## January 31, 2008

### Geometric Representation Theory (Lecture 24)

#### Posted by John Baez

This time in the Geometric Representation Theory Seminar, I finished my lightning review of the quantum harmonic oscillator. Then I moved on to a lightning review of how to groupoidify it!

I’ve already explained this stuff in vastly greater detail back in the Fall 2003, Winter 2004 and Spring 2004 sessions of the seminar — you can see extensive notes by clicking on the links. This time we’re whizzing through this material very fast. Then we’ll use it to groupoidify a bunch of representations of the Lie algebras $gl(n)$. Then we’ll try to $q$-deform the whole story! At that point, we’ll hook up with what Jim has been explaining about quiver representations and quantum groups.

• Lecture 24 (Jan. 17) - John Baez on groupoidifying the harmonic oscillator. Lightning review of the quantum harmonic oscillator, continued. The number operator. The basis of $L^2(\mathbb{R})$ given by eigenfunctions of the number operator. The isomorphism between $L^2(\mathbb{R})$ and ‘Fock space’, which is a Hilbert space completion of the polynomial algebra $\mathbb{C}[z]$. We will be algebraists and call $k[z]$ Fock space, where $k$ is any field of characteristic zero.

Lightning review of groupoidification. Groupoids and functors. How to get a vector space from a groupoid $X$: its zeroth homology $H_0(X)$. How to get two different linear operators from a functor $f: X \to Y$ between groupoids: the pushforward $f_* : H_0(X) \to H_0(Y)$ and the transfer $f^! : H_0(Y) \to H_0(X)$. Definition of zeroth homology, pushforward and transfer.

Groupoidifying Fock space and the annihilation and creation operators. If we let $FinSet_0$ be the groupoid of finite sets, $H_0(FinSet_0)$ is the Fock space $k[z]$. If we let $+1 : FinSet_0 \to FinSet_0$ be the functor “disjoint union with the 1-element set”, then its pushforward is the creation operator, while its transfer is the annihilation operator!

Posted at January 31, 2008 2:53 AM UTC

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### Re: Geometric Representation Theory (Lecture 24)

There was a particularly dramatic moment in one of the discussions between Jim and me, where we together bore witness to the successful groupoidification of a certain R matrix. At the time I was very, very impressed (still am). Has this sort of thing come up in seminar?

Posted by: Todd Trimble on January 31, 2008 9:54 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

We are slowly marching towards the groupoidification of the Hecke algebra of the $A_n$ Dynkin diagram — maybe that’s what you mean? So far Jim has quickly sketched the idea.

But, it could use a lot more detail. This $A_n$ Hecke algebra stuff, the $A_n$ Hall algebra stuff, and the $q$-deformed harmonic oscillator are all different aspects of the same thing - and that’s the thing I’m trying to explain this quarter. But, it’s taking a while!

Posted by: John Baez on January 31, 2008 9:57 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

It’s related, but I was thinking more about the representation of the Hecke algebra on tensor powers of the tautological representation of quantum $su(2)$, for example. For me, this example was a real litmus test of groupoidification as holding the key to many things.

Posted by: Todd Trimble on January 31, 2008 9:58 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

In lecture 28, Jim just talked about representations of the Hall algebra associated to the $A_2$ quiver. So, we’re getting close to the Hecke algebra representations… but it would be great if you could write up some stuff about them. (I assume that’s where that $R$ matrix is coming from.)

I can’t tell how much of this stuff should eventually find its way into HDA7. If you write up stuff and want to put it into that paper, it’d be great. You’ve already helped so much that you should be a coauthor…

Posted by: John Baez on January 31, 2008 11:48 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

I can’t open the stream with lecture; also, lecture notes links don’t work for me.

Posted by: sirix on February 1, 2008 1:58 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

I fixed the lecture note links — thanks. I’m not sure what’s up with the streaming video. It worked earlier for me; now it says the site is too busy. I’ll check it tomorrow.

Posted by: John Baez on February 1, 2008 6:36 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

UCR’s streaming video website has been down since last Thursday night — they know there’s a problem, and they’re trying to fix it. Sorry!

Posted by: John Baez on February 6, 2008 5:13 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 24)

I believe the streaming videos are working again! Someone try ‘em and let me know.

Posted by: John Baez on February 12, 2008 7:02 PM | Permalink | Reply to this

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