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January 18, 2008

Geometric Representation Theory (Lecture 22)

Posted by John Baez

This time in the Geometric Representation Theory Seminar, Jim introduces a new example: the Hall algebra of a quiver.

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 ADEA 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)H(A) of an abelian category AA.

As a set, this consists of formal linear combinations of isomorphism classes of objects of AA. 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][a] [b] be the sum of all isomorphism classes of objects [x][x] weighted by the number of isomorphism classes of short exact sequences

0axb00 \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)Rep(Q), where QQ 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 qq elements, where qq is a prime power.

Then, Ringel proved an amazing theorem about the Hall algebra H(Rep(Q))H(Rep(Q)) when QQ comes from a Dynkin diagram of type AA, DD, or EE:

  • 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 qq-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 nA_n is sl(n+1)sl(n+1), and the maximal nilpotent subalgebra consists of strictly upper triangular matrices. We’re qq-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 0axb0 0 \to a \to x \to b \to 0 to the subobject xx, the quotient object bb and the ‘total’ object xx, respectively. Degroupoidifying A 0A_0 we get a vector space H(A)H(A) — this consists of formal linear combinations of isomorphism classes of objects of AA. Ignoring possible divergences, degroupoidifying the trispan then gives a product

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

    The classic example arises when AA is the category of representations of a quiver on vector spaces over the field with qq elements, F qF_q. The simplest example: the quiver A 2A_2, which looks like this: \bullet \longrightarrow \bullet

Posted at January 18, 2008 1:54 AM UTC

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Re: Geometric Representation Theory (Lecture 22)

There’s now a downloadable version of the video for this seminar! Check it out and see if it works for you.

Posted by: John Baez on January 28, 2008 10:43 PM | Permalink | Reply to this
Read the post Geometric Representation Theory (Lecture 23)
Weblog: The n-Category Café
Excerpt: James Dolan on groupoidifying the Hall algebra of the A2 quiver.
Tracked: January 31, 2008 3:25 AM

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