The diagrammatic beauty of Rep(Uq

(1)

The diagrammatic beauty of Rep(U

q

(sl

n

)): Part II

Daniel Tubbenhauer

The categorified story

March 2014

= =

Daniel Tubbenhauer March 2014 1 / 37

(2)

1 What is categorification?

From the viewpoint of “natural” constructions From the viewpoint of topology

From the viewpoint of algebra

2 Thesl2-web algebra The algebra

Straightening helps again!

Categorifiedq-skew Howe duality

3 Connection to thesln-link-homology The Khovanov homology

Connection to the sl2-web algebra

The KL-R algebra and Khovanov homology

(3)

What is categorification?

Forced to reduce this presentation to one sentence, the author would choose:

Interesting integers are shadows of richer structures in categories.

The basic idea can be seen as follows. Take a“set-based”structureS and try to find a“category-based”structureC such thatS is just a shadow ofC.

Categorification, which can be seen as“remembering” or “inventing”information, comes with an “inverse” process calleddecategorification, which is more like

“forgetting” or “identifying”.

Note that decategorification should beeasy.

Daniel Tubbenhauer From the viewpoint of “natural” constructions March 2014 3 / 37

(4)

The underlying basic example

TakeC=FinVecK for a fixed fieldK, i.e. objects are finite dimensionalK-vector spacesV,V^′, . . . and morphisms areK-linear mapsf:V →V^′ between them.C categorifiesN: We can go back by taking thedimensiondimV ∈N.

Whatis the upshot? Note the following:

Much information is lost if we only considerN, i.e. we can only saythattwo objects are isomorphic (aka equal) instead ofhowthey are isomorphic. Thus,

n=n^′⇔V ∼=V^′.

A vector space can carryadditional structure as for example inner products.

We have the power oflinear algebrabetweenV andV^′, i.e. homK(V,W).

(5)

Never forget the original structure

Thestructure ofNisreflectedon a “higher” level!

The product and coproduct⊕and the monoidal structure ⊗K categorify addition and multiplication, i.e. dim(V ⊕V^′) = dimV + dimV^′ and dim(V⊗KV^′) = dimV ·dimV^′.

The zero object 0 and the identity of⊗K categorifythe identities, i.e V⊕0≃V andV⊗KK ≃V.

We haveV ֒→W iff dimV ≤dimW andV ։W iff dimV ≥dimW, i.e.

injections and surjectionscategorifythe order relation.

One can write down thecategorifiedstatements of other properties as “Addition and multiplication are associative and commutative”, “Multiplication distributes over addition” or “Addition and multiplication preserve order”.

Daniel Tubbenhauer From the viewpoint of “natural” constructions March 2014 5 / 37

(6)

Integer based invariants

A moretopologicalflavoured example goes back to Riemann (1857), Betti (1871) and Poincar´e (1895): TheBetti numbersbk(X) andEuler characteristicχ(X) of a reasonable topological spaceX. Noether, Hopf and Alexandroff (1925)

“categorified”these invariants as follows.

If we liftm,n∈Nto the twoK-vector spacesV,W with dimensions dimV =m,dimW =n, then the differencem−nlifts to the complex

0 //V ^d //W //0,

for any linear mapd andV in even homology degree. As before, some of the basic properties of the integersZcan be lifted to the categoryKomb(C).

Conclusion(Noether): Thehomology groupsHk(X,Z) categorifybk(X) andchain complexes(C(X),c_∗) categorifyχ(X).

(7)

Well-known upshots

We note the following observations.

The spaceHk(X,Z) is a graded abelian group and more information of the spaceX is encoded. Again, homomorphisms between the groups tellhow some groups are related.

Singular homology works for all topological spaces and the homological Euler characteristic can be defined for a huge class of spaces.

The homology extends to afunctorand provides information about continuous maps as well.

Moresophisticated constructionslike multiplication in cohomology provide even more information.

Although it isnotthe main point: TheHk(X,Z) are better invariants.

Daniel Tubbenhauer From the viewpoint of topology March 2014 7 / 37

(8)

Categorified symmetries

Another viewpoint comes fromrepresentation theory. LetAbe some algebra, M be aA-module andC be a suitable category.

“Usual” /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o //“Higher”

a7→fa∈End(M) /o /o /o /o /o /o /o /o /o /o /o //a7→ Fa∈End(C)

(fa1·fa2)(m) =fa1a2(m) /o /o /o /o /o /o /o //(Fa1◦ Fa2) ^X_ϕ∼=Fa1a2

X ϕ

A(weak) categorificationof theA-moduleM should be though of a categorical action ofAon a suitable categoryC with an isomorphismψsuch that

K0(C)⊗A ^[F^a^] //

ψ

K0(C)⊗A

ψ

M ^·a //M.

(9)

There is no direct minus

We haveseveralupshots again.

The natural transformations between functors give informationinvisiblein

“classical” representation theory. This gives a hint that we can go even

“higher”, e.g. actions of 2-categories on 2-categories.

IfCis suitable, e.g. module categories over an algebra, then its

indecomposable objectsX gives a basis [X] ofM withpositivity properties.

In particular, considerAas aA-module. Then [X] gives a basis ofAwith positivestructure coefficientsc_k^ij via

Xai⊗Xaj ∼=M

k

X^c

ij

akk aiaj=X

k

c_k^ijak, c_k^ij ∈N.

Daniel Tubbenhauer From the viewpoint of algebra March 2014 9 / 37

(10)

Natural transformations between sl

₂

-webs

Recall that we are interested in theintertwinerofRep(Uq(sl2)) or in pictures u: ¯Q(q)→Q¯²⊗Q¯²⊗Q¯²⊗Q¯²=

Howcan we describe higher structure between these intertwiners? That is, what can we say about

hom ,

!

?

Note that the intertwiners “are”1-dimensional. Thus, the natural transformations between them should be2-dimensional.

Moreover, we can again“restrict”toInv_U_q(sl₂)(N

2nQ¯²). Recall that the invariant tensors form a ¯Q(q)-vector space with basis Arc(n), that is all sl₂-arc diagrams with 2nboundary points.

(11)

sl

₂

-foams

Asl₂-pre-foamis a cobordism between twosl₂-webs. Composition consists of placing onesl₂-pre-foam ontopof the other. The following are called thesaddle up and downrespectively.

They havedotsthat can movefreelyabout the facet on which they belong. Define theq−degreeof asl₂-foamF withd dots andbboundary components as

qdeg(F) =−χ(F) + 2d+b 2.

Asl₂-foamis a formal ¯Q-linear combination of isotopy classes ofsl₂-pre-foams modulo the following (degree preserving!) relations.

Daniel Tubbenhauer The algebra March 2014 11 / 37

(12)

The sl

₂

-foam relations ℓ = (2 D , NC , S )

= 0 (2D)

= + (NC)

= 0, = 1 (S)

The relationsℓ= (2D,NC,S)sufficeto evaluatesl₂-foam without boundary!

= +

(13)

The sl

₂

-foam category

Foam2is theZ-graded,2-category ofsl2-foamsconsisting of:

Theobjectsare sequences of points in the interval [0,1].

The1-cellsare formal direct sums ofZ-gradedsl2-webs with boundary corresponding to the sequences of points for the source and target.

The2-cellsare formal matrices of ¯Q-linear combinations of degree-zero dottedsl₂-foams modulo isotopy andsl₂-foam relations.

Verticalcomposition◦v is stacking on top of each other andhorizontal composition◦h is stacking next to each other. We write

homFoam2(u,v) = hom(u,v).

Thesl2-foam homologyof a closedsl2-webw:∅ → ∅is defined by F(w) = homFoam2(∅,w) = hom(∅,w).

F(w) is aZ-graded, ¯Q-vector space.

(14)

Exempli gratia

Example

A saddles are 2-morphisms

∈hom

,

∈hom

,

Vertical composition gives anon-trivial“natural transformation” in hom( , )!

◦v = 6=

(15)

The sl

₂

-web algebra

Definition(Khovanov 2002)

Thesl2-web algebraH2(n) is defined by H2(n) = M

u,v∈Arc(n) uHv,

with

uHv =F(u^∗v){n}, i.e. allsl₂-foams:∅ →u^∗v.

Multiplicationis defined by compositionF(u^∗v)∼= hom(u,v).

Example

Since ^∗= , we haveH2(1) =h , iQ¯ ∼= ¯Q[X]/X²with multiplication

·

0

(16)

It’s Frobenius!

There is a trace form tr :H2(n)→Q¯ given by closing asl2-foamfu with1u. The trace isnon-degeneratedandsymmetric, i.e. tr(fg) = tr(gf):

f

g

=

g

f

Theorem(Khovanov 2002)

The algebraH2(n) is a graded, finite dimensional, symmetric Frobenius algebra.

(17)

Higher representation theory

Moreover, we define

W₍₂^ℓ₎ = M

~k∈Λ(2ℓ,2ℓ)3

W2(~k)∼= M

~k∈Λ(2ℓ,2ℓ)3

Inv_U_q₍sl₂)(~k)

on thelevelofsl2-webs and on the levelofsl2-foams we define (below the technicaldefinition, butthink: Take the module category overHn)

W₍₂^(p)^ℓ₎= M

~k∈Λ(2ℓ,2ℓ)2

H2(~k)- (p)Mod_gr.

With this constructions we obtain thecategorificationresult.

Theorem(Khovanov 2002, Brundan-Stroppel 2008)

K0(W(2^ℓ))⊗Z[q,q⁻¹]Q(q)¯ ∼=W₍₂^∗^ℓ₎ andK₀^⊕(W₍₂^p^ℓ₎)⊗Z[q,q⁻¹]Q(q)¯ ∼=W₍₂^ℓ₎. This is nice, but how togeneralizeton>2?

(18)

Recall: Rigid sl

₂

-spider

Recall that therigidversion ofSp(Uq(sl2)) consists of

k1 k2 k3 k4

k1 k2

k1±k k2∓k k

k=0,1,2

2 0

1 1

0 2

=

2 0

0 2

with labelski ∈ {0,1,2}. Weonlypicture edges labeled 1 in black and edges labeled 2 as a dotted leashes. Moreover, we picture a “left-plus-ladder” with an arrow to theleftandvice versafor a “right-plus-ladder”.

The advantage of this was that it was“easy” to generalize ton>2 and we were able to see anU˙q(sld)-actionon theUq(sl2)-webs!

(19)

Rigid sl

₂

-foams: Sloppy version

Instead of giving theformaldefinition of the rigidsl2-foam categoryFoam2let me just give someexamples.

Therigidversions of thesl₂-foams are locally generated by

where facet get the numbers of their incident edges. Facets labeled 0 are removed, facets labeled 1 really exists and facet labeled 2 are pictured using leashes as boundary (but they exist). Thus, these will besingularsurfaces!

The singular surfaces above are calledidentitiesandsingularsaddles.

Facets with label 1 are allowed to carry dots. Dots move freely on a facet but arenotallowed to cross singular lines.

There are some relations and the 2-category is graded by a slight rearrangement of thegeometrical Euler characteristic.

Daniel Tubbenhauer Straightening helps again! March 2014 19 / 37

(20)

Exempli gratia

Rigid examples

∈hom

,

∈hom

,

∈hom ,

!

= −

Think: Leash-facestake careof sign-issues coming from the fact that Λ⁰Q¯²and its dual Λ²Q¯²areonlyisomorphic. Moreover:“Easy”to generalize, since one needs singular surfaces already for non-rigidsl3-foams.

(21)

The overview

U(sld)

How it should be!

“Higher”q-skew Howe U(sl_d) acts

//

/o

K₀^⊕

Hn(~k)- (p)Mod_gr

K₀^⊕

U˙q(sld) q-skew Howe U˙q(sl_d) acts

//

/o

/o Wn(~k)

This is how it should be: There is anU˙q(sld)-actionon thesln-web spaces (for us it was mostly the casen= 2). Moreover,suitablemodule categories over the sln-web algebrasHn(~k)categorifythese spaces.

On the left side: There isKhovanov-Lauda’s categorificationof ˙Uq(sld) denoted byU(sld) (which Ibrieflyrecall on the next slides).

Conclusion: Thereshouldbe a 2-action of U(sld) on the top right!

Daniel Tubbenhauer Categorifiedq-skew Howe duality March 2014 21 / 37

(22)

Khovanov-Lauda’s 2-category U (sl

d

)

Idea(Khovanov-Lauda)

The algebra ˙Uq(sld) has a basis with surprisinglynice behaviour, e.g. positive structure coefficients. Thus, thereshouldbe a categorification of ˙Uq(sld) pulling the strings from the background!

Definition(Khovanov-Lauda 2008)

The 2-categoryU(sld) is defined by (everything suitablyZ-graded and ¯Q-linear):

The objects inU(sld) are the weights~k ∈Z^d−1.

The 1-morphisms are finite formal sums of the formEi1~k{t} andFi1~k{t}.

2-cells are graded, ¯Q-vector spaces generated by compositions of diagrams (additional ones with reversed arrows) as illustrated below plus relations.

i

~k

~k−αi

i

~k

~k−αi

i j

~k

i

~k

i

~k

(23)

sl

₂

-foamation (works for all n > 1!)

We define a 2-functor

Ψ : U(sld)→ M

~k∈Λ(2ℓ,2ℓ)2

H2(~k)- (p)Mod_gr =W₍₂^(p)ℓ)

calledsl2-foamation, in the following way.

On objects:The functor is defined by sending ansld-weight~k = (~k1, . . . , ~kd−1) to an object Ψ(λ) ofW₍₂^(p)ℓ) by

Ψ(λ) =S, S = (a1, . . . ,aℓ),ai ∈ {0,1,2}, λi =ai+1−ai,

ℓ

X

i=1

ai = 2ℓ.

On morphisms:The functor on morphisms is by glueing the ladder webs from before on top of thesl2-webs inW₍₂^ℓ₎=L

~k∈Λ(2ℓ,2ℓ)2W2(~k).

(24)

sl

₂

-foamation (Part 2)

On 2-cells:We define

i,~k

7→

~ki

~ki+1

i,~k

7→

~ki

~ki+1

i,i,~k

7→

~ki

~ki+1

i,i+1,~k

7→

~k_i

~ki+1

~ki+2

i+1,i,~k

7→

~k_i

~ki+1

~ki+2

And some others (that are not important today).

(25)

Everything fits

Theorem

The 2-functor Ψ :U(sld)→ W₍₂^(p)ℓ) categorifiesq-skew Howe duality.

Example without labels (One has to check well-definedness!)

↓

=

↓ ↓

(26)

Khovanov’s categorification of the Jones polynomial

Recall the rules for the Jones polynomial.

h∅i= 1 (normalization).

h i=h i −qh i (recursion step 1).

h ∐LDi= [2]· hLDi(recursion step 2).

[2]J(LD) = (−1)ⁿ⁻qⁿ⁺⁻²ⁿ⁻hLDi(Re-normalization).

Definition/Theorem(Khovanov 1999)

LetLD be a diagram of an oriented link. Denote byA= ¯Q[X]/X²the dual numbers with qdeg(1) = 1 and qdeg(X) =−1 - this is a Frobenius algebra with a given comultiplication ∆. We assign to it a chain complexJLDKofZ-graded Q¯-vector spaces using thecategorified rules:

J∅K= 0→Q¯ →0 (normalization).

J K= Γ

0→J K→^d J K→0

withd=m,∆ (recursion step 1).

J ∐LDK=A⊗Q¯JLDK(recursion step 2).

Kh(LD) =JLDK[−n−]{n+−2n−}(Re-normalization).

ThenKh(·) is aninvariantof oriented links whose graded Euler characteristic givesχq(Kh(LD)) = [2]J(LD).

(27)

This is better than the Jones polynomial

Khovanov’s construction can beextended to a categorification of the HOMFLY-PT polynomial.

It isfunctorial(in this formulation only up to a sign).

Kronheimer and Mrowka showed that Khovanov homologydetectsthe unknot. This is still anopenquestion for the Jones polynomial.

Rasmussen obtained from the homology an invariant that“knows”the slice genus and used it to give acombinatorial proofof the Milnor conjecture.

Rasmussen also gives a way tocombinatorial construct exoticR⁴. The categorification is not unique, e.g. the so-called “oddKhovanov homology”differs over ¯Q.

Before I forget: It is astrictly stronger invariant.

Historyrepeatsitself: After Khovanov lots of other homologies of

“Khovanov-type” were discovered. So we need to understand this better, e.g. how to extend this totangles?

Daniel Tubbenhauer The Khovanov homology March 2014 27 / 37

(28)

Resolutions are H

₂

( m ) − H

₂

( n )-bimodules

LetT_D^m,n be a oriented diagram of a tangle with numbered crossingsc1, . . . ,cr

and 2mbottom and 2ntop boundary points. AresolutionR(T_D^m,n)k ofT_D^m,n is a local replacement of theci by either or .

Definition

Define aH2(m)−H2(n)-bimodule for a=R(T_D^m,n)k by F(a) = M

u∈Arc(n),v∈Arc(m)

F(u^∗av),

that isallsl2-foams∅ →u^∗av forall suitableu,v. This is an

H2(m)−H2(n)-bimodule where the elements ofH2(m)act by stackingfrom the bottom and the elements ofH2(n)act by stackingfrom the top.

Example

H2(1)−H2(1)-bimodules: F( ) = hom(∅, ) and F( ) = hom(∅, ).

(29)

How to build a chain complex Kh( T

^m,n

D

)

For an oriented diagramT_D withr =n++n− crossings and resolutionsR(T_D^m,n)k

ordered intor+ 1-columns in a suitable way - for two consecutive columns the local differenceis → or → .

Fori= 0, . . . ,nthei−n₋ chain module is the formal direct sum of all H2(m)−H2(n)-bimodules for the resolutions of columni.

Between resolutions of columni andi+ 1 the morphisms should besaddles between the resolutions. These areH2(m)−H2(n)-bimodulehomomorphisms.

Extraformal signsto make everything well-defined - skipped today.

Shift everything suitable and obtainKh(T_D^m,n) - acomplex of H2(m)−H2(n)-bimodules.

Theorem(Khovanov 2002)

The complexKh(T_D^m,n) is a functorial invariant of oriented tangles.

Daniel Tubbenhauer Connection to thesl2-web algebra March 2014 29 / 37

(30)

Exempli gratia - Khovanov homology using sl

₂

-foams

u ww v

}

~:

❊❊

""

❊❊

●●

##

●●

✇✇

;;

✇✇

−

②②

<<

②②

F( ) ^d¹ //F( )⊕ F( ) ^d² //F( ) This is aH2(1)−H2(1)-bimodule!

(31)

Shouldn’t ˙ U

⁻_q

(sl

d

) be sufficient?

We gave a method to obtain theUq(sln)-link polynomials usingU˙q(sld)-highest weight representation theory because we could restrict toF’s: ˙U⁻_q(sld)suffices.

Moreover, thesln-foamationconnectsthesln-web algebrasHn(Λ) with Khovanov-Lauda’s categorificationU(sld).

Moreover, which I explain in a second, there is a (easier to work with) “version” of U(sld), called theKhovanov-Lauda and Rouquier (KL-R) algebraRd, and a cyclotomic quotientRΛ ,calledcyclotomicKL-R algebra, whichcategorifyU_q⁻(sld) and its highest weight representationVΛrespectively.

This gives two natural questions:

On the level ofsln-link polynomials onlyF’s suffice. Shouldn’t the“same”

hold for thesln-link homologies?

If so, how can we use the cyclotomic KL-R algebra to“explain”thesln-link homologies as instances ofU_q⁻(sld)-highest weight representation theory.

Daniel Tubbenhauer The KL-R algebra and Khovanov homology March 2014 31 / 37

(32)

The KL-R algebra

Definition/Theorem(Khovanov-Lauda, Rouquier 2008/2009)

LetRd be acertaindirect sum of subalgebras of homU(sl_d)(Fi1~k{t},Fj1~k^′{t}).

Thusonly downwardspointing arrows - akaonlyF’s. That is, working withRd

enables us to ignore orientations and consider only diagrams of the form

or or or

The KL-R algebra has the structure of aZ-graded, ¯Q-algebra. We have (note that this works for moregeneralg)

U˙⁻_q(sld)∼=K₀^⊕(Rd)⊗_Z[q,q⁻¹_]Q¯(q).

NOT allowed: But = 0 is the Nil-Hecke relation

(33)

The cyclotomic quotient

Definition(Khovanov-Lauda, Rouquier 2008/2009)

Fix a dominantsld-weight Λ. Thecyclotomic KL-R algebraRΛis the subquotient ofU(sld) defined by the subalgebra ofonly downward (onlyF’s!)pointing arrows and rightmost region labeled Λ modulo the so-calledcyclotomic relation

i_m i3 i2 i1

~km

Λ

Theorem(Brundan-Kleshchev, Lauda-Vazirani, Webster, Kang-Kashiwara,...>2008)

LetVΛbe the ˙Uq(sld)-module of highest weight Λ. We have VΛ∼=K₀^⊕(Rd)⊗_Z[q,q⁻¹_]Q(q)¯ as ˙Uq(sld)-modules (note that this works for more generalg).

(34)

Recall: Only F ’s suffices!

2 1 1 1 1 0 0 0 0 0 0 0

0 1 2 2 1 1 0 0 0 0 0

0 0 0 0 0 1 1 1 1 0 0

0 0 0 0 0 0 0 1 0 0 1 2

2 2 1 0 1 1 1 1 1 2 1 1 0

0 0 0 1 1 1 1 1 1 1 2 2 2

F1

F2 F3 F₄

F5

F2 F3 F4

F4

F5

F3

F4

T1

T2

Hopf:

F₄⁽²⁾F4F3F5F4T2T1F4F3F2F5F4F3F2F1F₄⁽²⁾F₃⁽²⁾F₂⁽²⁾v220000=FtT2T1Fbv220000

(35)

Exempli gratia (The Hopf link - part two)

The Hopf link example from before will give a complex FtF4F3F2F3Fbvh{5}

Ψ( )˜❚❚❚ ^:^F²^F³^→F³^F²

❚

))

❚❚

❚❚ FtF3F4F2F3Fbvh{4} L

Ψ( )˜ ^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

Ψ( )˜ ^:^F²^F³^→F²^F³

❚❚

))

❚❚

FtF3F4F2F3Fbvh{6}

FtF4F3F2F3Fbvh{5}

−Ψ( )˜ ^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

that, up to some degree conventions,agreeswith thesl2-link homology ofHopf, because the “are”the saddles.

Observation - a more “down to earth” point of view

One can use the Hu-Mathas basis for the cyclotomic KL-R algebra to write down a basis for each of thesl2-web algebra modules. The are homomorphisms:

Calculatingthe homology reduces to linear algebra because we only need to track the image of the basis elements!

(36)

The sl

_n

-homologies using sl

_d

-symmetries

Let ussummarizethe connection between sln-homologies and the higherq-skew Howe duality.

Khovanov, Khovanov-Rozansky and others: Thesln-link homology can be obtainedusing thesln-web algebras.

Only “F’s”: Thesln-foamsarepart of the (Karoubian) of some KL-R algebra.

Conclusion: Thesln-homologies areinstances of highestUq(sld)-weight representation theory!

IfLD is a link diagram, then its homology is obtained by“jumping via higher F’s” from a highestUq(sld)-objectvhto a lowestUq(sld)-objectvl!

Missing: Connection to Webster’s categorification of the RT-polynomials!

Missing: Is the module category of the cyclotomic KL-R algebra braided?

Missing: Details about coloredsln-homologies have to be worked out!

(37)

There is stillmuchto do...

(38)

Thanks for your attention!