say that multiplication is commutative is to say that we have an equality of maps: , where is the switching map .
This suggests a relatively easy solution to our problem. If we adjust the map T suitably (e.g. using the identity map), we can make any ring commutative. Unfortunately, in solving our problem, we’ve destroyed any meaningful content in the notion of commutativity. We can salvage some meaning by demanding that T satisfy some natural sounding conditions, such as . This conveniently eliminates the identity map from consideration, but we still don’t have any immediate guidance for making a good choice.
I’d like to say that T is more than just a map of abelian groups, i.e., T ought to have some structure that makes it natural in a strong sense. One clue to turning this vague idea into mathematics is the unit compatibility condition I gave above. Since the unit can be viewed as a ring homomorphism , we can say that T should produce isomorphisms like in a manner compatible with the unit maps. One possible first attempt to strengthen this is to demand that T produce isomorphisms for all abelian groups in a manner compatible with all abelian group homomorphisms.
We can give this a categorical interpretation. Tensor product takes a pair of abelian groups to an abelian group, and transforms maps in a compatible way. This means tensor product is a functor , and since it satisfies some conditions like associativity, we say that Ab is a monoidal category. There is also a switch functor , taking an object (B,C) to (C,B) and also switching homomorphisms. Our demand on T then translates to the assertion that T is a natural isomorphism .
Unfortunately, there are no natural isomorphisms T that make our group ring commutative in this sense, mostly because the category of abelian groups is really well-behaved. Tensor products commute with colimits, and any abelian group is a colimit of some diagram of copies of the integers. Since T is the switch map whenever one of its inputs is a copy of the integers, T is forced to be the usual switch map on all inputs.
Fortunately, we can view the group ring in a different category, namely, it is a G-graded abelian group. The category of G-graded abelian groups admits a monoidal structure via a graded tensor product that puts the tensor product of a degree g group with a degree h group in degree gh. However, if G is nonabelian, there is no longer an obvious switch transformation, which we need to describe commutativity, since the tensor product in the opposite order would place things in a different degree.
We can try to solve this problem in the following way. Recall that the center of a group or a ring is the set of elements that commute with everything else. We could define the center of a monoidal category to be the subcategory of objects for which the tensor product with anything on the left is isomorphic to the tensor product on the right, and hope the group ring fits in somehow. This category looks nice at first, but since we don’t know what the isomorphisms are, we don’t have a good choice of a natural transformation T.
There is a better version, discovered by Drinfeld, and written up by Majid and Joyal-Street. Given a monoidal category C, one can construct a new category Z(C), called the Drinfeld center. Its objects are pairs , where x is an object in C, and is a natural isomorphism called a braiding, satisfying a compatibility condition:
(to be precise, I should have inserted some associators). Z(C) then admits a natural monoidal structure induced by the monoidal structure on C, but it also comes with a canonical transformation T, made out of the isomorphisms . With this structure, Z(C) is called a braided monoidal category, and together with the “forget braiding” functor, it is in fact universal with respect to braided monoidal categories with a monoidal functor to C.
Our task is now to find our group ring in some form inside the Drinfeld center of the category of G-graded abelian groups, and hope that it is in some sense commutative. First, we should write down precisely what it means to have a commutative ring in a braided monoidal category. Baez calls these structures r-commutative in his Hochschild homology paper, so if things work out, we could say that group rings are r-commutative.
Given a braided tensor category C, with associator , commutor
T, and unit structures l and r, a commutative ring is an object A equipped
with a multiplication morphism and a unit
morphism , such that:
- Multiplication with the unit yields the identity: , and the corresponding statement on the right.
- Multiplication can be dragged across the commutor:
and the reflected version. (I’m omitting the associator here.)
- Associativity: .
- Commutativity: .
There are diagrammatic ways of representing these axioms, using bits of string on a table, tied together at trivalent vertices. There might even be a Youtube video about it.
Also, we need to understand the center itself. Not every G-graded abelian group admits a braiding transformation. If lies in the center, we can apply to a copy of the integers in degree g. Since the natural transformation has to respect degree, we find that we have an isomorphism from the part of A in degree h to the part in degree . In general, the compatibility condition induces an action of the group ring on A, conjugating degrees.
Objects of the Drinfeld center are then G-equivariant G-graded abelian groups. We can view these as sums of pairs (g,A), where g is a representative of a conjugacy class and A is a representation of the centralizer . The G-action on (g,A) is given by conjugating g, and acting on the G-module induced from A. From our choice of commutor notation, the G-action on the grading is from the left, and the braiding is given by .
Now, consider the sum . Under the “forget braiding” functor to G-graded vector spaces, this lands on the group ring. We can pull back the multiplication map to get a copy of the group ring in the center. To check that it is commutative, we compare the two maps and . In other words, group rings are commutative because .
I should note that this formalism works if we replace abelian groups with sets. G-graded abelian groups become sets with a map to G, and the center is the category of G-sets with a G-map to G (with action given by conjugation). G lives in this category in a straightforward way, and we can say that G is an abelian group object in this category. One reason I didn’t name this post “All groups are abelian” is that the current title seems slightly less blatantly false.
You might be a bit disappointed if you’ve read this far to find that I’ve just redefined commutativity to hold using a rather tautological-looking conjugation trick. I concede that the punch line is a bit anticlimactic. However, we can think of this example of hidden commutativity as a toy model of a deeper phenomenon, known as transmutation.
There is an old philosophy due to Tannaka that a group is determined by its representation theory, and this was put into a categorical framework by Saavedra-Rivano and later refined by Deligne, Krein, and possibly others. The statement of Tannaka-Krein duality is that any symmetric tensor category with well-behaved duals and a faithful monoidal functor to vector spaces is equivalent to the category of representations of a proalgebraic group that is unique up to isomorphism. We will consider the coordinate ring picture, which is that it is equivalent to the category of comodules of a commutative Hopf algebra. The Hopf algebra is constructed by considering the natural transformations from the faithful functor to itself. Since the functor is linear, we have an additive structure, and composition makes it an algebra. We also have a coproduct arising from the tensor struture on the categories, and the counit and antipode are unique if they exist (this tends to require some completeness assumptions).
In some work in the early 1990s, Majid and Lyubashenko pointed out that one doesn’t need the functor to go to vector spaces. If the functor goes to any symmetric tensor category, you can reconstruct a Hopf algebra object in
that category by the same precedure (assuming existence of suitable colimits). More generally, if you have two braided tensor categories and a faithful braided tensor functor between them, the Hopf algebra of natural transformations has comodule category equivalent to the first category. This is particularly useful in part because we can use the identity functor, so we don’t actually need the second category. Given a braided tensor category, it is the [co]representation category of a commutative Hopf algebra in itself.
This can be applied to familiar examples of braided tensor categories, such as representations of quantum enveloping algebras, and the Drinfeld center that we saw earlier. These particular cases have a nondegeneracy property, known as factorizability: there is a map, called the inverse quantum Killing form (or “braided Fourier transform”), that is an isomorphism between the Hopf algebra and its dual. In particular, if we recast as a Hopf algebra in its own category of representations, it becomes transmuted to something commutative and cocommutative, and it is isomorphic to its dual, the “braided coordinate ring”. This is a phenomenon that only happens in the braided world, since sending q to 1 degenerates the two into a universal enveloping algebra and the coordinate ring of an algebraic group, and they are no longer isomorphic. One can take an extreme interpretation of this fact, and claim that braided commutative groups are the natural objects in this picture, and that the algebraic groups that we know and love are degenerate manifestations.
Thanks to David Jordan for introducing me to transmutation in the pre-Talbot seminar. Also, I should point out a neat paper by John Francis and the Davids,that develops Drinfeld centers from the viewpoint of sheaves on loop space, along with other gems.