F F F F F F v { 5 } ) ) ❚❚❚❚ ❥❥❥❥ ˜Γ() ˜Γ() ❚❚❚❚ ❥❥❥❥ 5 5 F F F F F F v { 4 } F F F F F F v { 6 } ) ) ❚❚❚❚ ❥❥❥❥ ˜Γ() ˜Γ() ❚❚❚❚ ❥❥❥❥ 5 5 F F F F F F v { 5 } April2014 DanielTubbenhauer n q m sl -linkhomologiesusing˙ U ( sl )-highestweighttheory

(1)

sl

_n

-link homologies using ˙ U

_q

( sl

_m

)-highest weight theory

Daniel Tubbenhauer

Themis not a typo!

April 2014

FtF4F3F2F3Fbvh{5}

˜Γ( )^:^F²^F³^→F³^F²

❚❚

))

❚❚

❚❚ FtF3F4F2F3Fbvh{4} L

Γ(˜ )^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

Γ(˜❚❚❚ )^:^F²^F³^→F²^F³

❚

))

❚❚

FtF3F4F2F3Fbvh{6}

FtF4F3F2F3Fbvh{5}

−˜Γ( )^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

Daniel Tubbenhauer April 2014 1 / 36

(2)

1 A diagrammatic presentation sl2-webs

Connection toRep(Uq(sl2))

How can one prove the graphical representation?

2 Connection to thesln-link polynomials The Jones polynomial

Links as F’s

It’s Reshetikhin-Turaev’s coloredsln-link polynomial

3 Its categorification!

“Higher” representation theory Categorifiedq-skew Howe duality The Khovanov homology

Daniel Tubbenhauer April 2014 2 / 36

(3)

An old story: Rumer, Teller and Weyl (1932)

Daniel Tubbenhauer sl2-webs April 2014 3 / 36

(4)

Think topologically but write algebraically

Think:

Write:

Advantage: Decomposition `a la Morse intobasic pieces.

Ignore dotted red lines: We used them to solvesign issues(functoriality of Khovanov homology for example). Theyencodethe fact for quantum groups the antipode (dual representations) comes with asign.

(5)

The (rigid) sl

₂

-webs - the objects

Definition - Part I

The(rigid) sl2-web spiderSp(Uq(sl2))is the monoidal, ¯Q(q)-linear 1-category consisting of the following.

Theobjectsare ordered compositions~k ofd∈Nwith only 0,1,2 as entries.

Stated otherwise: The objects arem-tuples

~k= (k1, . . . ,km) such that Xm

j=1

kj =d,kj ∈ {0,1,2}.

Example:

d= 10 : ~k1= (2,2,0,1,2,0,1,2,0,0) and ~k2= (2,2,2,2,2,0,0,0,0,0) We call an object ahighest weightobject ifkj ∈ {0,2} andkj≥kj+1, e.g.~k2.

(6)

The (rigid) sl

₂

-webs - the generating 1-morphisms

Definition - Part II

Thegenerating 1-morphismsarew:~k →~k^′ areedge-labeled graphssuch that The vertices areeither1-valent and part of the bottom (where we place~k) or top (where we place~k^′) boundary or3-valent. The labels are from the set {0,1,2}and edges that end in a 1-valent vertex kj should have labelkj. Wedo notpicture edges labeled 0 and picture the edges labeled 2dotted.

The generators are eitheridentities

k1 k2 k3 k4

k1 k2 k3 k4 For example:

1 0 1 2

Orladders

k1 k2

k1±k k2∓k k

k=0,1,2

For example:

0 2

2 0

2 or

2 0

1 1

1

(7)

The (rigid) sl

₂

-webs - and all the rest

Definition - Part III

The ¯Q(q)-linear composition◦ isstacking(see below).

The monoidal structure⊗is given byjuxtaposition, e.g.

1 0 1 2

1 0 1 2 ⊗

1 2

2 1

=

1 0 1 2

1 2

2 1

All 1-morphisms should begeneratedby identities and ladders by◦ and⊗.

Relations are thecircle removalsandisotopies, e.g. ([2] =q+q⁻¹)

2 0

1 1

2 0

= [2]·

2 0

and

1 2

2 ◦ 1

2 1

1 2

=

1 2

2 1

1 2

=

1 2

=

1 2

1 ◦ 2

1 2

(8)

The quantum algebra U

_q

( sl

_m

)

Definition

Form∈N_>1thequantum special linear algebraUq(slm) is the associative, unital Q(q)-algebra¯ generated byK_i^±1 andEi andFi, fori= 1, . . . ,m−1 subject the followingrelations.

KiKj =KjKi, KiK_i⁻¹=K_i⁻¹Ki = 1, EiFj−FjEi=δi,j

KiK_i+1⁻¹−K_i⁻¹Ki+1

q−q⁻¹ , KiEj =q^(ǫⁱ^,α^j⁾EjKi,

KiFj =q^−(ǫⁱ^,α^j⁾FjKi,

E_i²Ej−[2]EiEjEi+EjE_i²= 0, if |i−j|= 1, EiEj−EjEi = 0, else,

F_i²Fj−[2]FiFjFi+FjF_i²= 0, if |i−j|= 1, FiFj−FjFi= 0, else.

Daniel Tubbenhauer Connection toRep(Uq(sl2)) April 2014 8 / 36

(9)

The idempotented version

Definition(Beilinson-Lusztig-MacPherson)

For each~k ∈Z^m−1 adjoin anidempotent1~k (think: projection to the~k-weight space!) toUq(slm) and add some relations, e.g.

1~k1~k^′ =δ~k,~k^′1~k and Fi1~k = 1~k−αFi and K±i1~k =q^±^~^kⁱ1~k (noK^′s anymore!).

Theidempotented quantum special linear algebrais defined by U˙q(slm) = M

~k,~k^′∈Z^m−1

1~kUq(slm)1~k^′.

Itslower part ˙U⁻_q(slm)is the subalgebra ofonlyF’s.

An important fact: The ˙Uq(slm) has the“same”representation theory asUq(slm) and ˙U⁻_q(slm) suffices to describe it.

(10)

The category Rep(U

q

( sl

₂

))

Definition

Therepresentation categoryRep(Uq(sl2))is the monoidal, ¯Q(q)-linear 1-category consisting of:

Theobjectsare finite tensor products of theUq(sl2)-representations Λ^kQ¯². Denote them by~k = (k1, . . . ,km) withki ∈ {0,1,2}.

The1-cellsw:~k→~k^′ areUq(sl2)-intertwiners.

Composition◦of 1-cells iscomposition of intertwinersand⊗is theordered tensor product.

It is worth noting that Λ⁰Q¯²= ¯Qis the trivialUq(sl2)-representation, Λ²Q¯²∼= ¯Q its dual and Λ¹Q¯²= ¯Q² is the (self-dual)Uq(sl2)-vector representation.

Example:~k = (2,0),~k= (1,1) and ¯Q²=hx+1,x−1i. Then

cup : ¯Q∼= Λ²Q¯²⊗Q¯ →Λ¹Q¯²⊗Λ¹Q¯²∼= ¯Q²⊗Q¯²,17→x+1⊗x−1−q⁻¹·x−1⊗x+1

forms a basis ofMor(~k, ~k^′).

(11)

Intertwiner are pictures

Theorem(Kuperberg 1997, n > 3: Cautis-Kamnitzer-Morrison 2012)

The 1-categoriesRep(Uq(sl2)) andSp(Uq(sl2)) areequivalent.

Example: cup = cup, i.e.

cup : Λ²Q¯²⊗Q¯ →Λ¹Q¯²⊗Λ¹Q¯² 7→

2 0

1 1

1

Question

How can one prove such a statement?

Finding the generators forRep(Uq(sl2)) isdoable, but...

Finding acompleteset of relations isvery hard!

(12)

An instance of q -skew Howe duality

The commuting actions of ˙Uq(slm) and ˙Uq(sl2) on M

a1+···+am=N

(Λ^a¹Q¯²⊗ · · · ⊗Λ^a^mQ¯²)∼= Λ^N( ¯Q^m⊗Q¯²)∼= M

a1+a2=N

(Λ^a¹Q¯^m⊗Λ^a²Q¯^m)

introducea ˙Uq(slm)-action on the left side and a ˙Uq(sl2)-action on the right side.

The left and right side are ˙Uq(slm)- and ˙Uq(sl2)-weight spaceswith weights

~kU˙q(sl_m) = (a1−a2, . . . ,am−1−am) and ~kU˙q(sl₂) = (a1−a2) respectively.

Here the Λ^kQ¯^l_qare irreducible ˙Uq(sll)-representations (l∈ {2,m}).

Daniel Tubbenhauer How can one prove the graphical representation? April 2014 12 / 36

(13)

Graphical quantum skew Howe duality

Theorem

There is anU˙q(slm)-actiononSp(Uq(sl2))^m (objects of lengthm)!

1~k 7→

... ...

k1 ki−1 ki ki+1 ki+2 km

Ei1~k, Fi1~k 7→

... ...

k1 ki−1 k_i ki+1 ki+2 k_m k1 ki−1 ki±1 ki+1∓1 ki+2 km

That is, we stack these pictures ontopof a givensl2-web.

Thus,Sp(Uq(sl2))^mis a U˙q(slm)-module andnot just aUq(sl2)-module.

(14)

Graphical quantum skew Howe duality - even better

Theorem

The ˙Uq(slm)-submodule W2((2^ℓ)) = M

~k∈Λ(m,2ℓ)

W2(~k) = M

~k∈Λ(m,2ℓ)

Mor_Sp(U_q₍sl₂))((2^ℓ), ~k),

called thesl2-web space, is a ˙Uq(slm)-module of highest weight (2^ℓ). Thus, it is generated by ˙U⁻_q(slm) (akaF’s suffice).

↑

U˙⁻_q(slm)−action

↓

Uq(sl2)−web →

←2 2 0 0

F2

2 1 1 0

F3

2 1 0 1

F1

1 2 0 1

F2

1 1 1 1

Λ²Q¯²⊗Λ²Q¯²⊗Λ⁰Q¯²⊗Λ⁰Q¯²

↑F2

Λ²Q¯²⊗Λ¹Q¯²⊗Λ¹Q¯²⊗Λ⁰Q¯²

↑F3

Λ²Q¯²⊗Λ¹Q¯²⊗Λ⁰Q¯²⊗Λ¹Q¯²

↑F1

Λ¹Q¯²⊗Λ²Q¯²⊗Λ⁰Q¯²⊗Λ¹Q¯²

↑F2

Λ¹Q¯²⊗Λ¹Q¯²⊗Λ¹Q¯²⊗Λ¹Q¯²

(15)

An instance of ˙ U

_q

( sl

_m

)-highest weight theory

What is theupshotof this?

“Explains”theUq(sl2)-intertwiner as instances of the (well developed) U˙q(slm)-highest weight theory.

The action of theF’s isexplicit and inductive- a powerful tool to prove statements.

Allthe relations follow from the well-known ones from ˙Uq(slm), e.g.

E1F1v20−F1E1v20

| {z }

=0

=K1K₂⁻¹−K₁⁻¹K2

q−q⁻¹

| {z }

=[2]120in ˙Uq(sl_m)

v20⇒

2 0

1 1

2 0

F1

E1

= [2]·

2 0

120

Even better: ˙U⁻_q(slm)sufficesfor everything!

(16)

The famous Jones polynomial

LetLD be a diagram of an oriented link. Set [2] =q+q⁻¹ and n+= number of crossings n₋= number of crossings

Definition/Theorem(Jones 1984, Kauffman 1987)

Thebracket polynomialof the diagramLD (without orientations) is a polynomial hLDi ∈Z[q,q⁻¹] given by the following rules.

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).

The polynomialJ(·)∈Z[q,q⁻¹] is an invariantof oriented links.

Daniel Tubbenhauer The Jones polynomial April 2014 16 / 36

(17)

Exempli gratia

* +

:

■■

▲▲

▲ rr rr r

✉✉

[2]² −2q·[2] +q²·[2]²

Thus,J(Hopf) =q⁵+q, i.e the Hopf link isnot trivial!

Daniel Tubbenhauer The Jones polynomial April 2014 17 / 36

(18)

Crossings measure the difference between F

_i

F

_i+1

and F

_i+1

F

_i

Define anUq(sl2)-intertwiner called positive crossingT₁⁺as follows.

T₁⁺

=

1 1

111

0-resolution

−q·

1 1

2 0

1 1

E1

F1

1-resolution

= 111v11−q·F1E1v11.

Wait: It is a ˙Uq(slm)-highest weight module:NoE’sare needed!

T₁⁺

=

1 1 0

1 0 1

0 1 1

F₂ F1

0-resolution

−q·

1 1 0

0 2 0

0 1 1

F2

F₁

1-resolution

= F1F2v110−q·F2F1v110.

Exercise: Do .

Daniel Tubbenhauer Links asF’s April 2014 18 / 36

(19)

U ˙

⁻_q

( sl

_m

) knows link diagrams

Using theseT_k⁺andT_k⁻ together with theF’s we can write link diagrams as

↑

↓

Uq(sl2)−“web”→

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

T₁⁺ T₂⁺

F4

F5

F3

F4

qH(Hopf) =F₄⁽²⁾F4F3F5F4T₂⁺T₁⁺F4F3F2F5F4F3F2F1F₄⁽²⁾F₃⁽²⁾F₂⁽²⁾v220000.

(20)

Jumping from a highest to a lowest weight

↑

↓

Uq(sln)−web →

2 ←

1 1 1 1 0 0 0 0 0 0 0 0

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

0 0 0 0 0 0 1 0 1 1 1 0 0

0 0 0 0 0 0 0 1 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

F1

F2

F3

F₂

F4

F5

F3

F₄

Resolutions are strings ofF’s jumping from a highest to a lowest ˙Uq(slm)-weight space. Both are1-dimensional, thus, this gives aquantum number!

(21)

It works fine!

Definition

Given a link diagramLD. Put it in a position `a la Morse and obtain qH(LD).

DefineP_RT² (LD)∈Z[q,q⁻¹] as theq-weighed, alternating sum over all resolutions multiplied by (−1)ⁿ⁻qⁿ⁺⁻²ⁿ⁻ (re-normalization).

Exercise: Check that this doesnotdepend on the choice of the position `a la Morse by using the relations from ˙Uq(slm).

There is asln-variantof this, denoted byP_RTⁿ (LD), that can also becolored with different fundamentalUq(sln)-representations Λ^kQ¯ⁿ.

Theorem

The (colored) polynomialP_RTⁿ (LD) is an invariant of links. Moreover, itis the (colored) Reshetikhin-Turaevsln-link polynomial.

In particular: The polynomialP_RT² (LD) colored with theUq(sl2)-tensor representations Λ¹Q¯² givesthe Jones polynomialJ(LD).

Daniel Tubbenhauer It’s Reshetikhin-Turaev’s coloredsln-link polynomial April 2014 21 / 36

(22)

The sl

_n

-link polynomials using sl

_m

-symmetries

Let ussummarizethe connection between (colored)sln-link polynomials and the U˙q(slm)- ˙Uq(sln)-skew Howe duality.

Reshetikhin-Turaev: Thesln-link polynomialsP_RTⁿ (·)areUq(sln)-intertwiner.

Uq(sln)-intertwinerarevectors in hom’s between ˙Uq(slm)-weight spaces.

OnlyF’s: ˙U⁻_q(slm)suffices. Conclusion: The (colored)sln-link polynomials P_RTⁿ (·) areinstances of ˙Uq(slm)-highest weight theory!

Even better: There exists a fixedmfor each linkLsuch that ˙Uq(slm)-highest weight theorygovernsall thesln-polynomials ofL.

IfLD is a link diagram, thenP_RTⁿ (LD) is obtained byjumping viaF’sfrom a highest ˙Uq(slm)-weightvh to a lowest ˙Uq(slm)-weightvl!

P_RTⁿ (LD) “measure” thedifferencebetween different “ways” from vh tovl.

(23)

Please, fasten your seat belts!

Let’scategorifyeverything!

(24)

Categorified symmetries

LetAbe some algebra,M be aA-module andCbe asuitablecategory. Denote by a,a1,a2some words in some generating set.

“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

a1∼a2 !

⇒fa1 =fa2 /o /o /o /o /o /o /o /o /o //a1∼a2 !

⇒ Fa1∼=Fa2

X ϕ

Moral:Liftmodules to categories, actions to functors, = to natural isomorphisms.

Daniel Tubbenhauer “Higher” representation theory April 2014 24 / 36

(25)

There is no direct minus

We haveseveralupshots.

The natural transformations between functors give informationinvisiblein

“classical” representation theory.

A categorical representation containsmoreinformation about the symmetries (or representations) ofA.

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

Xa_i⊗Xa_j ∼=M

k

X^c

ij

a_kk aiaj=X

k

c_k^ijak, c_k^ij ∈N.

(26)

How to get our hand on the natural transformations?

Reformulate: Let us seeAas acategoryAwith one object∗and a morphisma for eacha∈A. Then a categorical action can be seen as afunctor

R:A →End(C),∗ 7→ C anda7→ Fa.

But since End(C) is a2-category (2-morphisms are the natural transformations) one can expect that thereshouldbe a 2-categoryAthatcategorifiesAand a 2-functorR:A→End(C) thatcategorifiesR.

A ^2-action^R

This should exist!

//

forget 2-structur

End(C)

forget 2-structur

A ^1-action^R

This is given

//End(C)

(27)

The overview

U(slm)

How it should be!

Categorifiedq-skew Howe U(sl_m) acts

//

/o

K₀^⊕

????

K₀^⊕

U˙q(slm) q-skew Howe U˙q(sl_m) acts

//

/o

/o Sp(Uq(sln))^m

This is how it should be: There is anU˙q(slm)-actionon thesln-web spiders (for us it was mostly the casen= 2)

On the left side: There isKhovanov-Lauda’s categorificationof ˙Uq(slm) denoted byU(slm) (which Ibrieflyexplain soon).

Conclusion: Thereshouldbe a 2-action of U(slm) on the top right - a suitable 2-category of “natural transformations” betweenUq(sln)-intertwiners!

Daniel Tubbenhauer Categorifiedq-skew Howe duality April 2014 27 / 36

(28)

Rigid sl

₂

-foams: Hopefully illustrating examples

Instead of giving theformaldefinition of the rigidsl2-foam 2-categoryFoam2

(that fills the top right from before) let me just give someexamples.

Think←

→ Write

∈hom

,

∈hom

,

Think←

→ Write

∈hom ,

!

= −

(29)

Khovanov-Lauda’s 2-category U ( sl

_m

)

Definition/Theorem (Khovanov-Lauda 2008)

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

The objects ofU(slm) are the weights~k ∈Z^m−1.

The 1-morphisms are finite formal sums of the formEi1_~_k{t} andFi1_~_k{t}.

2-cells are graded, ¯Q-vector spacesgeneratedby compositions of diagrams (additional ones with reversed arrows) as illustrated belowplus relations.

i

~k

~k+αi

i

~k

~k−αi

i

~k

~k−αi

i j

~k

i i

~k

i i

~k

We have

U˙q(slm)∼=K₀^⊕(U(slm))⊗Z[q,q⁻¹]Q(q).¯ Roughly:Readthis as follows.

i i

~k ϕ:EiFi1~k →1~k

i i

~k ψ: 1~k →EiFi1~k

(30)

The KL-R algebra

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

LetRmbe acertain direct sum of subalgebras of homU(sl_m)(Fi1_~_k{t},Fj1_~_k′{t}).

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

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 U˙⁻_q(slm)∼=K₀^⊕(Rm)⊗Z[q,q⁻¹]Q(q).¯

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

(31)

The cyclotomic quotient

Definition(Khovanov-Lauda, Rouquier 2008/2009)

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

i_d i3 i2 i1

~kd

Λ

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

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

(32)

sl

₂

-foamation (works for all n > 1!)

We define a 2-functor

Γ :U(slm)→Foam^m₂ calledsl2-foamation, roughly in the following way.

On 2-cells:We define

i,~k

7→

~ki

~ki+1

i,~k

7→

~ki

~ki+1

i,i,~k

7→

~ki

~ki+1

And some others (that are not important today).

Theorem

The 2-functor Γ :U(slm)→Foam^m₂ categorifiesq-skew Howe duality.

It descents down to a 2-functor ˜Γ :RΛ(~k)- (p)Mod_gr →Foam^m₂.

(33)

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).

Daniel Tubbenhauer The Khovanov homology April 2014 33 / 36

(34)

Link diagrams are F ’s and differentials are KL-R crossings

Very roughly: Usecategorifiedq-skew Howe duality to express a link diagramLD

as a certain string ofonlyF_i^(j)’s. Obtain a complex as FtF4F3F2F3Fbvh{5}

˜Γ( )^:^F²^F³^→F³^F²

❚❚

))

❚❚

❚❚ FtF3F4F2F3Fbvh{4} L

Γ(˜ )^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

Γ(˜ )^:^F²^F³^→F²^F³

❚❚

))

❚❚

FtF3F4F2F3Fbvh{6}

FtF4F3F2F3Fbvh{5}

−˜Γ( )^:^F³^F⁴^→F⁴^F³

❥❥

55

❥❥

Theorem

This, under categorifiedq-skew Howe duality,givesthesln-link homology (because the “are”the “saddles”).

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The sl

_n

-homologies using sl

_m

-symmetries

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

Khovanov, Khovanov-Rozansky and others: Thesln-link homology can be obtainedusing certain “sln-foams”.

OnlyF’s: The (cyclotomic) KL-Rsuffices.

Conclusion: Thesln-link homologies areinstances of highestU(slm)-weight representation theory!

IfLD is a link diagram, then they are obtained byjumping viaF’sfrom a highestU(slm)-weightVh object to a lowestU(slm)-weight 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!

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There is stillmuchto do...

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Thanks for your attention!