The sl 3 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 Intertwiners sl k -Webs � U q ( sl k ) -Tensors � � � ���������������� � � � � � � � � � Kauff, Kup, MOY � Resh-Tur � � sl k -knot polynomials 3
� � � � � Diagrammatic categorification Intertwiners sl k -Webs � U q ( sl k ) -Tensors � � � ���������������� � � � � � � � � � Kauff, Kup, MOY � Resh-Tur � � sl k -knot polynomials ??? sl k -Foams/MF sl k -Cycl string diags � � � ����������������� � � � � � � � � � � Khov, Khov-Roz � Webster � � � sl k -Knot homologies 4
� � Howe dual picture Howe duality U q ( sl n ) -Irrep sl k -Webs � � � � � � � � � � � � � � � � � ������������� � � � � � � � � � Kauff, Kup, MOY Lusz � � sl k − B n -Action 5
� � � � Howe dual picture Howe duality U q ( sl n ) -Irrep sl k -Webs � � � � � � � � � � � � � � � � � ������������� � � � � � � � � � Kauff, Kup, MOY Lusz � � sl k − B n -Action Howe 2-duality sl k -Foams/MF sl n -Cycl KLR-algebra � � � � � � � � � � � � � � � � � � ����������������� � � � � � � � � � � Khov, Khov-Roz � Chuang-Rouq � � sl k − B n -2-Action 6
Skew Howe duality The natural actions of GL k and GL n on Λ p � C k ⊗ C n � are Howe dual ( skew Howe duality ). 7
Skew Howe duality The natural actions of GL k and GL n on Λ p � C k ⊗ C n � are Howe dual ( skew Howe duality ). This implies that � Λ p 1 � C k � ⊗ Λ p 2 � C k � ⊗ · · · ⊗ Λ p n � C k �� ∼ Inv SL k = W ( p 1 , . . . , p n ) , where W ( p 1 , . . . , p n ) denotes the ( p 1 , . . . , p n ) -weight space of the irreducible GL n -module W ( k ℓ ) , if n = kℓ . 8
First things first Let’s q -deform this (for k = 3 ) 9
Kuperberg’s sl 3 -webs Example (0.1) Any web can be obtained from the following elementary webs by gluing and disjoint union: (0.2) 10
Let S = ( s 1 , . . . , s n ) be a sign string . Definition (Kuperberg) W S := C ( q ) { w | ∂w = S } /I S where I S is generated by: = [3] (0.3) = [2] (0.4) = + (0.5) 11
Let S = ( s 1 , . . . , s n ) be a sign string . Definition (Kuperberg) W S := C ( q ) { w | ∂w = S } /I S where I S is generated by: = [3] (0.3) = [2] (0.4) = + (0.5) From (0.3), (0.4) and (0.5) it follows that any w ∈ W S is a linear combination of non-elliptic webs (no circles, digons or squares). The latter form a basis, B S . 12
Define w ∗ by w w* (0.6) Define uv ∗ by u v* (0.7) Define v ∗ u by v* u (0.8) 13
Representation theory of U q ( sl 3 ) Given S = ( s 1 , . . . , s n ) , we define V S = V s 1 ⊗ · · · ⊗ V s n . V + is the fundamental U q ( sl 3 ) representation and V − its dual + ∼ V − := V ∗ = V + ∧ V + . 14
Representation theory of U q ( sl 3 ) Given S = ( s 1 , . . . , s n ) , we define V S = V s 1 ⊗ · · · ⊗ V s n . V + is the fundamental U q ( sl 3 ) representation and V − its dual + ∼ V − := V ∗ = V + ∧ V + . Webs correspond to intertwiners Theorem (Kuperberg) W S ∼ = Hom( C ( q ) , V S ) ∼ = Inv( V S ) . B S is called the web basis of Inv( V S ) . 15
The general and special linear quantum groups Definition i) U q ( gl n ) is generated by K ± 1 1 , . . . , K ± 1 n , E ± 1 , . . . , E ± ( n − 1) , subject to ( α i = ε i − ε i +1 = (0 , . . . , 1 , − 1 , . . . , 0) ∈ Z n − 1 ): K i K − 1 = K − 1 K i K j = K j K i K i = 1 i i K i K − 1 i +1 − K − 1 K i +1 i E i E − j − E − j E i = δ i,j q − q − 1 K i E ± j = q ± ( ε i ,α j ) E ± j K i + some extra relations we won’t need today ii) U q ( sl n ) ⊆ U q ( gl n ) is generated by K i K − 1 i +1 and E ± i . 16
Idempotented quantum groups Definition (Beilinson-Lusztig-MacPherson) For each λ ∈ Z n , adjoin an idempotent 1 λ and add the relations 1 λ 1 µ = δ λ,ν 1 λ E ± i 1 λ = 1 λ ± α i E ± i K i 1 λ = q λ i 1 λ . Define � ˙ U ( gl n ) = 1 λ U q ( gl n )1 µ . λ,µ ∈ Z n Define ˙ U ( sl n ) similarly by adjoining idempotents 1 µ to U q ( sl n ) for µ ∈ Z n − 1 . 17
A finite-dimensional semi-simple quotient Lemma (Doty-Giaquinto) The q-Schur algebra S q ( n, d ) is generated by 1 λ , for λ ∈ Λ( n, d ) , and E ± i , for i = 1 , . . . , n − 1 , such that 1 λ 1 µ = δ λ,µ 1 λ � 1 λ = 1 λ ∈ Λ( n,d ) E ± i 1 λ = 1 λ ± α i E ± i � E i E − j − E − j E i = δ ij [ λ i − λ i +1 ]1 λ λ ∈ Λ( n,d ) 18
Back to q -skew Howe duality Definition An enhanced sign sequence is a sequence S = ( s 1 , . . . , s n ) with s i ∈ {∅ , − 1 , 1 , ×} , for all i = 1 , . . . n . The corresponding weight µ = µ S ∈ Λ( n, d ) is given by the rules 0 if s i = ∅ 1 if s i = 1 µ i = . 2 if s i = − 1 3 if s i = × Let Λ( n, d ) 3 ⊂ Λ( n, d ) be the subset of weights with entries between 0 and 3 . 19
Let n = d = 3 ℓ . For any enhanced sign string S , we define � S by deleting the entries equal to ∅ or × . 20
Let n = d = 3 ℓ . For any enhanced sign string S , we define � S by deleting the entries equal to ∅ or × . We define W S := W � S , B S := B � S . and 21
Let n = d = 3 ℓ . For any enhanced sign string S , we define � S by deleting the entries equal to ∅ or × . We define W S := W � S , B S := B � S . and Definition Define � W (3 ℓ ) := W S . µ S ∈ Λ( n,n ) 3 22
The action Define � � ϕ : S q ( n, n ) → End C ( q ) W (3 ℓ ) 1 λ �→ λ 1 λ 2 λ n λ i ± 1 λ i +1 ∓ 1 E ± i 1 λ �→ � �� �� � � � � � � � � � � � � � �� �� � � � � � � λ i − 1 λ i +1 λ i +2 λ 1 λ i λ n 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 3 1 E +1 1 (22) �→ 2 2 2 0 2 E − 2 E +1 1 (121) �→ 1 2 1 24
The isomorphism from q -skew Howe duality Recall S q ( n, n )1 (3 ℓ ) / [ µ > (3 ℓ )] ∼ = V (3 ℓ ) . Lemma The map ϕ gives rise to an isomorphism ϕ : V (3 ℓ ) → W (3 ℓ ) of S q ( n, n ) -modules. Note that the empty web w h := w (3 ℓ ) , which generates W ( × k , ∅ 2 k ) ∼ = C ( q ) , is a highest weight vector. 25
Please, fasten your seat belts! Let’s categorify everything 26
sl 3 Foams Consider formal C -linear combinations of isotopy classes of singular cobordisms, e.g. the zip and unzip : 27
sl 3 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
sl 3 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 = (3 D, NC, S, Θ) and the closure relation : 29
Khovanov’s local relations: ℓ = (3 D, 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 0 else (0.12) The relations in ℓ suffice to evaluate any closed foam! 30
The category of foams Let Foam 3 be the category of webs and foams. 31
The category of foams Let Foam 3 be the category of webs and foams. Other relations in Foam 3 are: = − (Bamboo) = − (RD) 32
More relations in Foam 3 = 0 (Bubble) = − (DR) = − − (SqR) 33
More relations in Foam 3 + + = 0 + + = 0 (Dot Migration) = 0 34
The grading The q -grading of a foam U is defined as q ( U ) := χ ( ∂U ) − 2 χ ( U ) + 2 d + b. This makes Foam 3 into a graded category. 35
Foam homology Definition The foam homology of a closed web w is defined by F ( w ) := Foam 3 ( ∅ , w ) . 36
Foam homology Definition The foam homology of a closed web w is defined by F ( w ) := Foam 3 ( ∅ , w ) . F ( w ) is a graded complex vector space, whose q -dimension can be computed by the Kuperberg bracket : � � w ∐ = [3] � w � 1 � � = [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 = ( s 1 , . . . , s n ) . The web algebra K S is defined by � K S := u K v , u,v ∈ B S with u K v := F ( u ∗ v ) { n } . Multiplication is defined as follows: u K v 1 ⊗ v 2 K w → u K w is zero, if v 1 � = v 2 . If v 1 = v 2 , use the multiplication foam , e.g. 38
The multiplication foam w* w* v v* v v 39
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