Diagram categories for Uq

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Diagram categories for U

_q

-tilting modules at q

^ℓ

= 1

Or: fun with diagrams!

Daniel Tubbenhauer

s s s t t s

s t s t s s

=

s s s t t s

s t s t s s

Joint work with Henning Haahr Andersen

October 2014

Daniel Tubbenhauer October 2014 1 / 30

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1 The why of diagram categories String calculus

Biadjoint functors

2 Categorification of Hecke algebras Hecke algebras and Soergel bimodules Soergel’s categorification

3 Let us use diagrams!

TheFi are selfadjoint functors Diagrammatic categorification

4 What about Uq-modules at roots of unity?

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String calculus for 2-categories - Part 1

Question: Can we interpretCat² usingdiagrams? Let us start withCat¹: Instead of

C ^F¹ //D use thePoincar´e dual

D F C

Composition

D ^F² //E ◦ C ^F¹ //D = C ^F²^◦^F¹//E becomes

E F2 D

◦ D F1 C

= E F2 D F1 C Not really spectacular...

Daniel Tubbenhauer String calculus October 2014 3 / 30

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String calculus for 2-categories - Part 2

Let us go toCat²now:

Think of a natural transformationsα, β,· · · as aproceeding in time:

D C

F G

α =α:F⇒G E C

D

F1

G2 G1

β =β:F1⇒G2◦G1

Wedo notdraw identities

D C

F G

=id:F ⇒G

C

D id

H2 H1

γ =γ:H2◦H1⇒id

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String calculus for 2-categories - Part 3

Compositions?Sure!Vertical:

E D

C F1

G2 G1

β^′ ◦v E

D C

F1

G2 G1

β = E D C

F1

β^′

β

and horizontal

E D

F^′ G^′

α^′ ◦^h D C

F G

α = E D C

F^′ F

G^′ G

α^′ α

That looks promising: 2-categoriesare like2-dimensional spaces.

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Adjoint functors abstract

Definition(Dan Kan 1958)

Two functorsF: C → DandG: D → C are adjoint iff there exist natural transformations calledunitι: idC ⇒GF andcounitε:FG ⇒idD such that

F ^id^F^◦ι//

idF

33

FGF ^ε◦id^F //F and G ^ι◦id^G //

idG

33

GFG ^id^G^◦ε//G

commute. HereF is theleftadjoint ofG.

Example

forget: Q-Vect→Sethas a left adjointfree:Set→Q-Vect.

In words: If you have lost your key, then the onlyguaranteedsolution is to search everywhere.

Daniel Tubbenhauer Biadjoint functors October 2014 6 / 30

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Adjoint functors such that I understand

Let usdrawsting pictures!

idF =D C F

F

, idG =C D

G

, ι=

D C

G F

idC

and ε= C D

F G

idD

Adjointness is juststraighteningof the strings

C D

F

=D C

F

and D C

G G

=C D

G

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Biadjoint functors = Isotopies

IfF is also the right adjoint ofG, then the picture gets topological.

Biadjointness is juststraighteningof the strings! First left

C D

F

=D C

F

and D C

G G

=C D

G

G then right

D C

F F

=D C

F

and D C

G

=C D

G

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Biadjoint functors in “nature” (do not ask me for details...)

Categories of modules over finite dimensional symmetric algebras and their derived counterparts haveplentyof built-in biadjoint functors (tensoring with certain bimodules).

Prominent examples are finite groups andinduction and restriction functors between them.

Various categories arising in representation theory of Hecke algebras and categoryO admitlotsof biadjoint functor. For example translation functors and Zuckerman functors.

Every (extended) TQFTF:Cobⁿ⁺²→Vec²gives abunchof biadjoint functors: (F(M),F(τ(M))) for anyn+ 1 manifoldM where τ flipsM.

Prominent examples come from commutative Frobenius algebras forn= 2, Witten-Reshetikhin-Turaev TQFT’s forn= 3, Donaldson-Floer forn= 4, andway more...

Otherfancystuff like Fukaya-Floer categories, derived categories of constructible sheaves on flag varieties...

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The Iwahori-Hecke algebra (for me n = 3 is enough)

Let us fixn= 3. Then the group ring of the symmetric groupQ[S3] has two generatorss1,s2. They satisfy

s₁²= 1 =s₂² and s1s2s1=s2s1s2.

Iwahori: TheHecke algebraH3=H[S3]is aq-deformationofQ[S3].

Definition/Theorem(Iwahori 1965)

The Hecke algebraH3has generatorsT1,T2and relations

T₁²= (q−1)T1,2+q=T₂² and T1T2T1=T2T1T2. The classical limitq→1 givesQ[S3].

Nowadays Hecke algebras `a la Iwahori appear “everywhere”, e.g. low dimensional topology, combinatorics, representation theory ofgl_netc.

Daniel Tubbenhauer Hecke algebras and Soergel bimodules October 2014 10 / 30

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Idempotents are better!

Recall that primitive idempotentsej ∈Ain any finite dimensionalQ-algebraA give rise toAej which is indecomposable.

The group algebraQ[S3]admits “idempotents”:i1= 1 +s1 andi2= 1 +s2, because they satisfy

i₁²= 2i1,2=i₂² and i1i2i1+i2=i2i1i2+i1.

For the Hecke algebra: Sett =√qanddefineb1,2=t⁻¹(1 +T1,2)(we see the Hecke algebra overQ[t,t⁻¹] now).

Theb1,b2satisfy

b₁²= (t+t⁻¹)b1,2=b₂² and b1b2b1+b2=b2b1b2+b1. Onlypositive coefficients? Suspicious...

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Bimodules do the job?

TakeR=Q[X1,X2,X3] (with degree of Xi= 2) and define thes1,2-invariantsas R^s¹={p(X1,X2,X3)∈R|p(X1,X2,X3) =p(X2,X1,X3)}

and

R^s²={p(X1,X2,X3)∈R|p(X1,X2,X3) =p(X1,X3,X2)}. For exampleX1+X2∈R^s¹, butX1+X26∈R^s².

The algebraR is a (left and right)R^s^1,2-module. Thus,

B1=R⊗R^s¹R{−1} and B2=R⊗R^s² R{−1}

areR-bimodules. Write shortB_ij forB_i⊗RB_j. Funny observation (i= 1,2):

Bii ∼=Bi{+1} ⊕Bi{−1} and B121⊕B2∼=B212⊕B1. We have seen thisbefore...

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The combinatoric of S

₃

Thebimodule world: Take tensor productsBi of theBi’s.

Theatomsof the bimodules world are the indecomposables: AllM such that M∼=M1⊕M2impliesM1,2∼= 0.

We haveB∅=R,B1=B1,B2=B2, B12=B12 andB21=B21 as atoms,but B121∼=B1⊕R⊗R^S³R{−3} and B212∼=B2⊕R⊗R^S³ R{−3}

B1

B21 R

B12

B2

B121

s1 s2

s2 s1

andB121=B212=R⊗R^S³R{−3}is indecomposable.

There areexactlyas many indecomposables as elements inS3.Suspicious...

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What is a morphism of elements of H

₃

?

Definition(Soergel 1992)

DefineSC(3) to be the category with the following data:

Objects are (shifted) direct sums⊕of tensor productsB_i ofB_i’s.

Morphismsare matrices of (graded) bimodule maps.

Theorem(Soergel 1992)

SC(3) categorifiesH3. The indecomposables categorify the Kazhdan-Lusztig basis elements ofH3.

Morally:SC(3) is thecategorical analogonofH3. The morphisms inSC(3) are invisible inH3.

Wait: What do you mean bycategorify?

Daniel Tubbenhauer Soergel’s categorification October 2014 14 / 30

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(Split) Grothendieck group

If you have a suitable categoryC, then we can easilycollapse structureby totally forgetting the morphisms:

The (split) Grothendieck groupK₀^⊕(C) ofC has isomorphism classes [M] of objectsM∈Ob(C) as elements together with

[M0] = [M1]+[M2]⇔M0∼=M1⊕M2,[M1][M2] = [M1⊗M2] and [M{s}] =t^s[M].

This is aZ[t,t⁻¹]-module.

Example

We have

K₀^⊕(Q-Vectgr)→^∼⁼ Z[t,t⁻¹], [Q{s}]7→t^s·1.

The whole power of linear algebra isforgottenby going toK0(Q-Vect)gr.

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Categorification?

We have two functorsF1=B1⊗^R·andF2=B2⊗^R·. These are additive endofunctors ofSC(3). Thus, the introduce an action [Fi] onK₀^⊕( ˙SC(3)).

We have a commuting diagram (we ignore to tensor withQ(t))

K₀^⊕( ˙SC(3)) ^[Fⁱ^] //

φ ∼=

K₀^⊕( ˙SC(3))

∼= φ

H3 ·bi

//H3.

Thus, the functorsF1,F2categorifythe multiplication inH3! Said otherwise: They categorify the action ofH3on itself.

Moreover, the indecomposables give agood basisofH3.

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The speaker is lost...

The speaker is lost: That was too abstract. Can we understand thistopological?

Observation(Elias-Khovanov 2009)

The functorsF1andF2areselfadjoint! Thus, there is a stringy calculus forSC(3).

As before: Well denote compositions likeF1F2F2F1F1by

1 SC(3) 2 SC(3) 2 1 1

SC(3) SC(3) SC(3) SC(3)

or simplified

1 2 2 1 1

Think: ApplyF1F2F2F1F1 toR on the right.

Daniel Tubbenhauer TheFiare selfadjoint functors October 2014 17 / 30

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Generators

We have the following one colorgenerators:

F1⇒id deg= +1

id⇒F1

deg= +1

F1⇒F1F1

deg=−1

F1F1⇒F1

deg=−1

F1F2F1⇒F1F2F1

deg= 0

F2⇒id deg= +1

id⇒F2

deg= +1

F2⇒F2F2

deg=−1

F2F2⇒F2

deg=−1

F2F1F2⇒F2F1F2

deg= 0

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Exempli gratia

F2F2F1F2F2F1F1⇒F2F2F2F1F2F1F1

deg= +8

These Soergel diagrams can getvery complicated, but this is an information completely invisiblein H3.

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Some relations - Part 1

We need some additionalrelationsto make the story work. Some are combinatorial (which we do not recall), but, due to biadjointness, some aretopological.

= = =

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Some relations - Part 2

Some arereally topological: There is more than planar isotopies. The functorsF1

andF2areFrobenius. This gives

= = = =

=

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The diagram category suffices

Definition(Elias-Khovanov 2009)

DefineDC(3) to be the category with the following data:

Objects are (shifted) formal direct sums⊕of sequences of the form F2F2F2F1F2F1F1.

Morphismsare matrices of (graded) Soergel diagrams module the local relations.

Theorem(Elias-Khovanov 2009)

There is an equivalence of graded, monoidal,Q-linear categories DC(3)∼=SC(3).

Conclusion: The (seemingly veryrigid) Hecke algebraH3has an overlying topologicalcounterpart!

Daniel Tubbenhauer Diagrammatic categorification October 2014 22 / 30

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Some upshots of Elias-Khovanov’s approach

Norestriction toS3: Any Coxeter system works.

Diagrammatic categorification is“low tech”. Playing with diagrams is fun, easy and the topological flavour gives new insights. For example, Elias and Williamson’s algebraic proof that the Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems wasdiscoveredusing the diagrammatic framework.

New insights into topology:

Elias used the topological behaviour to give anewcategorification of the Temperley-Lieb algebra.

Rouquier produced abraid group actionon (chain complexes of) Soergel diagrams. This isfunctorial: It also talks about braid cobordisms (these live in dimension 4!).

Rouquier’s results can beextendedto give HOMFLY-PT homology. Thisstill mysterioushomology is related to knot Floer homology.

More is to be expected!

Daniel Tubbenhauer Diagrammatic categorification October 2014 23 / 30

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Non-associative=bad

Recall thatsl2is [·,·]-spanned byF = ^{0 0}_{1 0}

,E = ^{0 1}_{0 0}

andH= ^{1 0}₀₋₁ . Non-associative: TakeU(·) :LieAlg→AssoQ-Algwhich is the left adjoint of [·,·] :AssoQ-Alg→LieAlg. Thus, theuniversal envelopeU(sl2)is the free, associativeQ-algebra spanned by symbolsE,F,H,H⁻¹modulo

HH⁻¹=H⁻¹H= 1, HE=EH and HF =FH. EF−FE =H.

By magic:sl2-Mod∼=U(sl2)-Mod.

Naively quantize:Uq(sl2) =Uqis the free, associativeQ(q)-algebra spanned by symbolsE,F andK,K⁻¹(think:K =q^H,K⁻¹=q⁻^H) modulo

KK⁻¹=K⁻¹K = 1, EK =q²KE and KF =q⁻²FK. EF−FE =K −K⁻¹

q−q⁻¹ (think: q^H−q⁻^H q−q⁻¹

q→1

−→H).

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What are the atoms?

Fact of life

Ifqis an indeterminate, thenUq has the “same” representation theory assl2. In particular,Uq-Modfin issemisimple: Atoms are the irreducibles.

Ifq^ℓ= 1, then this totally fails:Uq-Modfin is far away to be semisimple.

Why do wewantto study something so nasty?

Magic:Many similaritiesto the representation theory of a corresponding almost simple, simply connected algebraic groupG modulop.

Many similaritiesto the representation theory of a correspondingaffine Kac-Moody algebra.

It providesribboncategories (link invariants) which can be “semisimplified”

to providemodularcategories (2 + 1-dimensional TQFT’s).

It turns out that the “right” atoms are the so-calledindecomposableUq-tilting modules. The corresponding categoryTis what we want to understand.

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Translation functors

Principle(Bernstein-Gelfand-Gelfand 1970)

Do not study representations explicitly: That istoo hard. Study thecombinatorial andfunctorialbehaviour of their module categories!

So let us adopt the BGG principle from categoryO!

In particular, there are two endofunctors Θs,Θt ofTλ (there is a decomposition of Tinto blocksTλ) calledtranslation through thes,t-wall. These are selfadjoint Frobeniusfunctors with combinatorial behaviour governed by the∞-dihedral groupD∞={s,t |s²= 1 =t²}:

ΘsΘs ∼= Θs⊕Θs and ΘtΘt ∼= Θt⊕Θt.

We have seen something similarbefore: There should be a diagram category (inspired by the corresponding one forH(D∞)) that governsTand pEnd(T).

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Sorry: No tenure means I have to stress my own results

Definition/Theorem(Elias 2013)

There is a diagram categoryD(∞) that categorifiesH(D∞) (that is what we are looking for!). The indecomposables categorify the Kazhdan-Lusztig basis elements ofH(D∞).

Definition/Theorem

There is arediagram categoriesQD(∞) andMat^fs∞(dQD(∞))c forTand pEnd(T). The diagram categories are naturally graded whichintroduce a non-trivial gradingonT and pEnd(T).

We haveK₀^⊕(T^gr_λ)∼=B^∞: Thus,T^gr_λ categorifiesthe Burau representationB^∞of the braid groupB∞ in∞-strands (cut-offs are possible). The action ofB∞ is categorified using certain chain complexes of truncations of Θs,Θt.

We haveK₀^⊕(pEnd(T^gr_λ))∼=TL^q∞: Thus, pEnd(T^gr_λ)categorifies(a certain summand of) the Temperley-Lieb algebra in∞-strands (cut-offs are possible).

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Elias’ dihedral cathedral

The categoryD(∞) is almost as before, buteasier: No relations among the

“colors” reds and greent:

Neither nor

Pictures look like

OurQD(∞) looks similar plus some extra relations.

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Open connections

Question:What is the non-trivial grading (purely a root of unity phenomena) trying to tell us about the link and 3-manifold invariants deduced fromT?

Question:Similarly, what is the non-trivial grading (purely a root of unity phenomena) trying to tell us about algebraic groups modulop?

We argue that each blockT^gr_λ separatelycan be used to obtain invariants of links and tangles - there are very explicit relations to (sutured) Khovanov homology and bordered Floer homology.

Hence, each blockT^gr_λ separatelyyields information about link and tangle invariants in thenon-root of unitycase, while the ribbon/modular structure ofTyields the Witten-Reshetikhin-Turaev invariants.Question:What is going on here?

As in theH(Sn) case:Question:Is there a “cobordism” theory that explains the grading and the Frobenius structure topological?

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There is stillmuchto do...

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Thanks for your attention!