As before, this seminar is jointly run by John Baez and James Dolan, and we'll report on research we've done with Todd Trimble.
Also as before, here you can find videos and handwritten notes of the seminar, as well as links to blog entries at the n-Category Café, where you're encouraged to ask questions and make comments! If you can't figure out how the videos work, try this.
For much more on this subject, see:
Here are the lecture notes, videos and blog entries for this quarter:
Review of the harmonic oscillator and how to quantize it. The harmonic oscillator hamiltonian. Annihilation and creation operators.
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 F_{q}. The simplest example: the quiver A_{2}, which looks like this:
As usual, the Hall algebra has a basis consisting of isomorphism classes of representations. Every representation of A_{2} is a direct sum of copies of three indecomposable ones:
A = 0 → F_{q}
B = F_{q} → 0
and
1 C = F_{q} → F_{q}
where F_{q} is the field with q elements. Note that A and B are irreducible, while C is not: it is a "twisted sum" of A and B. In other words, there is a short exact sequence of quiver representations
0 → A → C → B → 0
which does not split. Computing the product in the Hall algebra: warmup.
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 → Y between groupoids: the pushforward f_{ *}: H_{0}(X) → H_{0}(Y) and the transfer f^{ !}: H_{0}(Y) → 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
be the functor "disjoint union with the 1-element set", then its pushforward is the creation operator, while its transfer is the annihilation operator!
Demonstration: an actual experiment proving these commutation relations! Weak pullbacks for composing spans of groupoids. Examples of weak pullbacks. Using weak pullbacks to compute AA^{*} and A^{*}A.
If you catch mistakes, let me know and I'll add them to the list of errata. You can also see LaTeX, encapsulated PostScript and xfig files to download if for some bizarre reason you want them. However, we reserve all rights to this work.