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 1element set”, then its pushforward is the creation operator, while its transfer is the annihilation operator!
 Answers to homework by John Huerta: the inner product on Fock space.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_01_17_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare
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?