There haven’t been many colloquium talks here this term, but there was one a week ago (Thursday) by Joel Kamnitzer from University of Toronto (and contributor to the Secret Blogging Seminar), who gave a talk called “Categorical $sl_2$ Actions and Equivalence of Categories”.

As it turns out, I have at least two things in common with Joel Kamnitzer. First, we were both President of the University of Waterloo Pure Math Club (which became the Pure Math, Applied Math, and Combinatorics and Optimization club ’round about my time, when we noticed the other two math faculties at Waterloo no longer had their own undergraduate clubs). Second, we both did math Ph.D’s in California.  And while that’s probably a coincidence, there were several themes in the talk that overlap things I’ve talked about here.

The basic idea behind the talk was roughly this: when there’s an action of the Lie algebra $sl_2(\mathbb{C})$ (i.e. trace-zero 2-by-2 matrices) on a space, that space can be decomposed into some eigenspaces, and one can get isomorphisms between certain pairs of them. So the question is whether this can be categorified: if there’s an action of a categorical $sl_2$ on a category, can it be decomposed into subcategories which generate it, such that certain pairs can be shown to be equivalent?

So first he reminded/informed us of some of the non-categorified examples. The main thing is to show an equivalent way of describing an $sl_2$ action. This uses that $sl_2$ is generated by three matrices:

$e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ and $h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

These satisfy some commutation relations: $(e,f) = h$, $(e,h) = 2e$ and $(f,h) = -2f$. These relations specify $sl_2$ up to isomorphism, so one can describe an action on a set by specifying what $e$, $f$, and $h$ do (satisfying the commutation relations, of course).  It’s a classical fact from Lie theory that representations of $sl_2$ all look similar: they’re direct sums $\bigoplus_r V(r)$ of eigenspaces of the generator $h$ (for integer eigenvalues $r$), and the generators $e$ and $f$ act as “raising” and “lowering” operators, $e: V(r) \rightarrow V(r+2)$ and $f : V(r) \rightarrow V(r-2)$.  (All of which is key to describing spins of fundamental particles, due to $SL_2(\mathbb{C})$ being the cover of the Lorentz group $SO(3,1; \mathbb{R})$, though that’s beside the point just at the moment.

We heard three examples, of which for me the most intuitively nice involves an action on the vector space $V_X = \mathbb{C}^{P(X)}$ generated by the power set of a fixed finite set $X$ of size $n$.  Then $h$ is a (modified) counting operator – its eigenspaces are the subspaces $V(r)$ generated by subsets of size $k$ (where $r = 2k -n$).  The operator $e$ takes a set $A \subset X$ of size $k$ and maps it to the sum $\sum B$ over all $A \subset B$ with $B$ of size $(k+1)$ (all ways to “add one element” to $A$);  $f$ takes $A$ to the sum of all subsets of size $(k-1)$ contained in $A$ (all ways to “remove one element” from $B$.  (This all seems very familiar to me from the combinatorial interpretation of the Weyl algebra, which I talk about here.)  These satisfy the commutation relations $ef - fe = h$.

Now, the “equivalences” in the talk will be categorified versions of some obvious isomorphisms here, namely $V(r) \cong V(-r)$ (that is, $k$ subsets are in bijection with $(n-k)$-subsets).  These turn out to be imposed by the fact that we have a representation of $sl_2$, which lifts to a representation of $SL_2(\mathbb{C})$ in $GL(V)$.  The isomorphism is given by restricting the action of $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ to $V(r)$.

There is a more algebraic-geometry version of this example which replaces the power set of a set with the union of the Grassman varieties of subspaces of $\mathbb{C}^n$.  Instead of the vector space generated by subsets of size $k$, one builds $V$ out of the cohomology of the tangent bundle to the variety, with $V(r) = H^{\bullet} ( T^{\star}Gr(k,\mathbb{C}^n))$.

Now, the thing I find interesting about this picture is that, as with the Weyl algebra setup I mention above, it represets the raising and lowering operators in terms of transfer through a span.  Since this seems to pop up everywhere, it’s important enough to think on for a moment.  The span in question goes from $T^{\star}Gr(k,\mathbb{C}^n)$ to $T^{\star}Gr(k+1,\mathbb{C}^n)$.

To say what goes in the middle, we use the fact that an element of the cotangent bundle $T^{\star}Gr(k,\mathbb{C}^n$ amounts to a pair $(W,X)$, where $W < \mathbb{C}^n$ is a $k$-dimensional subspace (a point on $Gr(k,\mathbb{C}^n)$) and $X$ is a tangent vector at $W$.  As it turns out $X$ amounts to a map $X : \mathbb{C}^n \rightarrow W$ which annihilates $W$ itself.  So then we have the variety $I = \{ (X,W_k,W_{k+1}) \}$ where $W_k < W_{k+1}$, and $(X,W_k)$ and $(X,W_{k+1})$ are cotangent vectors.  This has projection maps to the two cotangent bundles: $T^{\star}(Gr(k,\mathbb{C}^n)) \stackrel{\pi_k}{\leftarrow} I \stackrel{\pi_{k+1}}{\rightarrow} T^{\star}(Gr(k,\mathbb{C}^n))$.

Then the point is that the cohomology spaces $H^{\bullet}(T^{\star}(Gr(k,\mathbb{C}^n))$ are build from maps into $\mathbb{C}^n$, so we call “pull-push” them through the span by $e = (\pi_{k+1})_{\star} \circ \pi_k^{\star}$.  This defines $e$, and $f$ is similar, going the other way.

So much for actions of “old-school” $sl_2$: what about “categorical” $sl_2$?  To begin with, what does that even mean?  Well, Aaron Lauda has described a “categorified” version of $sl_2$ (actually, of Lusztig’s presentation of the enveloping algebra $U_q(sl_2)$ – a quantum version, though that won’t enter into this).  This is a categorification of the generators $E$, $F$, and $H$, and of their commutation relations (which now become isomorphisms, which may have to satisfy some coherence laws – the details here being incredibly important, but not very enlightening at first).  These $E$, $F$ and $H$ are now functors, rather than maps.

As a side note, this is not precisely a categorification of the Lie algebra $sl_2$, but actually a categorification of a particular presentation of $sl_2$.  Though, since I’m mentioning this, I’ll remark it’s much more like the categorification of the Weyl algebra which is involved in the groupoidification of the quantum harmonic oscillator.

In any case, Joel went on to describe categorical actions of $sl_2$.  Actually, he distinguished “weak” and “strong” versions, which is apparently a common usage, though not the one I’m used to.  “Weak” means things are specified up to unspecified isomorphisms required to exist, and “strong” means things are defined up to specified (presumably coherent) isomorphisms (which is what I usually understand “weak” to mean).  The strong ones are the ones which give the equivalences we’re looking for, though.

It turns out that an action of the categorical $sl_2$ on an additive category $D$ gives: (1) a way to split up $D = \bigoplus_r D(r)$ for integers $-n \leq r n$, and (2) the action of the generators $E$ and $F$ with $E : D(r) \rightarrow D(r+2)$ and $F : D(r) \rightarrow D(r-2)$, such that (3) there are commutation isomorphisms analogous to the commutator identities for regular $sl_2$.  I note that algebraic geometers prefer to use additive categories – where the $hom$-sets are abelian groups, rather than vector spaces, which is what they would be in a 2-vector space.  In fact, later in the talk we heard about generalizations to triangulated categories – even a weaker condition.  In the special case where the additive category happened to be a 2-vector space, we’d have a “2-linear representation of a 2-algebra”.

Now, the main example was similar to the one above involving Grassman varieties.  The difference is that one doesn’t of cooking up a vector space from $T^{\star}(Gr(k,\mathbb{C}^n))$ from the cohomology of its cotangent bundle, one cooks up an abelian category.  This is $D(r) = D Coh(T^{\star}(Gr(k,\mathbb{C}^n))$ where, again, $r = 2k - n$, for $r = -n ... n$.  This is the derived category of coherent sheaves on the cotangent bundle.  There seems to be some analogy between the two: cohomology involves maps into $\mathbb{C}$ (and the exterior algebra of forms), while coherent sheaves might be thought of as (algebraic) vector-space valued functions, a categorified version of functions.  Also, while the cohomology is a chain complex, the objects of the derived category are themselves chain complexes.  Exactly how the analogy works is something I can’t explain just now.

Anyway, the key result, due to Chuang and Rouqier, says that from a “strong” categorical $sl_2$ action (in the sense above) and the $E$ and $F$ are exact functors (in 2-vector spaces, they’d be “2-linear maps”), then there is an equivalence (given in terms of the $E$ and $F$) between the categories of complexes on $D(-r)$ and $D(r)$.  This isn’t quite what was wanted (we wanted an equivalence $D(-r) \cong D(r)$), so for the remainder of the talk we heard about work directed at this question: cases where it works, counterexamples when it doesn’t, some generalizations, and so on.