Higher representation theory in algebra and geometry ... et resume… · Higher representation theory in algebra and geometry: Lecture I Ben Webster UVA January 30, 2014 Ben Webster

Higher representation theory in algebra and geometry:Lecture I

Ben Webster

UVA

January 30, 2014

Ben Webster (UVA) HRT : Lecture I January 30, 2014 1 / 31

References

For this lecture, useful references include:

Beilinson, Lusztig and MacPherson, A geometric setting for the quantumdeformation of GLn (covers the appearance of sl2 by counting points inGrassmannians)

Chuang and Rouquier, Derived equivalences for symmetric groups andsl2-categorification (introduces notion of weak sl2 categorification andcovers the appearance of sl2 in cohomology of Grassmannian (briefly))

Lauda, A categorification of quantum sl(2) and Categorified quantumsl(2) and equivariant cohomology of iterated flag varieties (gives a moredetailed acount of connection to the cohomology of Grassmannians)

The slides for the talk are on my webpage at:http://people.virginia.edu/ btw4e/lecture-1.pdf


My justification

So, the first question I should address is this: why should you show up for thiscourse? Why is what I’m about to tell you worth 20 hours of your time, whenyou could be at home petting your dog?

I’ll get into the specific content of in a moment, but let me put in a pitch forthe whole field of “categorification.”

Definitioncat·e·gor·i·fic·a·tion nounthe philosophy that any number worth its salt is the size of a set or dimensionof a vector space, and any interesting set is the simple objects in category(even if it doesn’t know it yet).


My justification

I think this idea tends to sound like a scam when people first hear it. Theytend to say things like “wait, but that isn’t going to be unique...”

I’m contractually obligated to reply “categorification is an art, not a functor.”

Sometimes, things in life aren’t unique, but they can still be useful.

ExampleThe rational function

x1001 x73

2 − x1002 x73

1 + x1002 x73

3 − x1003 x73

2 + x1003 x73

1 − x1001 x73

3

x21x2 − x2

2x1 + x22x3 − x2

3x2 + x23x1 − x2

1x3

is actually a polynomial with non-negative integral coefficients; Weyl tells usthat these are the weight multiplicities of a simple representation of GL3.

Might seem like a forced example, but who would have thought to look atSchur functions if they hadn’t known about GLn?


My justification

Another example is the replacement of Euler characteristic by homology.

homology is a stronger invariant of topological spaces than Eulercharacteristic, for sure.

much more important, it’s functorial! You can talk about maps betweensets or vector spaces, not between numbers.

While algebraic topology certainly does a good job of telling topologicalspaces apart, it would be pretty reductive to suggest that was its main goal. Italso gives us a lens for thinking about the structure of topology whichcouldn’t even be phrased just taking about numbers.


My justification

So these are the themes I want to emphasize throughout.

Categorifications:

allow us to show properties of weaker structures that might be hard tosee otherwise. Positivity and integrality results are especially important,but there are other examples.

suggest facets of already understood structures that are interesting ontheir own, but we might not have noticed. In particular, I maintain thatthe theory of quantum groups makes absolutely no sense unless youcategorify it.

allow us to ask questions about previously known invariants andstructures that simply made no sense before. Think Euler characteristicvs. homology.


My plan

Ok, after that long prolegomenon, let me give a short plan for the course. Itneatly divides into two halves (at least in theory): one on the categorificationof structures related to sl2, and a second generalizing these notions to otherLie algebras.

1 in the first half, we’ll cover the definition of a categorical action of sl2,its connection to the cohomology of Grassmannians, how to categorifysimple representations and tensor products and connections to Khovanovhomology and the Temperley-Lieb category.

2 in the second half, we’ll extend this idea to other Lie algebras. This willrequire defining KLR algebras. As time permits, we’ll discuss differentapplications of this formalism, in knot theory, the theory of symmetricgroups, Hecke algebras and Cherednik algebras.


About sl2

Consider the Lie algebra g = sl2. This is the Lie algebra of 2× 2 trace 0matrices with the usual commutator.

If we let E =[

0 10 0

], F =

[0 01 0

]and H =

[1 00 −1

], then this algebra has a

presentation of the form

[H,E] = 2E [H,F] = −2F [E,F] = H

By definition, the universal enveloping algebra U(sl2) is the associativealgebra generated by the symbols E,F,H subject to the relations above(where [−,−] means commutator).


About sl2This algebra is fundamental for a lot of reasons, but let me emphasize aslightly less well-known one: it appears naturally from the structure of thecategory of finite sets.

Fix a finite set S, and consider the space V of functions 2S → C from thepower set of S. This is a C-vector space of dimension 2|S|.

Consider the operators E and F on V defined by

E · g(T) =∑

s∈S\T

g(T ∪ {s}) F · g(T) =∑t∈T

g(T \ {t})

A ridiculously easy, but still rather nice, theorem is:

TheoremThe operators E and F induce an action of sl2 on V. A function supported onsets of size k has weight (H-eigenvalue) |S| − 2k.


About sl2

This is one of these theorems that gets less and less impressive as you thinkabout it.

Proof. First, note that if S = S1 ∪ S2, then VS ∼= VS1 ⊗ VS2 , and that theendomorphisms E and F satisfy:

ES = E1 ⊗ 1 + 1⊗ E2 FS = F1 ⊗ 1 + 1⊗ F2.

Thus, if we have an sl2-action on VS1 and VS2 , then the action of VS is just thetensor product of these. Thus, we reduce to the case of a singleton: S = {s}.

In this case, V has a basis given by the characteristic functions v0 of ∅ and v1of {s}. In this case,

Ev0 = 0 Ev1 = v0 Fv1 = 0 Fv0 = v1.

This is how sl2 acts on C2, if we identify v0 7→ (1, 0) and v1 7→ (0, 1).


sl2 and Grassmannians

Ok, so this was just a fancy way of writing (C2)⊗|S|. But it’s not completelybankrupt; in fact, this is the kernel of the reason that sl2 has a categorification.

So, how can we upgrade this? Well, secretly, the power set 2S is “the points ofthe Grassmannians in S over a field with 1 element.” Maybe that’s an insanestatement to make, but certainly any interesting game involving inclusions offinite sets can be played using inclusions of vector spaces, especially of afinite field.

Let S be a vector space over a finite field Fp, let XS denote the union of theGrassmannians of subspaces in S (just as sets), and let VS denote the vectorspace of functions XS → C.



We can define some operators E and F on Vs much like before. For notationalpurposes, we set d = dim(S) and k = dim(W).

E · g(W) = p−1/2(k−1)∑

W′⊃Wdim(W′/W)=1

g(W ′)

F · g(W) = p−1/2(d−k−1)∑

W′⊂Wdim(W/W′)=1

g(W ′)

Ok, some of you know what the next theorem is. Some of you think you do,but you’re wrong. These don’t define an action of sl2 in a literal sense, thoughperhaps they do in a moral one.



In order to see what the problem is, let’s compute the commutator [E,F]:

[E,F] · g(W) = EF · g(W)− FE · g(W)

= p−1/2(d−1)∑

W′′⊂W′⊃Wdim(W′/W)=dim(W′/W′′)=1

g(W ′′)− p−1/2(d−1)∑

W′′⊃W′⊂Wdim(W/W′)=dim(W′′/W′)=1

g(W ′′)

If W ′′ 6= W, it appears exactly once in both sums, and thus its contributionscancel. On the other hand, W itself appears #P(S/W) many times in the firstterm, and #P(W) many times in the second. Thus, we have

[E,F] · g(W) = p−1/2(d−1)(#P(S/W)−#P(W))g(W)

This scalar is given by

p1/2d−k−1+p1/2d−k−2+· · ·+p−1/2(d−1)−pk−1/2d−1−pk−1/2d−2−· · ·−p−1/2(d−1)

=p1/2d−k − p−1/2d+k

p1/2 − p−1/2



This is a lot like a sl2 representation. After all, [E,F] acts by a scalar on thefunctions concentrated in a single dimension (which thus are like weightspaces).

However, for an sl2 representation, we would have expected that the scalar

would be d − 2k, not [d − 2k]p1/2 =p1/2d−k−p−1/2d+k

p1/2−p−1/2 .

As a function of a formal variable q, we call [n]q = qn−q−n

q−q−1 the quantizationor quantum number of n; note that limq→1[n]q = n.

Theorem (Beilinson-Lusztig-Macpherson)Actually E and F induce an action of the quantized universal envelopingalgebra Uq(sl2), a flat deformation of U(sl2) over C[q, q−1], specialized atq = p1/2.

However, I refuse to define Uq(sl2) before defining its categorification, sodon’t worry too much about this difference.


Convolution

The operations E and F can be thought of as having a geometric identitythemselves: for any finite set X, the set of functions C[X × X] has an algebrastructure under convolution:

f ? g(x1, x2) =∑x′∈X

f (x1, x′)g(x′, x2).

It’s easily shown that this is associative, and acts naturally on C[X] by

f ? h(x) =∑x′∈X

f (x, x′)h(x′).

Thus, the operations E and F are convolution with the functions

e(W,W ′) =

{p−1/2 dim W W ⊂ W ′, dim(W ′/W) = 10 otherwise

f (W,W ′) =

{p−1/2 dim(S/W) W ⊃ W ′, dim(W/W ′) = 10 otherwise


Back to categorification

I think you could rightly say “OK, that’s all lovely, but what does it have to dowith the price of tea in China?” (By which I mean categorification..)

Again, I suspect some of you know what’s coming. As a hint, the set I’vecalled XS isn’t really just a set. It’s the Fp points of an algebraic variety. So,functions on it are avatars of sheaves, under the function-sheafcorrespondence.

So, in order to categorify, I should replace every function by a sheaf.

Or not.This is the first class; maybe it’s a little early for the perverse sheaves.


Back to categorification

I think you could rightly say “OK, that’s all lovely, but what does it have to dowith the price of tea in China?” (By which I mean categorification..)

Again, I suspect some of you know what’s coming. As a hint, the set I’vecalled XS isn’t really just a set. It’s the Fp points of an algebraic variety. So,functions on it are avatars of sheaves, under the function-sheafcorrespondence.

So, in order to categorify, I should replace every function by a sheaf. Or not.This is the first class; maybe it’s a little early for the perverse sheaves.


Cohomology and bimodules

DefinitionThere is a decent replacement for them, though: the categories of modulesover cohomology rings. Let S be a complex vector space now, and let Gr(k, S)be the Grassmannian. We’ll want to consider the categories of modules overthe rings Hk := H∗(Gr(k, S);C) as a replacement for the vector space VS.

The maps E and F between these spaces should turn into functors, connectedto the subspace

Gr(k ⊂ k + 1, S) = {(W,W ′) ∈ Gr(k, S)×Gr(k + 1, S)|W ⊂ W ′}.



We have a diagram of projections:

Gr(k ⊂ k + 1, S)

Gr(k, S) Gr(k + 1, S)

π1 π2

DefinitionLet kHk+1 := H∗(Gr(k ⊂ k + 1, S)) as a left module over Hk via π∗1 and aright module over Hk+1 via π∗2 .

Let k+1Hk := H∗(Gr(k ⊂ k + 1, S)) with these actions reversed.

Consider the functors

Ek(M) := k−1Hk ⊗Hk M Fk(M) := k+1Hk ⊗Hk M.



Principle

The functors E and F categorify the operators E and F.

What the heck does this mean? I hope that intuitively, this seems reasonable:before, I had the characteristic function of a subvariety, and now I’ve replacedthat with its cohomology, a categorically richer object.

Theorem (Beilinson-Lusztig-Macpherson?, Chuang-Rouquier)

kHk+1 ⊗Hk+1 k+1Hk ∼= (kHk−1 ⊗Hk−1 k−1Hk)⊕ H⊕d−2kk (k ≤ d/2)

(kHk+1 ⊗Hk+1 k+1Hk)⊕ H⊕2k−dk

∼= kHk−1 ⊗Hk−1 k−1Hk (k ≥ d/2)

That is, a module over Hk behaves like a vector of weight d − 2k under theaction of E and F.



Principle

The functors E and F categorify the operators E and F.

What the heck does this mean? I hope that intuitively, this seems reasonable:before, I had the characteristic function of a subvariety, and now I’ve replacedthat with its cohomology, a categorically richer object.

Theorem (Beilinson-Lusztig-Macpherson?, Chuang-Rouquier)

Ek+1Fk ∼= Fk−1Ek ⊕ id⊕d−2k (k ≤ d/2)

Ek+1Fk ⊕ id⊕2k−d ∼= Fk−1Ek (k ≥ d/2)

That is, a module over Hk behaves like a vector of weight d − 2k under theaction of E and F.


A first categorical representation

Idea of proofIt’s a little difficult to give a proper proof without unpleasant formulae orgetting a bit more sheafy, but the underlying idea is simple.

First note that:

kHk+1 ⊗Hk+1 k+1Hk ∼= H∗(Gr(k ⊂ k + 1 ⊃ k, S)

kHk−1 ⊗Hk−1 k−1Hk ∼= H∗(Gr(k ⊃ k − 1 ⊂ k, S)

There’s a birational map between these varieties:

{W ′′ ⊃ W ′ ⊂ W} 7→ {W ′′ ⊂ W ′′ + W ⊃ W} when W ′′ 6= W

The “difference” between the two cohomologies is the same as the“difference” between the cohomologies of the bad sets for this map asmodules over H∗(Gr(k, S)).


A first categorical representation

One bad set is H∗(Gr(k ⊂ k + 1, S) and the other H∗(Gr(k ⊃ k − 1, S). Themaps

H∗(Gr(k ⊂ k+1, S)→ H∗(Gr(k, S)) H∗(Gr(k ⊂ k−1, S)→ H∗(Gr(k, S))

are fiber bundles with fiber Pd−k−1 in the first case and Pk−1 in the other.Thus, the cohomologies are free Hk-modules with ranks k and d − k.

Thus, the “difference” is

kHk+1 ⊗Hk+1 k+1Hk“− ”kHk−1 ⊗Hk−1 k−1Hk“ = ”(d − 2k) · Hk


Gradings

Thus far, everything I’ve done has been with ungraded modules, but thealgebras Hk are graded and we can make this whole constructionhomogeneous.

DefinitionAs a graded bimodules, let

k−1Hk := H∗(Gr(k ⊂ k + 1, S))(k − 1)

k+1Hk := H∗(Gr(k ⊂ k + 1, S))(d − k − 1)

where M(a) or qaM is M with its grading decreased by a.

Now, we should prove a graded version of the categorification theorem.


Gradings

Theorem (Beilinson-Lusztig-Macpherson?, Lauda)

Ek+1Fk ∼= Fk−1Ek⊕ id(d−2k)⊕ id(d−2k−2)⊕· · ·⊕ id(2k−d) (k ≤ d/2)

Ek+1Fk⊕ id(2k−d)⊕ id(d−2k−2)⊕· · ·⊕ id(d−2k) ∼= Fk−1Ek (k ≥ d/2)

That is, paying attention to the grading, we see that an Hk-module behaveslike a vector of weight d − 2k for the quantum group Uq(sl2). Setting q = 1corresponds to ignoring the grading.

This proof is almost the same as the ungraded proof; one just pays attention tothe grading on the cohomology of the bad loci. The term

[d − 2k]q = qd−2k + qd−2k−2 + · · ·+ q2k−d

thus comes from the difference of the cohomology of Pd−k−1 and Pk−1

(suitably shifted).Ben Webster (UVA) HRT : Lecture I January 30, 2014 23 / 31

Gradings

Theorem (Beilinson-Lusztig-Macpherson?, Lauda)

Ek+1Fk ∼= Fk−1Ek ⊕ id⊕[d−2k]q (k ≤ d/2)

Ek+1Fk ⊕ id⊕[2k−d]q ∼= Fk−1Ek (k ≥ d/2)

That is, paying attention to the grading, we see that an Hk-module behaveslike a vector of weight d − 2k for the quantum group Uq(sl2). Setting q = 1corresponds to ignoring the grading.

This proof is almost the same as the ungraded proof; one just pays attention tothe grading on the cohomology of the bad loci. The term

[d − 2k]q = qd−2k + qd−2k−2 + · · ·+ q2k−d

thus comes from the difference of the cohomology of Pd−k−1 and Pk−1

(suitably shifted).


Upgrade to sheaves

Don’t worry too much if you’re not familiar with these notions, but it wouldbe a failure on my part not to mention how to think about this action in asomewhat more sophisticated way.

The maps we’ve introduced don’t just induce maps on cohomology; theyinduce functors between any category of sheaves satisfying Grothendieck’ssix-functor formalism. Most importantly on:

constructible derived categories on Gr(k, S) (in any formalism you like:analytic, étale Q`-sheaves, etc.) or

derived categories of D-modules on Gr(k, S).Of course, any one who knows the Riemann-Hilbert correspondence will tellyou these are essentially the same, but it’s nice to have both.


Upgrade to sheaves

In either case, we let

Ek(M) = (π1)∗π∗2M[k − 1] Fk(M) = (π2)∗π

∗1M[d − k − 1]

Gr(k ⊂ k + 1, S)

Gr(k, S) Gr(k + 1, S)

π1 π2

Theorem (Beilinson-Lusztig-Macpherson)For constructible sheaves or D-modules, we have

Ek+1Fk ∼= Fk−1Ek⊕ id[d−2k]⊕ id[d−2k−2]⊕· · ·⊕ id[2k−d] (k ≤ d/2)

Ek+1Fk⊕ id[2k−d]⊕ id[d−2k−2]⊕· · ·⊕ id[d−2k] ∼= Fk−1Ek (k ≥ d/2).

Obviously, this is the “same” result we got for modules over the cohomology,but if there’s anything modern mathematics has taught me, it’s that you haveto be really careful about what “the same” means.


Weak sl2 actions

Still, now that we have two examples, we can make a definition.

DefinitionA weak sl2-action on an additive category C consists of:

A direct sum decomposition C ∼= ⊕n∈ZCn.

FunctorsEn : Cn → Cn+2 Fn : Cn → Cn−2

such that satisfy the relations

En−1Fn ∼= Fn+1En ⊕ id⊕n (n ≥ 0)

En−1Fn ⊕ id⊕−n ∼= Fn+1En (n ≤ 0)

(Chuang and Rouquier would require En and Fn+2 to be adjoint; I won’tbother with that for now.)


Weak sl2 actions

Examples:

The first example we gave is Cdd−2k

∼= Hk -mod. There are variations ofthis: you might only want projective modules, or various other naturalsubcategories.

Alternately, you can take Ddn∼= C -mod for n ∈ d, d− 2, . . . ,−d + 2,−d

and 0 otherwise. In this case, you should take E−d+2k(C) ∼= Ck+1 andFd−2k(C) ∼= Ck+1.

We’ll get into a lot more examples later; as you might guess, there’s a notionof strong action which we’ll want to discuss first.


Symmetric groups

Fix a field k, and consider the group algebras of the symmetric groups Sn overk. These are related by induction and restriction functors. LetXm =

∑i<m(im) by the mth Jucys-Murphy element.

Proposition

The action of Xm on ResSmSm−1

M commutes with the Sm−1-action. That is, it is a

natural transformation of the functor ResSmSm−1

.

Fix a ∈ k, and let

Ea(M) = {m ∈ ResSmSm−1

M | (Xm − a)Nm = 0 for N � 0}.

Fa(M) = {s⊗ m ∈ IndSmSm−1

M | s(Xm − a)N ⊗ m = 0 for N � 0}.

Proposition (Lascoux-Leclerc-Thibon; Chuang-Rouquier)

For each a ∈ k, the functors Ea and Fa define a weak sl2-action; under thesefunctors, k as S0-module “generates” all Sn-modules.


Weak sl2 actions

In order to talk in a clear way about categorification, we need the notion of aGrothendieck group. There are a few variations on this notion.

DefinitionGiven an additive category C, the split Grothendieck group (sGG) K(C) of Cis the abelian group generated by symbols [C] for C an object in C, modulothe relation:

[C1 ⊕ C2] = [C1] + [C2].

Let KC(C) ∼= K(C)⊗Z C.

TheoremIf C is an additive category with a weak sl2-action, the maps

[E] : KC(C)→ KC(C) [F] : KC(C)→ KC(C)

generate an sl2-action on KC(C).


Weak sl2 actions

Theorem

Both the categories of projective Hk-modules, and the example Ddn we

discussed have KC given by the unique irreducible sl2 representation ofdimension d + 1.

On the other hand, if we take d = 2, and consider all Hk-modules, we’llcategorify C2 ⊗ C2.

Philosophically, this theorem is a bit problematic. If there are a bunch ofdifferent ways to categorify the simple modules, then where are we? How willwe ever get control over the general situation? For me, this is a big strikeagainst the “weak” definition.

In fact, examples like Hk -mod show up a lot more often than ones like Ddn ;

thus, we’ll want a better definition that rules out the latter.


Next time...

There’s an even more basic philosophical issue here: you never want adefinition just saying things are isomorphic. You want to have control aboutwhat the isomorphisms are and what relations they satisfy.

All the hints we need about how to do this are hiding in the Grassmannians,but we’re going to have to look deeper to find them.

Next time we’ll get to the definition and examples of strong sl2-actions. Thiswill show us how rule out Dd, and get a better behaved theory of categoricalactions.


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Higher representation theory in algebra and geometry ... et resume… · Higher representation theory in algebra and geometry: Lecture I Ben Webster UVA January 30, 2014 Ben Webster