sl 3 -web bases, categorification and link invariants

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1 sl 3 -web bases, categorification and link invariants Daniel Tubbenhauer Joint work with Marco Mackaay and Weiwei Pan November 2013 F F (2) F F (2) 3 F (2) Daniel Tubbenhauer November / 40

2 1 Categorification What is categorification? 2 Webs and representations of U q (sl 3 ) sl 3 -webs Intermediate crystals q-skew Howe duality and growing of webs Knot polynomials and webs 3 The Categorification An algebra of foams Growing of foams Foamation Knot homologies and foams Daniel Tubbenhauer November / 40

3 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 structure S and try to find a category-based structure C such that S is just a shadow of C. Categorification, which can be seen as remembering or inventing information, comes with an inverse process called decategorification, which is more like forgetting or identifying. Note that decategorification should be easy. Daniel Tubbenhauer What is categorification? November / 40

4 Exempli gratia Examples of the pair categorification/decategorification are: Bettinumbers of a manifold M categorify decat=rank( ) Homology groups Polynomials in Z[q,q 1 ] categorify decat=χ gr( ) complexes of gr.vs The integers Z categorify decat=k 0( ) K vector spaces An A module categorify decat=k 0 ( ) ZA additive category Usually the categorified world is much more interesting. Today decategorification = Grothendieck group! Daniel Tubbenhauer What is categorification? November / 40

5 Kuperberg s sl 3 -webs Definition(Kuperberg) A sl 3 -web w is an oriented trivalent graph, such that all vertices are either sinks or sources. The boundary w of w is a sign string S = (s 1,...,s n ) under the convention s i = + iff the orientation is pointing in and s i = iff the orientation is pointing out (we also need 0,3 later - but they are not drawn). Example Daniel Tubbenhauer sl 3 -webs November / 40

6 Kuperberg s sl 3 -webs Definition(Kuperberg) The C(q)-web space W S for a given sign string S = (±,...,±) is generated by {w w = S}, where w is a web, subject to the relations = [3] = [2] = + Here [a] = qa q a q q 1 = q a 1 +q a 3 + +q (a 1) is the quantum integer. Daniel Tubbenhauer sl 3 -webs November / 40

7 A connection to knot theory Let L D be a link projection. Assign to it a polynomial P 3 (L D ) (it is in Z[q,q 1 ]) by local and inductive rules as follows. P 3 ( ) = q 2 P 3 ( ) q 3 P 3 ( ) (recursion rule 1). P 3 ( ) = q 2 P 3 ( ) q 3 P 3 ( ) (recursion rule 2). The Kuperberg relations. Theorem(Murakami, Ohtsuki and Yamada) The polynomial P 3 ( ) is uniquely determined by the rule and a link invariant. Moreover, it agrees with the so-called HOMFLY-PT polynomial under a certain substitution of variables and normalization. For example P 3 ( ) = q 2 P 3 ( ) q 3 P 3 ( ) = q 2 [3] 2 q 3 [2][3] = [3]. Daniel Tubbenhauer sl 3 -webs November / 40

8 Representation theory of U q (sl 3 ) A sign string S = (s 1,...,s n ) corresponds to tensors V S = V s1 V sn, where V + is the fundamental U q (sl 3 )-representation and V is its dual, and webs correspond to intertwiners. Theorem(Kuperberg) W S = HomUq (sl 3)(C(q),V S ) = Inv Uq (sl 3)(V S ) In fact, the so-called spider category of all webs modulo the Kuperberg relations is equivalent to the representation category of U q (sl 3 ). As a matter of fact, the sl 3 -webs without internal circles, digons and squares form a basis B S, called web basis, of W S! Daniel Tubbenhauer sl 3 -webs November / 40

9 Kuperberg s sl 3 -webs Example wt=0 wt= 1 Webs can be coloured with flow lines. At the boundary, the flow lines can be represented by a state string J. By convention, at the i-th boundary edge, we set j i = ±1 if the flow line is oriented downward/upward and j i = 0, if there is no flow line. So J = (0,0,0,0,0,+1, 1) in the example. Given a web with a flow w f, attribute a weight to each trivalent vertex and each arc in w f and take the sum. The weight of the example is 4. Daniel Tubbenhauer sl 3 -webs November / 40

10 Representation theory of U q (sl 3 ) Theorem(Khovanov-Kuperberg) Pairs of sign S and a state strings J correspond to the coefficients of the web basis relative to tensors of the standard basis {e± 1,e0 ±,e+1 ± } of V ±. Example wt= wt= 4 w S = +(q 2 +q 4 )(e 0 + e 0 e 0 + e 0 e 0 + e +1 + e 1 + )±. Daniel Tubbenhauer sl 3 -webs November / 40

11 What kind of basis is B S? Theorem(Khovanov-Kuperberg) Given (S,J), we have (with v = q 1 and es J = ej1 s 1 es jn n ) w J S = ej S + J <J c(s,j,j )e J S for c(s,j,j ) N[v,v 1 ]. In general we have B S dcan(w S ), but the web basis is bar-invariant. Theorem(MPT) We proved, by categorification, that the change-of-basis matrix from Kuperberg s web basis B S to the dual canonical basis dcan(w S ) is unitriangular. Question: The web basis B S is a somehow special basis of W S. But it is not the dual canonical. So what kind of basis is it? Daniel Tubbenhauer sl 3 -webs November / 40

12 The quantum algebra U q (sl d ) Definition For d N >1 the quantum special linear algebra U q (sl d ) is the associative, unital C(q)-algebra generated by K ±1 i and E i and F i, for i = 1,...,d 1, subject to some relations (that we do not need today). Definition(Beilinson-Lusztig-MacPherson) For each λ Z d 1 adjoin an idempotent 1 λ (think: projection to the λ-weight space!) to U q (sl d ) and add some relations, e.g. 1 λ 1 µ = δ λ,ν 1 λ and K ±i 1 λ = q ±λi 1 λ (no K s anymore!). The idempotented quantum special linear algebra is defined by U(sl d ) = λ,µ Z d 1 1 λ U q (sl d )1 µ. Daniel Tubbenhauer Intermediate crystals November / 40

13 Intermediate crystals Let d = 3l and let V Λ be the irreducible U(sl d )-module of highest weight Λ = (3 l ). Kashiwara-Lusztig s lower global crystal (or canonical) basis can(v Λ ) = {b T T Std(3 l )} has nice properties, but is in very hard to find. Leclerc and Toffin have defined an intermediate crystal basis B Λ of V Λ by an explicit algorithm that can be used to compute can(v Λ ) inductively, i.e. B Λ has some nice properties, but is still trackable enough to be written down. Example (with l = 3) T = LT(T) = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3 v Λ. Daniel Tubbenhauer Intermediate crystals November / 40

14 Sitting in-between {b T } and {x T } The intermediate crystal basis sits in-between the canonical can(v Λ ) and the tensor basis {x T Λ l q (Cd q ) 3 V Λ T Col(3 l )}. Theorem(Leclerc-Toffin) We have (for T Col(3 l ), T Std(3 l )) LT(T) = x T + T T α T (v)x T and b T = LT(T)+ T T β T T(v)LT(T ) with certain α T (v) N[v,v 1 ] and β T T(v) Z[v,v 1 ] (with v = q 1 ). Moreover, the intermediate crystal basis is bar-invariant. We have seen this before! Daniel Tubbenhauer Intermediate crystals November / 40

15 An instance of q-skew Howe duality The commuting actions of U(sl d ) and U(sl 3 ) on Λ q(c d q) 3 = Λ q (C d C 3 ) = Λ q(c 3 q) d introduce an U(sl d )-action on Λ q (C3 q ) d and an U(sl 3 )-action on Λ q (Cd q ) 3. Here l Λ q(c l q) = Λ k q(c l q) k=1 and all the Λ k q (Cl q ) are irreducible U(sl l )-representations. Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

16 The sl 3 -webs form a U(sl d )-module We defined an action φ of U(sl d ) on W (3 l ) = S Λ(n,n) 3 W S by 1 λ λ 1 λ 2 λ d λ i±1 λ i+1 1 E i 1 λ, F i 1 λ λ 1 λ i 1 λ i λ i+1 λ i+2 λ d We use the convention that vertical edges labeled 1 are oriented upwards, vertical edges labeled 2 are oriented downwards and edges labeled 0 or 3 are erased. Think: 0,3 indicates the trivial, 1 the V + and 2 the V -representation. Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

17 Exempli gratia 3 1 E 1 1 (22) F 2 E 1 1 (121) Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

18 An intermediate crystal basis Proposition(MT) The Kuperberg web basis B S is Leclerc-Toffin s intermediate crystal basis under q-skew Howe duality, i.e. LT(T) = {F (rs) i s (No K s and E s anymore!) F (r1) i 1 v 3 l T Std(3 l )} shd ws. J Corollary The change-of-basis matrix from Kuperberg s web basis B S to the dual canonical basis dcan(w S ) is unitriangular, it is bar-invariant and w J S = e J S + J <J c(s,j,j )e J S for c(s,j,j ) N[v,v 1 ]. We also defined a growth algorithm for flows - but we do not need it today. Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

19 Exempli gratia Example w = T = From T we obtain the string LT(T) = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3. Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

20 Exempli gratia LT(T)v 3 3 = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3 v F F F (2) F F F F F F (2) F (2) F (2) Daniel Tubbenhauer q-skew Howe duality and growing of webs November / 40

21 Crossings as F (i) j Define the following braiding operators B j (and similar ones for ). : (q 2 F j F j+1 q 3 F j+1 F j )v 110 : (q 2 F (2) j+1 F(2) j q 3 F j+1 F (2) j F j+1 )v 210 : (q 2 F j+1 F j q 3 F j F j+1 )v 332 : (q 2 F j+1 F j q 3 F j F j+1 )v 120 Example : (q 2 F j+1 F j q 3 F j F j+1 )v 120 gives (for j = 1) q q Daniel Tubbenhauer Knot polynomials and webs November / 40

22 Links as F (i) j Proposition(T) Let L D be a diagram of a link. Then (under q-skew Howe duality): LT(L D )v h = k F (i k) j k v h, with F = F, F = B for some highest weight vector v h. That is, L D can be realized combinatorial as sums of certain tableaux. Observation(T) Since the above construction agrees (up to some normalization/shifts) with the construction of the sl 3 -link polynomial (MOY-calculus), the combinatoric of tableaux can be used to calculate these invariants (note: works for all n > 1). Wish(T) There should be a way to get the colored sl 3 -link polynomial from this approach as well, since they correspond to arbitrary tensor products instead of V +,V. Daniel Tubbenhauer Knot polynomials and webs November / 40

23 Exempli gratia (The Hopf link - part one) LT(L D )v 3 3 = F 5 F 3 F 2 F 1 B 3 B 2 F (2) 5 F (2) 4 F (2) 3 F (2) 1 F (3) 2 v F F F F B B F (2) F (2) F (2) F (2) F (3) The corresponding four webs can always be evaluated using tableaux. Daniel Tubbenhauer Knot polynomials and webs November / 40

24 Please, fasten your seat belts! Let s categorify everything! Daniel Tubbenhauer Knot polynomials and webs November / 40

25 sl 3 -foams A pre-foam is a cobordism with singular arcs between two webs. Composition consists of placing one pre-foam on top of the other. The following are called the zip and the unzip respectively. They have dots that can move freely about the facet on which they belong, but we do not allow dot to cross singular arcs. A foam is a formal C-linear combination of isotopy classes of pre-foams modulo relations, e.g. α 1, (α,β,δ) = (1,2,0) or a cyclic permutation, β = 1, (α,β,δ) = (2,1,0) or a cyclic permutation, (Θ) δ 0, else. Daniel Tubbenhauer An algebra of foams November / 40

26 Involution on webs and foams Definition There is an involution on the webs and foams. That is for webs and for foams = A closed foam is a foam from to a closed web u v. Daniel Tubbenhauer An algebra of foams November / 40

27 The sl 3 -foam category Foam 3 is the category of foams, i.e. objects are webs w and morphisms are foams F between webs. The category is graded by the q degree deg q (F) = χ( F) 2χ(F)+2d +b, where d is the number of dots and b is the number of vertical boundary components. 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 (that is counting all flows on w and their weights). Daniel Tubbenhauer An algebra of foams November / 40

28 A higher connection to knot theory Let L D be a link projection. Assign to it a complex L D in the category of formal chain complexes of Foam 3 locally as follows. = 0 {2} d {3} 0. = 0 { 3} d { 2} 0. The differentials d are (un)zips. Tensor everything together. Theorem(Khovanov) The complex L D a link invariant. Moreover, it decategorifies to P 3 (L D ). For example = 0 {2} {3} 0. Daniel Tubbenhauer An algebra of foams November / 40

29 The sl 3 -web algebra Definition(MPT) Let S = (s 1,...,s n ). The sl 3 -web algebra K S is defined by K S = uk v, u,v B S with uk v = F(u v){n}, i.e. all foams: u v. Multiplication is defined as follows. uk v1 v2 K w u K w is zero, if v 1 v 2. If v 1 = v 2, use the multiplication foam m v, e.g. Daniel Tubbenhauer An algebra of foams November / 40

30 The sl 3 -web algebra Theorem(s)(MPT) The multiplication is well-defined, associative and unital. The multiplication foam m v has q-degree n. Hence, K S is a finite dimensional, unital and graded algebra. Moreover, it is a graded Frobenius algebra. Daniel Tubbenhauer An algebra of foams November / 40

31 Higher representation theory Moreover, for n = d = 3l we define W (3 l ) = µ s Λ(n,n) 3 W S on the level of webs and on the level of foams we define W (p) (3 l ) = K S (p)mod gr. µ s Λ(n,n) 3 With this constructions we obtain our first categorification result. Theorem(MPT) K 0 (W (3l )) Z[q,q 1 ] Q(q) = W (3l ) and K 0 (Wp (3 l ) ) Z[q,q 1 ] Q(q) = W (3l ). Daniel Tubbenhauer An algebra of foams November / 40

32 Categorification of the LT-algorithm As a remainder, the LT-algorithm gives LT(T) = x T + T T α T (v)x T. Thus, we need column-strict tableaux and 3-multitableaux (for x T and α T (v)). What do we expect to gain? Since Leclerc-Toffin also showed b T = LT(T)+ β T T(v)LT(T ) T T we expect that we get a method to compute the projective indecomposable of K S, since they should decategorify to the dual canonical basis. Daniel Tubbenhauer Growing of foams November / 40

33 A growth algorithm for foams Definition(T) Given a pair of a sign string and a state string (S,J), the corresponding 3-multipartition λ and two Kuperberg webs u,v B S that extend J to f u and f v receptively. We define a foam by Theorem(T) F λ T(ufu ), T(v = F fv ) σ }{{} u e( λ) }{{} Topology Idempotent d( λ) }{{} Dots F σ v }{{} Topology The growth algorithm for foams is well-defined, the only input data are webs and flows on webs, works inductively and gives a graded cellular basis of K S.. Daniel Tubbenhauer Growing of foams November / 40

34 Connection to U q (sl d ) Khovanov and Lauda s diagrammatic categorification of U q (sl d ), denoted U(sl d ), is also related to our framework! Roughly, it consist of string diagrams of the form i λ: E i E j 1 λ E j E i 1 λ {(α i,α j )}, j λ α i λ: F i 1 λ F i 1 λ {α ii } i with a weight λ Z n 1 and suitable shifts and relations like i λ = j i λ and j i λ = j i λ, if i j. j Daniel Tubbenhauer Foamation November / 40

35 sl 3 -foamation We define a 2-functor called foamation, in the following way. Ψ: U(sl d ) W (p) (3 l ) On objects: The functor is defined by sending an sl d -weight λ = (λ 1,...,λ d 1 ) to an object Ψ(λ) of W (p) (3 l ) by Ψ(λ) = S, S = (a 1,...,a l ), a i {0,1,2,3}, λ i = a i+1 a i, l a i = 3 l. i=1 On morphisms: The functor on morphisms is by glueing the ladder webs from before on top of the sl 3 -webs in W (3l ). Daniel Tubbenhauer Foamation November / 40

36 sl 3 -foamation On 2-cells: We define i,λ λ i λ i+1 i,λ λ i λ i+1 i,i,λ λ i λ i+1 i,i+1,λ ( 1) λi+1 λ i λ i+1 λ i+2 i+1,i,λ λ i λ i+1 λ i+2 And some others. Daniel Tubbenhauer Foamation November / 40

37 Everything fits Theorem(MPT) The 2-functor Ψ: U(sl d ) W (p) categorifies q-skew Howe duality. (3 l ) Corollary( Almost directly) We have ψ p ([D λ p]) = b λ under the isometry ψ p : K 0 (Wp (3 l ) ) Z[q,q 1 ] Q(q) W (3l ), that is projective covers Dp λ (who give a complete list of all projective, irreducible K S -modules) of the simple heads D λ of the cell modules C λ categorify the upper global crystal basis b λ of W (3 ). In principle, the D λ l p are computable from the extended growth algorithm. Daniel Tubbenhauer Foamation November / 40

38 Crossings as complexes In order to make sense of the minus signs from before we have to go to the category of complexes on the categorified level. That is, define higher braiding operators B j (and other, similar ones) = { 3} { 2} where the sign is categorified using the language of complexes and the q using degree shifts, e.g. { 3}. Daniel Tubbenhauer Knot homologies and foams November / 40

39 Exempli gratia (The Hopf link - part two) The Hopf link example from before will give a complex F F 4 F 3 F 3 F 2 Fv 3 2{5} : F 2 F 3 F 3 F 2 F F 4 F 3 F 2 F 3 Fv 3 2{4} : F 4F 3 F 3F 4 F F 3 F 4 F 2 F 3 Fv 3 2{5} : F 4F 3 F 3F 4 : F 2F 3 F 3F 2 F F 3 F 4 F 3 F 2 Fv 3 2{6} that, up to some degree conventions, agrees with sl 3 -Khovanov homology of L D. Here F = F (2) 5 F (2) 4 F (2) 3 F (2) 1 F (3) 2 and F = F 5 F 4 F 3 F 1. Wish(T) Since the above construction agrees (up to some normalization/shifts) with the construction of the sl 3 -link homologies, the higher combinatoric of tableaux should tell us how to calculate these invariants (note: this should work for all n > 1). Moreover, the colored sl 3 -link homologies should fit in this framework. Daniel Tubbenhauer Knot homologies and foams November / 40

40 There is still much to do... Daniel Tubbenhauer Knot homologies and foams November / 40

41 Thanks for your attention! Daniel Tubbenhauer Knot homologies and foams November / 40

sl 3 -web bases, intermediate crystal bases and categorification

sl 3 -web bases, intermediate crystal bases and categorification Daniel Tubbenhauer Joint work with Marco Mackaay and Weiwei Pan October 2013 F 1 1 1 2 2 F (2) 2 2 0 2 2 F 1 2 2 0 2 F (2) 3 F (2) 2 3 1