## Geometric Representation Theory Seminar - Winter 2008

#### John Baez and James Dolan

This winter, our seminar continues what we began in last quarter: studying geometric representation theory with the help of groupoidification. Last quarter we developed the basic idea of groupoidification, starting from scratch. This time 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, 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!

All the videos are on Google drive, here, listed by date. For example, the first, data January 8th, is the file baez_01_08_stream.mov.

For much more on this subject, see:

Here are the lecture notes, videos and blog entries for this quarter:

• 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 C[z1, ..., zn]. 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 Cq[z1, ..., zn]. If both groupoidify and q-deform it, what do we get? A guess: the groupoid of finite-dimensional vector spaces with n-stage flag over the finite field with q elements, Fq

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.

• Lecture 22 (Jan. 10) - James Dolan on groupoidifying the Hall algebra of an abelian category. Any abelian category A gives a "trispan" of groupoids: three functors from the groupoid of short exact sequences in A to the underlying groupoid of A, say A0. Degroupoidifying A0 we get a vector space H, the zeroth homology of A0. Ignoring possible divergences, degroupoidifying the trispan gives a product

H ⊗ H → H

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 Fq. The simplest example: the quiver A2, which looks like this:

• → •

• Lecture 23 (Jan. 15) - James Dolan on groupoidifying the Hall algebra of an abelian category. Spans as "nondeterministic maps". The "twisted sum" of objects in an abelian category as a nondeterministic map. Calculating the Hall algebra coming from the category of representations of this quiver, known as A2 because it has two dots in a row:

• → •

As usual, the Hall algebra has a basis consisting of isomorphism classes of representations. Every representation of A2 is a direct sum of copies of three indecomposable ones:

```A  =  0 → Fq
```

```B  =  Fq → 0
```

and

```      1
C  =  Fq → Fq
```

where Fq 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.

• 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 L2(R) given by eigenfunctions of the number operator. The isomorphism between L2(R) and "Fock space", which is a Hilbert space completion of the polynomial algebra 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 H0(X). How to get two different linear operators from a functor f: X → Y between groupoids: the pushforward f *: H0(X) → H0(Y) and the transfer f !: H0(Y) → H0(X). Definition of zeroth homology, pushforward and transfer.

Groupoidifying Fock space and the annihilation and creation operators. If we let FinSet0 be the groupoid of finite sets, H0(FinSet0) is the Fock space k[z]. If we let

+1 : FinSet0 → FinSet0

be the functor "disjoint union with the 1-element set", then its pushforward is the creation operator, while its transfer is the annihilation operator!

• Lecture 25 (Jan. 22) - John Baez on groupoidifying the harmonic oscillator. The annihilation and creation operators as spans of groupoids. The groupoidified commutation relations:

A A* ≅ A* A + 1

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.

#### Errata

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.

© 2008 John Baez and James Dolan
baez@math.removethis.ucr.andthis.edu