Geometric Representation Theory (Lecture 21)
Posted by John Baez
We’re back! In the fall quarter of the Geometric Representation Theory Seminar, James Dolan and I developed the basic idea of groupoidification. In the winter quarter we’ll apply it to examples, starting with three closely related ones:
 the $q$deformed harmonic oscillator,
 the Hall algebra of a quiver,
 the Hecke algebra of a Dynkin diagram.
As before, we’ll report on research we’ve done with Todd Trimble. Also as before, you’ll be able to see videos and handwritten notes of the seminar, and discuss them here at the $n$Category Café.

Lecture 21 (Jan. 8)  John Baez on groupoidifying and $q$deforming the quantum harmonic oscillator: overall battle
plan. The quantum harmonic oscillator is all about the polynomial algebra $\mathbb{C}[z_1, \dots, z_n]$. If we groupoidify this polynomial algebra, we get the groupoid of
$n$tuples of finite sets, which is also the groupoid of finite sets equipped with $n$stage flag.
If we $q$deform the polynomial algebra, we get a certain noncommutative algebra $\mathbb{C}_q[z_1, \dots, z_n]$. If both groupoidify and $q$deform it, what do we
get? A guess: the groupoid of finitedimensional vector spaces
with $n$stage flag over the finite field with $q$ elements, $F_q$.
Review of the harmonic oscillator and how to quantize it. The harmonic oscillator hamiltonian. Annihilation and creation operators.
 Answers to homework by John Huerta: the ground state of the harmonic oscillator Hamiltonian; the commutation relations between annihilation operators, creation operators, and the harmonic oscillator hamiltonian.
 Answers to homework by Christopher Walker.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_01_08_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare
Re: Geometric Representation Theory (Lecture 21)
When I download the linked .mov file, my browser just retrieves a 57byte file which points to the streaming video. Playing that in Quicktime then just tries to stream the video from the server, so I face the original bandwidth problems.
But I have to confess I’m only halfway through the last quarter’s lectures anyway!