A Hopf Algebra Structure on Hall Algebras
Posted by John Baez
My student Christopher Walker is groupoidifying Hall algebras. What’s a Hall algebra? We get such an algebra starting from any category with a sufficiently wellbehaved concept of ‘short exact sequence’. In this algebra, the product of an object $A$ and an object $B$ is a cleverly weighted sum over all objects $X$ that fit into a short exact sequence
$0 \to A \to X \to B \to 0$
The ‘clever weighting’ is neatly explained by groupoidification, as sketched here. And if we pick our category in a nice way, the algebra we get is part of a quantum group!
But the Hall algebra is more than a mere algebra. It’s also a coalgebra! The algebra and coalgebra want to fit together to form a Hopf algebra, and they do, but only after a peculiar sort of struggle. Lately Christopher has been thinking about this, and he’s written a paper:

Christopher Walker, A Hopf algebra structure on Hall algebras.
Abstract: One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category $Vect$. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a different underlying category $Vect^K$ of vector spaces graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the $K$ grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication, and will also become a Hopf algebra object.
Comments, anyone? Corrections?
The point is that a Hopf algebra is a bialgebra, so the multiplication and comultiplication need to get along nicely. As we’ve recently discussed, they need to obey a condition sort of like this:
where the green blob is the multiplication and the red blob is the comultiplication.
And this condition involves a braiding: in the diagram at left, one wire needs to cross over the other! It turns out that the Hall algebra becomes a Hopf algebra very neatly if we choose the right braiding. Otherwise we need to do a bunch of ad hoc mucking around.
Actually, in the picture above, you’ll note that the two wires just cross, without one visibly going over the other. That style of drawing is fine if we’re in a symmetric monoidal category, like the category of vector spaces with its usual tensor product and usual braiding. But the Hall algebra becomes a Hopf algebra in a braided monoidal category that’s not symmetric. So we need to draw the compatibility condition a bit more carefully — see Christopher’s paper.
All these ideas can be groupoidified, and that’s what Christopher is doing now.
Posted at October 4, 2010 3:58 AM UTC
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Re: A Hopf Algebra Structure on Hall Algebras
Hi Chris. I just had a quick look at your paper over breakfast.
My first question is about your equation for the set $P^E {}_{M\,N}$ on p4 (equation numbers would be helpful here!). Won’t its cardinality be the continuum in general? Also, I have some typographical comments: I think the distinction between the two types of P is too subtle, something like $P^E _{M\,N}$ might be better for the cardinality; also, in the line beginning “and we call” after the displayed equation, you’re missing some punctuation.
Does all this generalize to other quivers which aren’t simply laced?
Re: A Hopf Algebra Structure on Hall Algebras
Hi Chris,
This is the “other Chris”.
Some typos:
p.1: “…the lie algebra…”
p.2: “We first to draw…”
p.3: “We then accomplishes…”
p.4: “Where we call aut(M) the set cardinality of the group Aut(M)” (Not a sentence)
p.4 “These are the correct factor…”
A small comment/question: You say a few times in the introduction that the incompatibility between the multiplication and comultiplication is a “problematic feature”. Why should I think of this as a problem, exactly? It seems more like an “interesting feature”, rather than a problem. (You essentially say this yourself on page 2.)
Another stupid comment: You say “As is standard, we will write multiplication…” and then write your diagrams backward in my opinion. When I’m thinking about an operad, I always write the diagrams so the inputs are at the bottom. I thought that was the standard way, no? (I have a feeling this is the kind of question that starts heated debates…my apologies if this is the case.)
Anyway, I like your paper so far. I will read more.
Re: A Hopf Algebra Structure on Hall Algebras
Maybe my question will be answered in the paper — I’m so far responding only to the abstract. I will begin by recalling some standard facts that you probably know and reference in your paper, and then ask for clarification of the final sentence in the abstract, where you write: “the Hall algebra … and will also become a Hopf algebra object.”
Let $(A,\Delta,\epsilon)$ be any coassociative counital coalgebra and $(B,m,1)$ any associative unital algebra (both objects in the same monoidal category). Then $\operatorname{Hom}(A,B)$ (hom in the underlying category) is an associative unital monoid under the “convolution product”: $f\star g = m\circ (f\otimes g) \circ \Delta$, and the identity element for $\star$ is $1\circ \epsilon$.
Now suppose that $B=A$ as objects, but I don’t impose any bialgebra condition. Then I do get a distinguished elements $\operatorname{id} \in \operatorname{Hom}(A,A)$. Inventing a word (maybe someone else has also invented it?), I would say that the data $(A,m,1,\Delta,\epsilon)$ is antipodal if $\operatorname{id}$ is left and rightinvertible in the monoid $(\operatorname{Hom}(A,A),\star)$. Notice that there is no need for any braidings, compatibility conditions, etc.
So: it seems from your sentence that the point is that the Hall algebra just is antipodal, whereas for you a Hopf algebra is the data $(A,m,1,\Delta,\epsilon)$ such that it is both antipodal and satisfies the bialgebra compatibility condition.
Is there any other content to that line that I’m missing?
Re: A Hopf Algebra Structure on Hall Algebras
Very neat! Of course you have to make a choice regarding which string to cross over and which under in drawing the bialgebra compatibility diagram in a braided monoidal category. Does that mean there are multiple distinct notions of “bialgebra” in a braided monoidal category, and the Hall algebra is only one of them?
Re: A Hopf Algebra Structure on Hall Algebras
Shahn Majid has studied intensively Hopf algebras in braided categories and called them braided groups. One of the first papers was this, but look also for later papers and his book Foundations of quantum group theory.
He has lots of recipes of how to make new examples out of old. For example, making braided groups out of quasitriangular Hopf algebras by a process of “transmutation”. Majid developed also associated “braided” geometry.
Re: A Hopf Algebra Structure on Hall Algebras
Another typo: on page 4, “we define set:”
Re: A Hopf Algebra Structure on Hall Algebras
If I understood right, the main result of this paper is long known to the experts. In
Kapranov says:
Note that because of the twist (1.4.4), the theorem does not mean that $R(A)$ is a bialgebra in the ordinary sense; it can be interpreted, however, by saying that $R(A)$ is a bialgebra in braided monoidal category of $K_0(A)$graded vector spaces.
Re: A Hopf Algebra Structure on Hall Algebras
The good news is that I’ve finally found out that I like Hopf algebras and would like to go to grad school at a place that they’re covered in the US. The bad news is that it’s already too late to apply at most schools. Any ideas?
Last year I published a paper that can be rewritten in Hopf algebra language. It showed that when you resum the long time propagators for the Hopf algebra (defined by the mutually unbiased bases of the Pauli algebra), you get the usual spin1/2 propagators (i.e. projection operators), but three copies that you can think of as generations.
The paper I’m working on now shows that when you find the propagators for the group (Hopf) algebra defined by the permutation group on three elements you discover that they map nicely on to the weak hypercharge and weak isospin quantum numbers of the elementary fermions.
Right now I understand Hopf algebra at a very low, intuitive level, but it’s clear to me that this is where I’m going to continue to work. The challenge of getting numbers out of the algebra is incredibly attractive. I’ve been working on my own, but it would be a lot easier if I were in a department; any suggestions?
Read the post
Christopher Walker on Hall Algebras
Weblog: The nCategory Café
Excerpt: Christopher Walker has successfully defended his thesis, A Categorification of Hall Algebras.
Tracked: June 11, 2011 8:01 AM
Re: A Hopf Algebra Structure on Hall Algebras
Hi Chris. I just had a quick look at your paper over breakfast.
My first question is about your equation for the set $P^E {}_{M\,N}$ on p4 (equation numbers would be helpful here!). Won’t its cardinality be the continuum in general? Also, I have some typographical comments: I think the distinction between the two types of P is too subtle, something like $P^E _{M\,N}$ might be better for the cardinality; also, in the line beginning “and we call” after the displayed equation, you’re missing some punctuation.
Does all this generalize to other quivers which aren’t simply laced?