The n-Category Café

I talked about Hall algebras back in “week230”; now we’re going to groupoidify them. Hall algebras are exciting because they’re a slick way to get a quantum group from an $A D E$ type Dynkin diagram — or at least the top half of a quantum group.

Let me recall some stuff from “week230”, where I explained the the Hall algebra $H(A)$ of an abelian category $A$.

As a set, this consists of formal linear combinations of isomorphism classes of objects of $A$. No extra relations! It’s an abelian group with the obvious addition. But the cool part is, with a little luck, we can make it into a ring by letting the product $[a] [b]$ be the sum of all isomorphism classes of objects $[x]$ weighted by the number of isomorphism classes of short exact sequences

$$0 \to a \to x \to b \to 0$$

This only works if the number is always finite.

The fun starts when we take the Hall algebra of $Rep(Q)$, where $Q$ is a quiver. We could look at representations in vector spaces over any field, but let’s use a finite field - necessarily a field with $q$ elements, where $q$ is a prime power.

Then, Ringel proved an amazing theorem about the Hall algebra $H(Rep(Q))$ when $Q$ comes from a Dynkin diagram of type $A$, $D$, or $E$:

• C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-592.

He showed this Hall algebra is a quantum group! More precisely, it’s isomorphic to the $q$-deformed universal enveloping algebra of a maximal nilpotent subalgebra of the Lie algebra associated to the given Dynkin diagram.

That’s a mouthful, but it’s cool. For example, the Lie algebra associated to $A_n$ is $sl(n+1)$, and the maximal nilpotent subalgebra consists of strictly upper triangular matrices. We’re $q$-deforming the universal enveloping algebra of this. One cool thing here is that the "q" of q-deformation, familiar in quantum group theory, gets interpreted here as a prime power - something we’ve already seen in "week185" and subsequent weeks.

• Lecture 22 (Jan. 10) - James Dolan on groupoidifying the Hall algebra of an abelian category. Any abelian category A gives a “trispan” of groupoids: namely, three functors from the groupoid of short exact sequences in A to the underlying groupoid of A, say A₀. These three functors send any exact sequence $$0 \to a \to x \to b \to 0$$ to the subobject $x$, the quotient object $b$ and the ‘total’ object $x$, respectively. Degroupoidifying $A_0$ we get a vector space $H(A)$ — this consists of formal linear combinations of isomorphism classes of objects of $A$. Ignoring possible divergences, degroupoidifying the trispan then gives a product

  $$H \otimes H \to H$$ A magical fact: this product is associative, making $H$ into an associative algebra called the Hall algebra of $A$. So, we have groupoidified the Hall algebra.

  The classic example arises when $A$ is the category of representations of a quiver on vector spaces over the field with $q$ elements, $F_q$. The simplest example: the quiver $A_2$, which looks like this: $$\bullet \longrightarrow \bullet$$

  • Streaming video in QuickTime format; the URL is
    http://mainstream.ucr.edu/baez_01_10_stream.mov
  • Downloadable video
  • Lecture notes by Alex Hoffnung
  • Lecture notes by Apoorva Khare