Categoriﬁcation and applications in topology and representation theory

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Categorification and applications in topology and representation theory

Daniel Tubbenhauer

July 2013

Daniel Tubbenhauer July 2013 1 / 29

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1 Categorification

What is categorification?

2 Virtual knots and categorification The virtualsl₂polynomial The virtual Khovanov homology

3 Thesl₃ web algebra (joint work with Mackaay and Pan) Webs and representation theory

An algebra of foams

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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 What is categorification? July 2013 3 / 29

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Exempli gratia

Examples of the pair categorification/decategorification are:

The integersZ

categorify

//

decat=χ(·)

oo complexes of VS

Polynomials inZ[q,q⁻¹] ^categorify //

decat=χgr(·)

oo complexes of gr.VS

The integersZ

categorify

//

decat=K0(·)

oo K−vector spaces

AnA−module ^categorify //

decat=K₀^⊕(·)⊗_ZA

oo additive category

Thefirst/secondpart is related to thefirst/lasttwo examples.

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Classical and virtual knots and links

Definition

Avirtual knot or link diagramL_D is a four-valent graph embedded in the plane.

Moreover, every vertex is marked with an overcrossing , an undercrossing or a virtual crossing .

Anoriented virtual knot or link diagramis defined by orienting the projection, i.e.

crossings should look like , and .

Avirtual knot or linkLis an equivalence class of virtual knot or link diagrams modulo the so-calledgeneralised Reidemeister moves.

Daniel Tubbenhauer The virtualsl2 polynomial July 2013 5 / 29

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Classical and virtual knots and links

Generalised Reidemeister moves

RM1 RM2

vRM1 vRM2

mRM

vRM3 RM3

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Virtual links and topology

Theorem(Kauffman, Kuperberg)

Virtual links are acombinatorialdescription of copies ofS¹embedded in a thickened surface Σg of genusg. Such links are equivalent iff their projections to Σg arestable equivalent, i.e. up to homeomorphisms of surfaces, adding/removing

“unimportant” handles, classical Reidemeister moves and isotopies.

Example(Virtual trefoil and virtual Hopf link)

=

Daniel Tubbenhauer The virtualsl2 polynomial July 2013 7 / 29

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The famous (virtual) Jones polynomial

LetL_D be an oriented link diagram. Thebracket polynomialhL_Di ∈Z[q,q⁻¹] can berecursivelycomputed by the rules:

h∅i= 1 (normalisation).

h i=h i −qh i(recursion step 1).

hUnknot∐L_Di= (q+q⁻¹)hL_Di(recursion step 2).

TheKauffman polynomial isK(LD) = (−1)ⁿ⁻qⁿ⁺⁻²ⁿ⁻hLDi, withn+=number andn−=number of .

Theorem(Kauffman)

The Kauffman polynomialK(L) is an invariant of virtual links andK(L) = ˆJ(K), where ˆJ(K) denotes the unnormalised Jones polynomial.

Let us categorify this!

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A cobordism approach

The pre-additive, monoidal, graded categoryuCob²_R(∅) ofpossible unorientable, decoratedcobordisms has:

Objects are resolutions of virtual link diagrams, i.e. virtual link diagrams without classical crossings.

Morphisms aredecoratedcobordismsimmersedintoR²×[−1,1] generated by (last one is a two times puncturedRP²)

+

+ +

-

+

+ +

+

+ +

ε⁺ ι+

id⁺₊ Φ⁻₊ m⁺⁺+ ∆⁺₊₊ τ++⁺⁺ θ

Somerelationslike (last two are two times punctured Klein bottles)

u

l l₁ l₂ u₁ u₂

= =

l₁ l₂

u₁ u₂

=

= + + l₁ l₂

u₁ u₂

=

l₁ l₂

u₁ u₂

=

= l1

l₂ u₁ u₂

=

= +

+ - - -

- u

l

=

The monoidal structure is given by the disjoint union and the grading by the Euler characteristic.

Daniel Tubbenhauer The virtual Khovanov homology July 2013 9 / 29

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How to form a chain complex

DefineMat(uCob²_R(∅)) to be thecategory of matricesoveruCob²_R(∅), i.e.

objects are formal direct sums of the objects ofuCob²_R(∅) and morphisms are matrices whose entries are morphisms fromuCob²_R(∅).

DefineuKob_b(∅)R to be thecategory of chain complexes overMat(uCob²_R(∅)).

The category is pre-additive. Hence, the notiond²= 0makes sense.

As a reminder, to every virtual link diagramL_D we want toassignan object in uKob_b(∅)R that is aninvariantof virtual links. By our construction, this invariant willdecategorifyto the virtual Jones polynomial.

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How to form a chain complex

For a virtual link diagramLD withn=n++n− crossings the topological complex JLDKshould be:

Fori= 0, . . . ,nthei−n− chain module is the formal direct sum of all resolutions of lengthi.

Between resolutions of lengthi andi+ 1 the morphisms should besaddles between the resolutions.

The decorations for the saddles can be read of bychoosing an orientation for the resolutions. Locally they look like , which is calledstandard. Now compose with Φ iff the orientations differ or iff both are non-alternating

we useθ.

Extraformal signs- placement is rather technical and skipped today.

Note that itnotobvious why this definition gives awell-definedchain complex independentof all choices involved.

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Some results

Theorem(s)(T)

The topological complexesJ·Kof two equivalent virtual link diagrams are the same inuKob_b(∅)^hl_R, i.e. the complex is an invariant up to chain homotopy and

so-calledlocal relations. Moreover, it is a well-defined chain complex independent of all choices involved and can be extended to virtual tangles.

LetF denote a uTQFT, i.e. asuitablefunctorF: uCob²_R(∅)→R-Mod.

Theorem(s)(T)

LetF be an uTQFT that satisfies the local relations. Then the homology groups of the algebraic complexF(J·K) are virtual link invariants. Moreover, the category of uTQFT is equivalent to the category of skew-extended Frobenius algebras.

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Exempli gratia

00 11

01

10

(q+q⁻¹)² −2q(q+q⁻¹) q²(q+q⁻¹) Let us show how the calculation works. We consider the virtual trefoil and suppressgrading shifts and sign placement. First let usaddsome orientations.

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Exempli gratia

00 11

01

10

Let us show how the calculation works. We consider the virtual trefoil and suppressgrading shifts and sign placement. First let usaddsome orientations.

Now we canreadof the cobordisms.

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Exempli gratia

-

+

-

+

-

+

00 11

01

10

Let us show how the calculation works. We consider the virtual trefoil and suppressgrading shifts and sign placement. First let usaddsome orientations.

Now we canreadof the cobordisms.

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Exempli gratia

-

+

-

+

-

+

00 11

01

10

Note that thisisthe topological complex.

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Exempli gratia

-

+

-

+

-

+

00 11

01

10

Now we have totranslate(usingone particularuTQFT) the objects to graded Q-vector spaces and the cobordisms toQ-linear maps between them. Then the objects areA⊗A,A⊕AandAwithA=Q[X]/X².

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Exempli gratia

-

+

-

+

-

+

00 11

01

10

The two right maps are 0 and the two multiplications are given by 1⊗1→1,X⊗1→ ±X,1⊗X → −X andX⊗X →0

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Exempli gratia

-

+

-

+

-

+

00 11

01

10

The homologycanbe computed now and it turns out to be (up to shifts)

q⁻²t⁰+q²t⁻¹+qt⁻²+q³t⁻². Settingt=−1givesthe virtual Jones polynomial (q⁻¹−q+q²)(q+q⁻¹).

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Kuperberg’s sl

₃

webs

Definition(Kuperberg)

TheC(q)-web spaceW_S for a given sign stringS = (±, . . . ,±) is generated by {w |∂w =S}, wherew is a web, i.e. anoriented, trivalentgraph such that any vertex is either a sink or a source, with boundaryS subject to the relations

= [3]

= [2]

= +

Here [a] = ^q_q−qâ^−q−1^−a =qâ−1+qâ−3+· · ·+q^−(a−1) is thequantum integer.

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Kuperberg’s sl

₃

webs

Example

wt= 0

wt= 0 wt= 0

wt= 0

wt= 0 wt=−1

wt= 0

wt= 0 wt=−2

wt=−1

wt= 1 wt= 0

wt=−1

wt= 1 wt= 1

wt=−1

wt= 1 wt= 2

Webs can becolouredwith flow lines. At the boundary, the flow lines can be represented by astate stringJ. By convention, at thei-th boundary edge, we set ji =±1 if the flow line is oriented upward/downward andji = 0, if there is no flow line. SoJ = (0,0,0,0,0,−1,1) in the example.

Given a web with a floww_f, attribute aweightto each trivalent vertex and each arc inw_f and take the sum. The weight of the example is−3.

Daniel Tubbenhauer Webs and representation theory July 2013 15 / 29

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Representation theory of U

_q

( sl

₃

)

A sign stringS = (s1, . . . ,sn) corresponds to tensors VS =Vs1⊗ · · · ⊗Vsn,

whereV+ is the fundamental representation andV− is its dual, and webs correspond tointertwiners.

Theorem(Kuperberg)

WS ∼=homU_q(sl₃)(C(q),VS)∼=InvU_q(sl₃)(VS)

The set ofnon-elliptic webs, i.e. without circles, digons or squares, ofWS, denotedBS, is calledweb basisofInvU_q(sl₃)(VS). In fact, the so-called spider category of all webs modulo the Kuperberg relations isequivalentto the representation category ofU_q(sl₃).

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Representation theory of U

_q

( sl

₃

)

Theorem(Khovanov, Kuperberg)

Pairs of signS and a state stringsJ correspond to the coefficients of the web basis relative totensors of the standard basis{e₋₁^± ,e₀^±,e₊₁^±} ofV±.

Example

wt=−1 wt=−3

wS =· · · −(q⁻¹+q⁻³)(e₀⁺⊗e₀⁻⊗e₀⁺⊗e₀⁻⊗e₀⁺⊗e₋₁⁺ ⊗e₊₁⁺)± · · ·. Let us categorify this!

Daniel Tubbenhauer Webs and representation theory July 2013 17 / 29

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sl

₃

-foams

Apre-foamis a cobordism with singular arcs between two webs. Composition consists of placing one pre-foam ontopof the other. The following are called the zipand theunziprespectively.

They havedotsthat can movefreelyabout the facet on which they belong, but we donotallow dot to cross singular arcs.

Afoamis a formalC-linear combination of isotopy classes of pre-foams modulo the following relations.

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The foam relations ℓ = (3 D , NC , S , Θ)

= 0 (3D)

=− − − (NC)

= = 0, =−1 (S)

α β

δ =







1, (α, β, δ) = (1,2,0) or a cyclic permutation,

−1, (α, β, δ) = (2,1,0) or a cyclic permutation, 0, else.

(Θ)

Adding a closure relation toℓsufficeto evaluate foams without boundary!

Daniel Tubbenhauer An algebra of foams July 2013 19 / 29

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Involution on webs and closed webs

Definition

There is aninvolution^∗ on the webs.

w

w*

Aclosed webis defined by closing of two webs.

u v*

Aclosed foamis a foam from∅ to a closed web.

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The sl

₃

-foam category

Foam₃is thecategory of foams, i.e.objectsare websw andmorphismsare foams F between webs. The category isgradedby theq−degree

deg_q(F) =χ(∂F)−2χ(F) + 2d+b,

whered is the number of dots andb is the number of vertical boundary components. Thefoam homologyof a closed webw is defined by

F(w) =Foam₃(∅,w).

F(w) is a graded, complex vector space, whose q-dimension can be computed by theKuperberg bracket.

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The sl

₃

web algebra

Definition(MPT)

LetS = (s1, . . . ,sn). Thesl₃ web algebraKS is defined by KS = M

u,v∈BS

uKv,

with

uKv :=F(u^∗v){n}, i.e. all foams : ∅ →u^∗v.

Multiplication is defined as follows.

uKv1⊗v2Kw →uKw

is zero, ifv16=v2. Ifv1=v2, use themultiplication foamm_v, e.g.

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The sl

₃

web algebra

v w*

v v*

w*

v

Theorem(s)(MPT)

The multiplication isassociative and unital. The multiplication foamm_v only dependson the isotopy type ofv and hasq-degreen. Hence,K_S is a finite dimensional, unital and graded algebra. Moreover, it is agraded Frobenius algebra of Gorenstein parameter 2n.

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Exempli gratia

Every web has a homogeneous basisparametrisedby flow lines.

wt=3

wt=1

wt=−1

wt=−3

q−deg=0

q−deg=2

q−deg=4

q−deg=6

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En(c)hanced sign strings

Definiton

Anenhanced sign sequenceis a sequenceS = (s1, . . . ,s_n) withs_i∈ {◦,−,+,×}, for alli= 1, . . . ,n. The correspondingweightµ=µS ∈Λ(n,d) is given by the rules

µi =











0, ifsi=◦, 1, ifs_i= 1, 2, ifsi=−1, 3, ifs_i=×.

Let Λ(n,d)3⊂Λ(n,d) be the subset of weights with entries between 0 and 3.

GivenS, we define bS by deleting the entries equal to◦ or×.

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En(c)hanced sign strings

Moreover, forn=d= 3^k we define

WS =W_b_S and BS =B_b_S and W₍₃^k₎ = M

µs∈Λ(n,n)3

WS

on thelevelof webs and on the levelof foams, we define K_S=K_b_S and W₍₃k)= M

µs∈Λ(n,n)3

K_S−pMod

gr.

With this constructions we obtain ourcategorificationresult.

Theorem(MPT)

K₀^⊕(W₍₃^k₎)⊗_Z[q,q⁻¹_]C(q)∼=W₍₃^k₎.

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Connection to U

_q

( sl

_n

)

Khovanov and Lauda’s diagrammatic categorification of ˙U_q(sl_n), denoted U(sl_n), is alsorelatedto our framework! Roughly, it consist of string diagrams of the form

i j

λ:EiEj1_λ⇒ EjEi1_λ{(αi, αj)},

i

λ+αi λ:Ei1_λ⇒ Ei1_λ{(αi, αi)}

with a weightλ∈Zⁿ⁻¹and suitable shifts and relations like

i j

λ =

i j

λ and

i j

λ =

i j

λ.

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Connection to U

_q

( sl

_n

)

Letλ∈Λ(n,n)⁺be a dominant weight. Define thecyclotomic KL-R algebraRλto be the subquotient ofU(sl_n) defined by the subalgebra of only downward pointing arrows modulo the so-calledcyclotomic relationsand setVλ=R_λ−pMod

gr.

Theorem(s)(MPT)

There exists an equivalence of categoricalU(sl_n)-representations Φ :V(3^k) → W(3^k).

The two algebrasR₃^ℓ andK₃^ℓ are Morita equivalent. Moreover, the set {[Q_u]|Q_ugraded, indecomposable, projectiveK_S−module,u∈B_S} is the dual canonical basis forInvU_q(sl₃)(VS)∼=K₀^⊕(KS)⊗_Z[q,q⁻¹_]C(q).

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There is stillmuchto do...

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Thanks for your attention!