The sl 3 web algebra Marco Mackaay CAMGSD and Universidade do - - PowerPoint PPT Presentation

The sl3 web algebra

Marco Mackaay

CAMGSD and Universidade do Algarve

September, 2012 1

Credits

This is joint work with Weiwei Pan and Daniel Tubbenhauer 2

Diagrammatic categorification

slk-Webs

Intertwiners

• Kauff, Kup, MOY
• Uq(slk)-Tensors

Resh-Tur

• slk-knot polynomials

3

Diagrammatic categorification

slk-Webs

Intertwiners

• Kauff, Kup, MOY
• Uq(slk)-Tensors

Resh-Tur

• slk-knot polynomials

slk-Foams/MF

Khov, Khov-Roz

• slk-Cycl string diags
• ???
• Webster
• slk-Knot homologies

4

Howe dual picture

slk-Webs

Howe duality

• Kauff, Kup, MOY
• Uq(sln)-Irrep

Lusz

• slk − Bn-Action

5

Howe dual picture

slk-Webs

Howe duality

• Kauff, Kup, MOY
• Uq(sln)-Irrep

Lusz

• slk − Bn-Action

slk-Foams/MF

Khov, Khov-Roz

• sln-Cycl KLR-algebra
• Howe 2-duality

Chuang-Rouq

• slk − Bn-2-Action

6

Skew Howe duality

The natural actions of GLk and GLn on Λp Ck ⊗ Cn are Howe dual (skew Howe duality). 7

Skew Howe duality

The natural actions of GLk and GLn on Λp Ck ⊗ Cn are Howe dual (skew Howe duality). This implies that InvSLk

• Λp1

Ck ⊗ Λp2 Ck ⊗ · · · ⊗ Λpn Ck ∼ = W(p1, . . . , pn), where W(p1, . . . , pn) denotes the (p1, . . . , pn)-weight space of the irreducible GLn-module W(kℓ), if n = kℓ. 8

First things first

Let’s q-deform this (for k = 3) 9

Kuperberg’s sl3-webs

Example (0.1) Any web can be obtained from the following elementary webs by gluing and disjoint union: (0.2) 10

Let S = (s1, . . . , sn) be a sign string. Definition (Kuperberg) WS := C(q) {w | ∂w = S} /IS where IS is generated by: = [3] (0.3) = [2] (0.4) = + (0.5) 11

Let S = (s1, . . . , sn) be a sign string. Definition (Kuperberg) WS := C(q) {w | ∂w = S} /IS where IS is generated by: = [3] (0.3) = [2] (0.4) = + (0.5) From (0.3), (0.4) and (0.5) it follows that any w ∈ WS is a linear combination of non-elliptic webs (no circles, digons or squares). The latter form a basis, BS. 12

Define w∗ by

w w*

(0.6) Define uv∗ by

u v*

(0.7) Define v∗u by

u v*

(0.8) 13

Representation theory of Uq(sl3)

Given S = (s1, . . . , sn), we define VS = Vs1 ⊗ · · · ⊗ Vsn. V+ is the fundamental Uq(sl3) representation and V− its dual V− := V ∗

+ ∼

= V+ ∧ V+. 14

Representation theory of Uq(sl3)

Given S = (s1, . . . , sn), we define VS = Vs1 ⊗ · · · ⊗ Vsn. V+ is the fundamental Uq(sl3) representation and V− its dual V− := V ∗

+ ∼

= V+ ∧ V+. Webs correspond to intertwiners Theorem (Kuperberg) WS ∼ = Hom(C(q), VS) ∼ = Inv(VS). BS is called the web basis of Inv(VS). 15

The general and special linear quantum groups

Definition i) Uq(gln) is generated by K±1

1 , . . . , K±1 n , E±1, . . . , E±(n−1),

subject to (αi = εi − εi+1 = (0, . . . , 1, −1, . . . , 0) ∈ Zn−1): KiKj = KjKi KiK−1

i

= K−1

i

Ki = 1 EiE−j − E−jEi = δi,j KiK−1

i+1 − K−1 i

Ki+1 q − q−1 KiE±j = q±(εi,αj)E±jKi + some extra relations we won’t need today ii) Uq(sln) ⊆ Uq(gln) is generated by KiK−1

i+1 and E±i.

16

Idempotented quantum groups

Definition (Beilinson-Lusztig-MacPherson) For each λ ∈ Zn, adjoin an idempotent 1λ and add the relations 1λ1µ = δλ,ν1λ E±i1λ = 1λ±αiE±i Ki1λ = qλi1λ. Define ˙ U(gln) =

• λ,µ∈Zn

1λUq(gln)1µ. Define ˙ U(sln) similarly by adjoining idempotents 1µ to Uq(sln) for µ ∈ Zn−1. 17

A finite-dimensional semi-simple quotient

Lemma (Doty-Giaquinto) The q-Schur algebra Sq(n, d) is generated by 1λ, for λ ∈ Λ(n, d), and E±i, for i = 1, . . . , n − 1, such that 1λ1µ = δλ,µ1λ

• λ∈Λ(n,d)

1λ = 1 E±i1λ = 1λ±αiE±i EiE−j − E−jEi = δij

• λ∈Λ(n,d)

[λi − λi+1]1λ 18

Back to q-skew Howe duality

Definition An enhanced sign sequence is a sequence S = (s1, . . . , sn) with si ∈ {∅, −1, 1, ×}, for all i = 1, . . . n. The corresponding weight µ = µS ∈ Λ(n, d) is given by the rules µi =            if si = ∅ 1 if si = 1 2 if si = −1 3 if si = × . Let Λ(n, d)3 ⊂ Λ(n, d) be the subset of weights with entries between 0 and 3. 19

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. 20

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. We define WS := W

S,

and BS := B

S.

21

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. We define WS := W

S,

and BS := B

S.

Definition Define W(3ℓ) :=

• µS∈Λ(n,n)3

WS. 22

The action

Define ϕ: Sq(n, n) → EndC(q)

• W(3ℓ)
• 1λ →

λ1 λ2 λn

E±i1λ →

• λ1

λi−1 λi λi+1 λi±1 λi+1∓1 λi+2 λn

Conventions: vertical edges labeled 1 are oriented upwards, vertical edges labeled 2 are oriented downwards and edges labeled 0 or 3 are erased. 23

Examples

E+11(22) →

2 2 3 1

E−2E+11(121) →

1 2 1 2 2

24

The isomorphism from q-skew Howe duality

Recall Sq(n, n)1(3ℓ)/[µ > (3ℓ)] ∼ = V(3ℓ). Lemma The map ϕ gives rise to an isomorphism ϕ: V(3ℓ) → W(3ℓ)

• f Sq(n, n)-modules.

Note that the empty web wh := w(3ℓ), which generates W(×k,∅2k) ∼ = C(q), is a highest weight vector. 25

Please, fasten your seat belts!

Let’s categorify everything 26

sl3 Foams

Consider formal C-linear combinations of isotopy classes of singular cobordisms, e.g. the zip and unzip: 27

sl3 Foams

Consider formal C-linear combinations of isotopy classes of singular cobordisms, e.g. the zip and unzip: We also allow dots, which cannot cross singular arcs. 28

sl3 Foams

Consider formal C-linear combinations of isotopy classes of singular cobordisms, e.g. the zip and unzip: We also allow dots, which cannot cross singular arcs. Mod out by the ideal generated by ℓ = (3D, NC, S, Θ) and the closure relation: 29

Khovanov’s local relations: ℓ = (3D, NC, S, Θ)

= 0 (0.9) = − − − (0.10) = = 0, = −1 (0.11) =    1 (α, β, γ) = (1, 2, 0) or a cyclic permutation −1 (α, β, γ) = (2, 1, 0) or a cyclic permutation else (0.12) The relations in ℓ suffice to evaluate any closed foam! 30

The category of foams

Let Foam3 be the category of webs and foams. 31

The category of foams

Let Foam3 be the category of webs and foams. Other relations in Foam3 are: = − (Bamboo) = − (RD) 32

More relations in Foam3

= 0 (Bubble) = − (DR) = − − (SqR) 33

More relations in Foam3

+ + = + + = 0 = 0 (Dot Migration) 34

The grading

The q-grading of a foam U is defined as q(U) := χ(∂U) − 2χ(U) + 2d + b. This makes Foam3 into a graded category. 35

Foam homology

Definition The foam homology of a closed web w is defined by F(w) := Foam3(∅, w). 36

Foam homology

Definition The foam homology of a closed web w is defined by F(w) := Foam3(∅, w). F(w) is a graded complex vector space, whose q-dimension can be computed by the Kuperberg bracket:

1

• w ∐
• = [3]w

2

• = [2]
• 3
• =
• +
• The relations above correspond to the decomposition of F(w)

into direct summands. 37

The web algebra

Definition (M-P-T) Let S = (s1, . . . , sn). The web algebra KS is defined by KS :=

• u,v∈BS

uKv,

with

uKv := F(u∗v){n}.

Multiplication is defined as follows:

uKv1 ⊗ v2Kw → uKw

is zero, if v1 = v2. If v1 = v2, use the multiplication foam, e.g. 38

The multiplication foam

v w* v v* w* v

39

The multiplication foam

v w* v v* w* v

Lemma (M-P-T) The multiplication foam mv only depends on the isotopy type of v and has q-degree n, so KS is a graded algebra. 40

For any enhanced sign string, define KS := K

S.

41

For any enhanced sign string, define KS := K

S.

Define W(3ℓ) :=

• µS∈Λ(n,n)3

KS-pmodgr. 42

For any enhanced sign string, define KS := K

S.

Define W(3ℓ) :=

• µS∈Λ(n,n)3

KS-pmodgr. I will explain that categorified Howe duality implies Proposition (M-P-T) K0(W(3ℓ)) ⊗Z[q,q−1] C(q) ∼ = W(3ℓ). 43

Khovanov and Lauda’s categorification of ˙ U(sln)

Definition (Khovanov-Lauda) The 2-category U(sln) consists of

• bjects: λ ∈ Zn−1.

a 1-morphism from λ to λ′ is a formal finite direct sum of 1-morphisms Ei1λ{t} = 1λ′Ei1λ{t} for any t ∈

λ′ = λ + iX. The 2-morphisms are Q-linear combinations of composites

• f:

44

The generating 2-morphisms

For any i, the identity 1Ei1λ{t} 2-morphism is represented as · · · i1 i2 im i1 i2 im λ λ + iX The strand labelled iα is oriented up if εα = + and oriented down if εα = −. 45

The generating 2-morphisms

For each λ ∈ Zn−1 the 2-morphisms

• i
• λ

λ + iX

• i
• λ

λ + iX

• i

j

λ

• i

j

λ i · i i · i −i · j −i · j

• i

λ

• i

λ

• i

λ

• i

λ 1 + λi 1 − λi 1 + λi 1 − λi 46

The relations on 2-morphisms

The one color relations:

i) Planar isotopies ii) Nil-Hecke relations such as (recall E2

i = [2]E(2) i

)

• λ

i i

=

• i

i

λ −

• i

i

λ iii) All dotted bubbles of negative degree are zero and a dotted bubble of degree zero equals ±1 iv) (Recall EiFi1λ = FiEi1λ + [λi]1λ)

• i

i

λ λ =

• λ

i i

−

λi−1

• f=0

f

• g=0

λ

• f−g
• λi−1−f

i

• −λi−1+g

47

More color relations, e.g: for i = j (recall EiE−j = E−jEi)

• λ

i j

=

• λ

i j

• λ

i j

=

• λ

i j

48

More color relations, e.g: for i = j (recall EiE−j = E−jEi)

• λ

i j

=

• λ

i j

• λ

i j

=

• λ

i j

Theorem (Khovanov-Lauda) Let ˙ U(sln) be the Karoubi envelope of U(sln). Then K0( ˙ U(sln)) ⊗Z[q,q−1] C(q) ∼ = ˙ U(sln). 49

gln-weights instead of sln-weights

Idea (M-Stoˇ si´ c-Vaz): i) Categorify ˙ U(gln) by taking U(sln) and changing to gln-weights. ii) Then define a 2-category S(n, d) by modding out by the ideal generated by all diagrams with regions labeled with weights not in Λ(n, d). 50

gln-weights instead of sln-weights

Idea (M-Stoˇ si´ c-Vaz): i) Categorify ˙ U(gln) by taking U(sln) and changing to gln-weights. ii) Then define a 2-category S(n, d) by modding out by the ideal generated by all diagrams with regions labeled with weights not in Λ(n, d). Theorem (M-Stoˇ si´ c-Vaz) Let ˙ S(n, d) be the Karoubi envelope of S(n, d). Then K0( ˙ S(n, d)) ⊗Z[q,q−1] C(q) ∼ = Sq(n, d). 51

Projections

Lemma There exists a full and essentially surjective 2-functor Ψn,d : U(sln) → S(n, d) which categorifies the projection ψn,d : ˙ U(sln) → Sq(n, d). 52

The cyclotomic KLR algebras

Let λ ∈ Λ(n, n)+. The finite-dimensional cyclotomic KLR-algebra Rλ is defined by taking all downward diagrams in S(n, n)1λ and modding out by the ideal generated by

• i1 i2 i3

im λm

λ 53

The cyclotomic KLR algebras

Let λ ∈ Λ(n, n)+. The finite-dimensional cyclotomic KLR-algebra Rλ is defined by taking all downward diagrams in S(n, n)1λ and modding out by the ideal generated by

• i1 i2 i3

im λm

λ Definition Vλ := Rλ-pmodgr. 54

The cyclotomic quotient theorem

The following result was conjectured by Khovanov and Lauda in 2008: Theorem (Brundan-Kleshchev, Lauda-Vazirani, Webster, Kang-Kashiwara,...) There is a graded categorical action of U(sln) on Vλ and K0(Vλ) ⊗Z[q,q−1] C(q) ∼ = Vλ as ˙ U(sln)-modules. 55

Universality

Rouquier’s showed that, in a certain sense, Rλ is the universal categorification of Vλ. Theorem (Rouquier) Let V be any additive idempotent complete category, which allows an integrable graded categorical action by U(sln). Suppose Vh is a highest weight object in V of weight λ, i.e 1µVh ∼ = δλ,µVh, Vh is killed by E+i1λ, for all i ∈ I, and EndV(Vh) ∼ = C. Suppose also that any object in V is a direct summand of XVh, for some object X ∈ U(sln). There exists an equivalence of categorical U(sln) representations Φ: Vλ → V. 56

The categorical action

We define a categorical action of S(n, n) on W(3ℓ). 57

The categorical action

We define a categorical action of S(n, n) on W(3ℓ). On objects: use ϕ: Sq(n, n) → End

• W(3ℓ)
• 58

The categorical action

We define a categorical action of S(n, n) on W(3ℓ). On objects: use ϕ: Sq(n, n) → End

• W(3ℓ)
• On morphisms: we give a list of the foams associated to the

generating morphisms of S(n, n). Warning: facets labeled 0 or 3 have to be removed. 59

The categorical action on morphisms

• i,λ

→

• λi

λi+1

• i,λ

→

• λi

λi+1

• i,i,λ →

−

• λi

λi+1

60

The categorical action on morphisms

• i,i+1,λ →

(−1)λi+1

λi λi+1 λi+2

• i+1,i,λ →

λi λi+1 λi+2

• i,j,λ →
• λi

λi+1 λj λj+1

61

The categorical action on morphisms

• j,i,λ →
• λi

λi+1 λj λj+1

• i,λ

→

λi λi+1

• i,λ

→ (−1)⌊ λi

2 ⌋+⌈ λi+1 2

⌉ λi λi+1

62

The categorical action on morphisms

• i,λ → (−1)⌈ λi

2 ⌉+⌊ λi+1 2

⌋ λi λi+1

i,λ →

λi λi+1

63

The categorical action is OK

Proposition (M-P-T) The foams on the previous slides determine a well-defined graded categorical action of S(n, n) on W(3ℓ). 64

Harvest time

Note that we can pull back the categorical action on W(3ℓ) via Ψn,n : U(sln) → S(n, n). By Rouquier’s universality theorem, we get Theorem (M-P-T) There exist an equivalence of U(sln) 2-representations Φ: V(3ℓ) → W(3ℓ). 65

Harvest time

Note that we can pull back the categorical action on W(3ℓ) via Ψn,n : U(sln) → S(n, n). By Rouquier’s universality theorem, we get Theorem (M-P-T) There exist an equivalence of U(sln) 2-representations Φ: V(3ℓ) → W(3ℓ). Corollary (M-P-T) K0(Φ): K0(V(3ℓ)) ⊗Z[q,q−1] C(q) → K0(W(3ℓ)) ⊗Z[q,q−1] C(q) is an Sq(n, n)-module isomorphism. 66

Harvest time

Checking all the definitions, we get V(3ℓ)

γ(3ℓ)

− − − − → K0(V(3ℓ)) ⊗Z[q,q−1] C(q)

ϕ

 

• K0(Φ)

 

• W(3ℓ)

ψ

− − − − → K0(W(3ℓ)) ⊗Z[q,q−1] C(q) . Corollary (M-P-T) We have WS ∼ = K0(KS) ⊗Z[q,q−1] C(q). 67

The End

THANKS!!! 68