Higher representation theory in algebra and geometry ... et resumes... · Higher representation...

Higher representation theory in algebra and geometry:Lecture VII

Ben Webster

UVA

March 13, 2014

Ben Webster (UVA) HRT : Lecture VII March 13, 2014 1 / 33

References

For this lecture, useful references include:Khovanov and Lauda, A diagrammatic approach to categorification ofquantum groups. IRouquier, 2-Kac-Moody algebras (both present definitions of KLRalgebra. I’ll use style of former, and substance of latter)Lauda and Vazirani, Crystals from categorified quantum groups(introduced crystal structure on simples)Varagnolo and Vasserot, Canonical bases and KLR-algebras (exactlywhat the title says)Tingley and B.W., Mirkovic-Vilonen polytopes andKhovanov-Lauda-Rouquier algebras (exactly what the title says)

The slides for the talk are on my webpage at:http://people.virginia.edu/~btw4e/lecture-7.pdf

You can also find some proofs that I didn’t feel like going through in class at:https://pages.shanti.virginia.edu/Higher_Rep_Theory/


http://people.virginia.edu/~btw4e/lecture-7.pdf

https://pages.shanti.virginia.edu/Higher_Rep_Theory/

Cartan matrices

DefinitionA symmetrizable Cartan matrix A is a matrix with entries in Z such that

aii = 2.

aij ≤ 0 and aij = 0⇔ aji = 0.

there are relatively prime numbers di such that djaij = diaji.

This allows us to define a symmetric bilinear form on the formal span ofsymbols {αi} (the root lattice) via 〈αi, αj〉 = diaij = djaji.

We’ll also fix a field k and polynomials Qij(u, v) ∈ k[u, v] such that

Qij(u, v) = tiju−aij + · · ·+ tjiv−aji

We’ll often want this to be homogeneous when deg(u) = di and deg(v) = dj.


KLR algebras

As discussed last time, attached to this data, we have aKhovanov-Lauda-Rouquier (KLR) or quiver Hecke algebra Rm.

The algebra Rm is generated over k by these elements with m strands modulothe relations on the next slide.

i1 i2 in

· · ·

ei

deg = 0

i1 ij in

· · ·· · ·

yj

deg = 2dij

i1 ij ij+1 in

· · ·· · ·

ψj

deg = −〈αij , αij+1〉


Diagrams

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i j

=

ji

Qij(y1, y2)

ki j

=

ki j

unless i = k = j± 1

i i

= 0

ii j

=

ii j

−

ii j

Qij(y3, y2)− Qij(y1, y2)

y3 − y1


Diagrams

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i j

=

ji

Qij(y1, y2)

ki j

=

ki j

unless i = k = j± 1

i i

= 0

ii j

=

ii j

−

ii j

Qij(y3, y2)− Qij(y1, y2)

y3 − y1


KLR algebras

This algebra is graded as long as Qij(u, v) is homogeneous via the rule

degi j

= −〈αi, αj〉 degi

= 〈αi, αi〉 = 2di

Last time we defined a special case of this algebra in the context ofcategorical actions, and we can use this one similarly:

DefinitionA categorical action of g is a collection of categorical sl2 actions (Fi,Ei) fori ∈ Γ, together with an extension of the nilHecke action to an action of Rm on(⊕Fi)

m.

However, this algebra is worth studying on its own terms.


Induction and restriction

DefinitionLet Rν be the subalgebra of diagrams where the sum of the labels on strandsis ν =

∑νiαi.

We have an horizontal composition map Rν ⊗ Rµ → Rν+µ, and thus inductionand restriction functors

M ◦ N = Rµ+ν ⊗Rν⊗Rµ M × N Resµ+νν;µ (L) = ResRµ+ν

Rν⊗Rµ(L).

The functor ◦ makes⊕

ν Rν−gmod into a monoidal category.The monoidal structure makes the Grothendieck groupK = K0(

⊕ν Rν−gmod) into a ring: [M][N] = [M ◦ N], graded by ν.

Grading shift makes it into a Z[q, q−1]-module: [M(1)] = q[M].The functor Res also endows K with a comultiplication:

∆(M) =∑

ν′+ν′′=ν

[Resνν′;ν′′(M)].



Proposition

[Resνν′,ν′′(M ◦ N)] =∑

µ=µ′−wt(M)+wt(N)

q〈µ,µ′〉[Reswt(M)−µ,µ M ◦ Resµ′,wt(N)−µ′ N]

The picture behind this is simple: you filter Resνν′,ν′′(M ◦ N) by the number ofstrands that had pass from the left factor to the right (or vice versa).

This formula almost shows that ∆ is an algebra homomorphism; there’s thatannoying power of q in the way.

Actually, K is a Hopf algebra in the category of root lattice graded vectorspaces with the braiding

σ(v⊗ w) = q〈wt(v),wt(w)〉w⊗ v.



We know another such Hopf algebra, UZq (n), generated over Z[q, q−1] by the

quantum divided powers F(n)i =

FiF(n−1)i

qn−1+qn−3+···+q1−n modulo the quantum Serrerelations

−aij+1∑i=0

(−1)iF(i)i FjF

(−aij+1−i)i

with the coproduct ∆(Fi) = Fi ⊗ 1 + 1⊗ Fi.

Theorem (Khovanov-Lauda)

K0(⊕

ν Rν−proj) ∼= UZq (n).


Simples over KLRA

In particular, the classes of the simple modules give a basis of Uq(n), as dothe classes of indecomposable projectives.

Theorem (Varagnolo-Vasserot)If A is a symmetric Cartan matrix, the field k has characteristic 0 and

Qij(u, v) = (u− v)#{i→j}(v− u)#{j→i}

then the classes of projectives (simples) give the (dual) canonical basis ofUq(n).

Note, if any of these conditions fail, there are counter-examples. In fact, for Anon-symmetric, the canonical basis doesn’t have positive structurecoefficients, which is obviously needed for the basis of projectives.


Canonical bases

A few words about canonical bases. The algebra Uq(n) hasa unique semi-linear automorphism satisfying Fi = Fi, q = q−1.a unique sesquilinear form satisfying

〈vw, u〉 = 〈v⊗ w,∆(u)〉 (Fi,Fj) = δij/(1− q−2).

These have categorical interpretations:

[M] = [HomR(M,R)] 〈[M], [N]〉 = dimq Hom(M,N)

Theorem (Lusztig)

Assume A symmetric. The canonical basis of UZq (n) is the unique basis B with

b = b,

multiplication has positive structure coefficients in this basis.

〈b, b′〉 ∈ δb,b′ + q−1Z[q−1] (quasi-orthogonality),


Canonical bases

It’s easy to work out that indecomposable projectives have a uniquegrading shift where P ∼= Hom(P,R).

It’s clear that the the projectives will have positive structure coefficientssince P ◦ P′ is a sum of projectives.

Quasi-orthogonality is essentially equivalent to the Morita equivalence ofR with a positively graded algebra A with A0 semi-simple.

This last item is by far the hardest (and the one that doesn’t work ifchar(k) 6= 0 or A is non-symmetric).

At the moment, there’s no purely algebraic argument I know, only a geometricone.


Canonical bases

Eν = ⊕i→jHom(Cνi ,Cνj) Gν =∏

i

GL(Cνi)

For a sequence i ∈ Γn, let

F`i = {Fi,1 ⊂ · · · ⊂ Fi,n ⊂ Cνi | dim(Fi,k/Fi,k−1) = δi,ik}

Xi = {(x•,F•) ∈ Eν × F`i | xe(Fi,k) ⊂ Fj,k−1}

Theorem (Varagnolo-Vasserot)

eiRnej ∼= HBM,Gν∗ (Xi ×Eν Xj), intertwining the product with convolution in

homology.

This is Morita equivalent to a positively graded algebra by the Decompositiontheorem, which is sort of Hodge theory on steroids.


Crystals

One of the interesting things about this theorem is that proving it doesn’treally require indexing the basis, and indexing the canonical basis of Uq(n) istricky.

Theorem (Kashiwara/Lusztig)

The set of canonical basis vectors in Uq(n) is endowed with a collection ofKashiwara operators fi, f ∗i such that fib (f ∗i b) is the “leading term” of bFi

(Fib). The maps fi and f ∗i are injective, and we let ei, e∗i be their (partiallydefined) inverse.

We call such operators together with some other data (in particular, the weightfunction on basis vectors) a bicrystal. We let B(∞) denote the crystal ofcanonical basis vectors in Uq(n).


Crystals

Theorem (Lauda-Vazirani)

The simple modules over Rν for all ν form a bicrystal isomorphic to B(−∞),where fiL is the unique simple quotient of L ◦ Rαi and f ∗i L the unique simplequotient of Rαi ◦ L.

This bijection is uniquely characterized by the fact that 1 7→ R0.

These crystal operators are simultaneously very simple, and devilishly tricky.It’s easy to prove that there’s a unique simple quotient, and very hard toactually construct the kernel or say what the dimension is.

Note that this statement doesn’t depend at all on the field k or the polynomialsQij. However, things like the dimension of the simples do!


Crystals

It turns out that B(∞) with Kashiwara operators shows up all over the place.

Theorem (Kashiwara-Saito)If A is symmetric (i.e. comes from a graph), there is a canonical bijectionbetween B(∞) and the components of the moduli space of nilpotentrepresentations of the corresponding preprojective algebra.

Theorem (Berenstein-Zelevinsky)

If A is finite type, the set B(∞) is in canonical bijection with the points of thegroup N of the semiring (Z; max,+).

Theorem (Kamnitzer/Lusztig)

If A is finite type, the set B(∞) is in canonical bijection with a collection ofpolytopes called Mirkovic-Vilonen polytopes defined by combinatorialconditions on the structure of faces.


MV polytopes

MV polytopes are particularly interesting. We take their definition in rank 2as an axiom:

type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2

a rectangle with sides parallel to simple roots

type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2

a hexagon with sides parallel to roots, at least one coloreddiagonal connects vertices.

a octagon with sides parallel to roots, two non-intersecting colored di-agonals connect vertices (+ε).

type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2

α12α1 + α2

2α1 + 2α2

α2


MV polytopes


type A1 × A1

α1 α2

2α1

0

4α2

2α1 + 4α2

2α1

4α2


type A2

α1 α2

α1 + α2

4α1

0

2α1 + 2α2

3α2

6α1 + 5α2

3α1

3α1 + 3α2

2α2



type B2

α1

α2 α1 + α2 2α1 + α2

2α10

2α1 + α2

α1 + α2

2α2

4α1 + 4α2α12α1 + α2

2α1 + 2α2

α2


MV polytopes

For other semi-simple Lie algebras of rank at least 3, we can define MVpolytopes inductively:

DefinitionA polytope whose faces are all parallel to faces of Weyl chambers in h∗ is aMirkovic -Vilonen polytope if each of its faces is an MV polytope for alower rank Lie algebra.

Of course, it is enough to check this for 2-dimensional faces.

Depending on how you look at it, the bijection to B(∞) is either dead easy, orcompletely unworkable. To apply fi, you “just” hold all the stuff “below” theαi-edge from the top constant, extend that edge αi-further, and then fill in toget an MV polytope in the only way possible.


MV polytopes

For other semi-simple Lie algebras of rank at least 3, we can define MVpolytopes inductively:

DefinitionA polytope whose faces are all parallel to faces of Weyl chambers in h∗ is aMirkovic -Vilonen polytope if each of its faces is an MV polytope for alower rank Lie algebra.

Of course, it is enough to check this for 2-dimensional faces.

Depending on how you look at it, the bijection to B(∞) is either dead easy, orcompletely unworkable. To apply fi, you “just” hold all the stuff “below” theαi-edge from the top constant, extend that edge αi-further, and then fill in toget an MV polytope in the only way possible.


MV polytopes for sl3










Crystals

So, MV polytopes and KLR simples give two different parameterizations ofB(∞); by abstract nonsense, there is a unique crystal isomorphism betweenthem.


Characters

One interesting question about KLR simples is their “character”

χL =∑

i

dimq eiL · w[i]

considered as a formal sum of words in the simple roots; this coincides withthe “q-shuffle expansion” of [L], thought of as an element of Uq(n).

Understanding these characters explicitly is a very interesting question;unfortunately, it’s a very hard one, much too hard for me.

DefinitionLet PL be the convex hull of the points ν ′ where Resνν′;ν−ν′ L 6= 0.

If you think of i as a path, this is the convex hull of all the paths in thecharacter.


Characters

One interesting question about KLR simples is their “character”

χL =∑

i

dimq eiL · w[i]

considered as a formal sum of words in the simple roots; this coincides withthe “q-shuffle expansion” of [L], thought of as an element of Uq(n).

Understanding these characters explicitly is a very interesting question;unfortunately, it’s a very hard one, much too hard for me.


If you think of i as a path, this is the convex hull of all the paths in thecharacter.


The polytope PL

For example, for sl3, the cases of R0 and Rαi are boring. For R2αi ,Rα1+α2 , weget

Now consider simples for Rα1+2α2 .

This is a sort of graphical expression of the Serre relation.


The polytope PL

For example, for sl3, the cases of R0 and Rαi are boring. For R2αi ,Rα1+α2 , weget

Now consider simples for Rα1+2α2 .

This is a sort of graphical expression of the Serre relation.


KLR polytopes for sl3










MV polytopes and PL

Theorem (Tingley-W.)The crystal isomorphism between KLRA simples and MV polytopes sendsL 7→ PL.

The proof is strikingly “high-level;” almost all you need is the crystalstructure.

The extra leverage that the algebra gives you is that you can “multiply”simples using the induction functor (though sometimes you will fail and notget a unique answer) and the compatibility of this multiplication with crystaloperations.

It seems likely that this is a purely combinatorial structure that can beformalized, which will make it easy to see how MV polytopes fall out thecrystal (+ε) structure alone.


MV polytopes for other types

Now, cool as this theorem is, the really exciting part is that KLRA makessense for all symmetrizable Kac-Moody algebras; the conventional MVpolytopes do not. Thus, we can use PL as the definition of MV polytopes forother types.


Obviously, we’re interested in how much of the finite type theory carries over:

Do these polytopes uniquely determine the crystal element?

Do they have a combinatorial characterization? That is, can we see onpurely combinatorial grounds what these representations “look like?”


MV polytopes for sl2Unfortunately, things don’t seem to work quite as well any more; for sl2, itseems our rule that MV polytopes are uniquely characterized by knowing oneside of them seems to fail.

α1 α2δ

α1 + δ α2 + δ2δ...

... ...

(2) (1) (1, 1)

Even worse, there are different simples with the same polytope!

But these edges are not “the same;” one of them is forced to make contactwith a word in the middle and the other isn’t.

Baumann, Dunlap, Kamnitzer and Tingley suggest a solution: edges parallelto δ must carry the extra data of a partition.



α1 α2δ

α1 + δ α2 + δ2δ...

... ...

(2) (1) (1, 1)






α1 α2δ

α1 + δ α2 + δ2δ...

... ...(2) (1) (1, 1)






In our context, we look for an idempotent (i.e. a path) which touches thatedge at top and bottom, but a minimal number of times in between; theattached partition is just the sizes of the gaps between touches.

Theorem (Tingley-W.)The assignment L→ PL enriched with this partition data is precisely thecrystal isomorphism to sl2 MV polytopes defined by Baumann, Dunlap,Kamnitzer and Tingley.



These are defined, as in finite type, in terms of non-intersecting diagonalsalong root directions.

(2, 1)(5, 2, 1)




(2, 1)(5, 2, 1)




(2, 1)(5, 2, 1)




(2, 1)(5, 2, 1)


MV polytopes for affine g

For a general affine Lie algebra, one has to attach a partition to each facetparallel to δ.

You should think of these as telling you the longest “leaps” you can take inthe δ-direction when you try to “walk” from the top of the facet to the bottom.

Theorem?The 2-dimensional faces of PL are polytopes for KLR representations of arank 2 algebra?

The two imaginary edges of an affine 2-face come from the 2 facets thatintersect it without containing it. So the edges should be labelled with thosepartitions. But they’re too long!


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅ (2)(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅ (2)(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2

2α1

3α0

2α0 + 2α1

α0

(2)∅ (2)(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2

2α1

3α0

2α0 + 2α1

α0

(2)∅ (2)(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅

(2)(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅

(2)

(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅

(2)

(1)


An sl3 example

α0α1α2

δ

α12α2

α1 + α2

δ

α0

2α1 + 2α2

2δ

2α02α1

α2

3α0

2α0 + 2α1

α0 + α2

3δ

α1

2α1 + 2α2

α22α1

α1 + α2

2α2

2α0

α0 + α23α0

α2 2α1

3α0

2α0 + 2α1

α0

(2)∅ (2)

(1)



Theorem (Tingley-W.)The polytopes PL for g of affine type are uniquely determined by the propertythat any 2-face is

a finite type rank 2 MV polytope if it’s real (not parallel to δ) or

the Minkowski sum of nδ with an affine rank 2 MV polytope if it’simaginary (parallel to δ; here n is the number of boxes in the partitionslabeling the facets containing that face).

CorollaryIf g has symmetric Cartan matrix, it coincides (after transposing partitions)with the affine MV polytopes defined using quiver varieties by Baumann,Kamnitzer and Tingley.

The transpose thing is silly; it’s the question of whether you should indexnilpotents by their Jordan type, or the type of the coarsest flag they preserve.



Theorem (Tingley-W.)The polytopes PL for g of affine type are uniquely determined by the propertythat any 2-face is

a finite type rank 2 MV polytope if it’s real (not parallel to δ) or

the Minkowski sum of nδ with an affine rank 2 MV polytope if it’simaginary (parallel to δ; here n is the number of boxes in the partitionslabeling the facets containing that face).

CorollaryIf g has symmetric Cartan matrix, it coincides (after transposing partitions)with the affine MV polytopes defined using quiver varieties by Baumann,Kamnitzer and Tingley.

The transpose thing is silly; it’s the question of whether you should indexnilpotents by their Jordan type, or the type of the coarsest flag they preserve.


Labeling

In general, the edges of our polytope are labeled with representations; fixingone edge e, a simple L of weight ν can be restricted to the subalgebra

Rα(e) ⊗ Re ⊗ Rν−ω(e).

· · · · · · · · ·

ω(e)wt(e)α(e)

This restriction will be of the form L′ � Le � L′′ for simples of the smalleralgebras. We associate Le to the edge e.

This representation will always be “semi-cuspidal;” that is, its character staysinside the polytope. If e is always parallel to a real root, there is only one ofthese; if parallel to an imaginary, there can be many more.


Labeling

In the finite type case, the convex paths from 0 to ν encode previously knowncombinatorial information: the Lusztig data.

Each of these corresponds to a convex order on roots, and the lengths of thevarious edges describe how to get the canonical basis vector in question froma PBW basis constructed from that order.

Theorem (Tingley-W.)The representation L is uniquely determined by PL and the representations Le;in fact, it can be reconstructed from any convex path along edges from 0 to νwith its labels.

Thus there exist Lusztig data for crystals in all types, with semi-cuspidalrepresentations replacing the lengths of edges.


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