A categorification of the ribbon element in quantum sl(2)

We define a bicomplex whose Euler characteristic is the idempotented version of the ribbon element of quantum sl(2). We show that properties of this bicomplex descend to the centrality, invertibility and symmetries of the ribbon element after decategorification.

The objects ofU are natural numbers, interpreted as the integral weight lattice of sl 2 . The 1morphisms ofU are generated by E (a) 1 n t and F (b) 1 n t , for all a, b ∈ AE, n, t ∈ , which are lifts of the Lusztig divided powers. The 2-morphisms are -linear combinations of planar diagrams modulo local relations. The split Grothendieck group K 0 (U) satisfies K 0 (E (a) 1 n t ) = q t E (a) 1 n and K 0 (F (a) 1 n t ) = q t F (a) 1 n and coincides with the integral idempotented version of the quantum enveloping algebra U q (sl 2 ) constructed in [1]. On the other hand, there exists a universal knot invariant [14] (see also [7]) which takes values in the center of quantum sl 2 and dominates all colored Jones polynomials. This paper can be considered as a first step towards a categorification of the universal link invariant.
The most challenging problem of this program is to find a categorical equivalent of the R-matrix, associated by the universal invariant to a crossing in a link diagram. Here we resolve a simpler problem, we categorify the ribbon element, which is the universal invariant of a self-crossing. The bicomplex we construct in this paper is conjecturally an element of the Drinfeld center of Com(U ) and is related to a Serre functor on category O associated with the longest braid [16] by Conjecture 3. Here we denote by Kom(U ) the category of bicomplexes over the 2-categoryU and by Com(U) its homotopy version.
The ribbon element r is an element of the h-adic version U h (sl 2 ) of the quantized enveloping algebra U q (sl 2 ). It is given by where (q a ; q b ) k = (1 − q a )(1 − q a+b ) . . . (1 − q a+(k−1)b ). Thus, the idempotented version of r is
Recall that the ribbon element is central and invertible in U q (sl 2 ) with the inverse The definition of the bicomplex r1 n ∈ Kom(U ) categorifying the ribbon element is outlined as follows. First, we construct a bicomplex C •,• in Kom(U) where the 1-morphism C k,l : n → n inU (i.e. an object C k,l in the additive categoryU(n, n)) is defined by where W k = Span {w 1 , . . . , w k } with deg(w j ) = −2j. Note that C 00 = F (0) E (0) 1 n 0 ⊗ Λ 0 W 0 ∼ = 1 n . The differentials d V k,l , d H k,l are defined in Section 4. Figure 1 there shows the beginning of this bicomplex. The Euler characteristic of the bicomplex C •,• is equal to the sum part of (1.1) ∞ k=0 (−1) k q −kn−k (q −2 ; q −2 ) k F (k) E (k) 1 n .
Finally, we shift our bicomplex by − n 2 2 − n in q-degree and by [n/2, n/2] in homological bi-degree to obtain The q-degree shift − n 2 2 − n corresponds to the factor q − n 2 2 −n in (1.1). The homological degree shift [n/2, n/2] is added to make r1 n commute with the 1-morphisms inU up to homotopy equivalence (see Theorem 1). We use the convention that the Euler characteristic is not affected by a global homological shift.
The bicomplex r −1 1 n is given by inverting all arrows in (1.3), rotating diagrams representing differentials by 180 degree, and replacing C k,l with 1 C L k,l := F (k) E (k) 1 n kn + k ⊗ Λ lW k ,W k = Span {w 1 , . . . ,w k } where deg(w j ) = 2j. Finally, we have It is an easy check that the Euler characteristics of r1 n and r −1 1 n in Kom(U) coincide with r1 n and r −1 1 n , respectively. Let us list some properties of these bicomplexes, which follow directly from the definition.
• For each k, the horizontal complex C k,• = ⊕ l∈Z C k,l forms a k-dimensional cube and hence is bounded. • The total complex Tot(C •,• ) is a well-defined complex inU, i.e., for each integer p, k+l=p C k,l ∈ Ob(U ) is a finite direct sum. • The horizontal differentials are given by certain central elements in End(F (k) E (k) 1 n ) defined in Section 3 and generated by dots. Recall thatU is the Karoubi envelope of the 2-category U defined in [13]. The generators for 1morphisms in U are E1 n t and F 1 n t for n, t ∈ . The involutive 2-functors ω, σ and ψ, generating the symmetry group G = ( /2 ) 3 of U were also introduced in [13]. We recall these definitions and extend them toU in Section 10. Let us denote by G 1 := {1, σω} the subgroup of G.
Throughout this paper we adopt the notation rr −1 1 n for the tensor product of two complexes, i.e. we omit tensor sign, since on 1-morphisms ofU tensor product is just a concatenation.
Our main result is the following: (Invertibility) The bicomplexes rr −1 1 n and r −1 r1 n in l Kom(U) are homotopy equivalent to 1 n .
Note that since the bicomplexes r1 n and r −1 1 n are bounded from above and below respectively, anḋ U does not admit infinite direct sums, their tensor product rr −1 1 n does not belong to Kom(U). Instead we are using l Kom(U ) which is the inverse limit of the categories of bounded bicomplexes Kom b (U N ), whereU N is the Schur quotient ofU defined by setting 1 N +2 = 0 (see Section 12 for more details).
Properties listed in Theorem 1 are natural lifts of the properties of the ribbon element to higher categorical level. We would also expect the following to hold.

Conjecture 2. (Naturality)
The chain maps κ X and η X are natural, i.e. they commute with all 2-morphisms ofU.
(Decomposability) For n ≥ 0 the bicomplex r1 n is indecomposable in Kom(U), and the bicomplex ω(r1 −n ) is isomorphic to a direct sum of r1 n and a contractible complex. For n ≤ 0 the bicomplex ω(r1 −n ) is indecomposable in Kom(U ), and the bicomplex r1 n is isomorphic to a direct sum of ω(r1 −n ) and a contractible complex. For n = 0 the bicomplexes r1 n and ω(r1 −n ) are isomorphic.
Observe that we could use σ(r1 −n ) instead of ω(r1 −n ), since by Theorem 1 they are isomorphic. Let us comment on this conjecture. We expect the isomorphisms κ X and η X to be natural, and hence, to be defined for any X ∈ Com(U). This would imply that the bicomplexes r1 n and r −1 1 n belong to the Drinfeld center of Com(U) viewed as an additive monoidal category. Here we regard 1-morphisms in Com(U ) as objects of the monoidal category. The monoidal structure is given by composition of 1-morphisms and horizontal composition of 2-morphisms. The collection of chain maps κ X define then an invertible natural transformation κ : −r =⇒ r− between endofunctors of Com(U ) given by tensoring on the left and on the right with the complex r1 n for an appropriate n.
The first column of our bicomplex C 00 → C 10 → C 20 → . . . is an example of so-called Rickard complex introduced by Chuang and Rouquier in [5] and intensively studied by Cautis and Kamnitzer [4], [3]. The Rickard-Rouquier complex 1 −n T1 n categorifies the action of the Weyl group on the finite-dimensional representations and satisfies the braid relation. It can be defined as follows: where the differential are non-zero maps.
Conjecture 3 (Cautis). The total complex of the ribbon bicomplex Tot(r −1 1 n ) is homotopy equivalent to Note that the decategorified version of this conjecture holds. For n ≥ 0, the Euler characteristic coincides with r −1 1 n after multiplying with q n 2 /2+n . The case n ≤ 0 can be obtained similarly, after replacing n by −n and exchanging E's and F 's.
1.1. Strategy of the proof of Theorem 1. It is enough to check centrality on the generators. This is because any "chain group" of X is a composition of E's and F 's and the maps η E and η F are adjoint to κ F and κ E , respectively.
Here rE1 n is the bicomplex obtained by composing r1 n+2 to the left of E1 n . The differentials are those of r1 n+2 extended by identity on E1 n . The other bicomplexes are defined analogously.
To construct κ E we will proceed as follows. We will define an intermediate bicomplex (rE1 n ) ′ as an indecomposable summand of rE1 n , whose "chain groups" are . . , w k+1 }, the total q-degree shift − n 2 2 − 3n − 4 and the homological shift [n/2+1, n/2+1]. Then we show that Er1 n and rE1 n retract to (rE1 n ) ′ (Theorems 7.1, 5.3). Composing the corresponding homotopy equivalences we will get κ E . The construction of the chain maps between Er1 n , rE1 n and (rE1 n ) ′ , and the proofs of Theorems 7.1 and 5.3 is the most involved technical part of the paper.
To define κ F we use the invariance of r1 n under σω. Indeed, we have However, σω(Er1 n ) = σω(r)F 1 n+2 ≃ rF 1 n+2 and similarly, σω(rE1 n ) is isomorphic to F r1 n+2 . The proof of invertibility is based on the next theorem computing the action of the ribbon complex on the category of complexes over Flag N defined in [13] and the main result of [2]. Let us define the endofunctors r L N and r R N of Com(Flag N ) by tensoring with Γ N (r1 n ) on the left and right, respectively, i.e. r L N (X) = Γ N (r1 n )X. Analogously, the endofunctors (r −1 ) L N and (r −1 ) R N are defined by tensoring with Γ N (r −1 1 n ).
Theorem 5. For any natural number N , r L N and r R N are the identity endofunctors of Com(Flag N ) up to degree shift. Their inverses are (r −1 ) L N and (r −1 ) R N , respectively. The main result of [2] says thatU is the inverse limit of Flag 2-categories. Hence, we conclude that tensoring with r ±1 r ∓1 1 n is the inverse limit of the identity functor in Flag N , which is the identity endofunctor of l Com(U).
To prove symmetry, we construct a bicomplex r1 n which is 1) isomorphic to r1 n in Kom(U), and 2) invariant under σω. This is done in Section 11. Then, given the isomorphism H : r1 n → r1 n , the composition σω(H −1 ) • H : r1 n → σω(r1 n ) is the required isomorphism.
The paper is organized as follows. After some preliminaries, we define the central elements c λ indexed by partitions, r1 n , r −1 1 n and the intermediate bicomplex (rE1 n ) ′ . The next four Sections are devoted to the definition of the chain maps and homotopies and to the proofs of Theorems 7.1, 5.3 and Theorem 1 (Centrality). After that we recall the definitions of the symmetry 2-functors and define the images of r1 n under those symmetries. Section 11 is devoted to the construction of r1 n . In the last section we prove Theorem 5 and Theorem 1 (Invertibility).
In Appendix we collect identities needed for the proofs.

General facts
2.1. Definitions and conventions. We refer to [13] and [11] for the definitions of the 2-categories U and its Karoubi envelopeU. The 2-morphisms in these 2-categories are given by diagrams modulo some local relations. The right most region in all our diagrams is labeled by n. For any 1-morphism x ∈ HomU (n, m), we denote by Dot(x) ⊂ EndU (x) the subspace of its 2-endomorphisms generated by dots. A thick line labeled with a positive integer k denotes the identity 2-morphism of E (k) : if it is oriented upwards; and the identity 2-morphism of F (k) := (F k , ′ k ) k(k−1) 2 iṅ U otherwise, where k is the idempotent defined in [11] and ′ k is its image under 180-degree rotation. The case k = 1 will be represented by a thin line without any label for a better visibility.
In this paper, for any 2-category C, we denote by Kom(C) the 2-category of bicomplexes over the 2-category C. The objects of Kom(C) coincide with objects of C, 1-morphisms are bicomplexes of 1morphisms in C, and 2-morphisms are chain maps, constructed from 2-morphisms in C. Let Com(C) be the 2-category with the same objects and 1-morphisms as Kom(C) but whose 2-morphisms are chain maps up to homotopy. We will denote by Kom b (C) and Com b (C) corresponding bounded versions.
The degree of a 2-morphism in U is defined as degree of the target minus degree of the source plus degree of the diagram. Remark. From the four equalities including g it is enough to show that f g = 1 and hd + dh = 1 − gf . The other two equalities(dg = gd ′ , hg = 0) follow from them.
Let (C, d V , d H ) and (C ′ , d ′V , d ′H ) be two bicomplexes. We say that the second bicomplex is a strong deformation retract of the first one if there exist Remark. The equalities d H g = gd ′H , d V g = gd ′V , h H g = 0 and h V g = 0 follow from the others.
2.3. Symmetric functions. Let us denote by S k the symmetric group and A k = [x 1 , . . . , x k ] S k the ring of symmetric polynomials. Let A be the ring of symmetric functions, defined as the inverse limit of the system (A k ) k∈AE . For a partition λ = (λ 1 , λ 2 , . . . , λ a ) with λ 1 ≥ λ 2 ≥ · · · ≥ λ a ≥ 0 let |λ| := a i=1 λ i . We denote by P (a) the set of all partitions λ with at most a parts (i.e. with λ a+1 = 0). Moreover, the set of all partitions (i.e. the set P (∞)) we denote simply by P .
The dual (conjugate) partition of λ is the partition λ t = (λ t 1 , λ t 2 , . . .) with λ t j = ♯{i|λ i ≥ j} which is given by reflecting the Young diagram of λ along the diagonal.
The Schur polynomials {s λ | λ ∈ P (k)} form a basis of A k , as well as Schur functions {s λ | λ ∈ P } is a base of A. The multiplication in this basis is given by the following formula where N λ µν are the Littlewood-Richardson coefficients. The elementary symmetric functions The ring A has a natural Hopf algebra structure with comultiplication

The center of the 2-categoryU
After recalling the general definition of a center for any linear category, we construct central elements inU(n, m).

3.1.
Center of a category. For a linear category C, the center Z(C) of C is the ring of endo-natural transformations on the identity functor 1 C : C −→ C. Thus, an element σ of Z(C) is a collection of endomorphisms σ x : x −→ x for objects x in C such that we have for any morphisms f : x −→ y in C. Multiplication of two elements σ and τ in Z(C) is defined by It is easily seen that Z(C) is commutative. We call σ a central element of C.
Let C be a linear 2-category. For each pair (x, y) of objects in C, one can consider the center Z(C(x, y)) of the category C(x, y) whose objects are the 1-morphisms between x and y and morphisms are 2-morphisms. The center Z(C(x, y)) is a commutative ring.

Central elements inU . We have natural ring homomorphisms
In this subsection, we define for each 1-morphism f : n −→ m inU a ring homomorphism We also adopt the notation (c λ ) f for c f (s λ ) and draw: Note that, for f = E a 1 n and f = F a 1 n we have and . . .
The following proposition is the direct consequence of the definitions.
In particular, for λ ∈ P d , and Proof. We need to prove that for every 2-morphism α : f → g inU(n, m) we have Splitting the thick lines and moving the dots as follows . . .
we see that it is enough to prove the proposition for 2-morphisms of U, generated by dots, crossings and turns. For dots the proposition is clear.
Note that (c 1 d ) E 2 1n = 0 only for d = 1 or d = 2 and Using the NilHecke relations we can easily check that both upper 2-morphisms commute with the crossing. The same is true for downward oriented arrows. Similarly, we have We see that by multiplying with the turn from below, we get 0, which coincides with (c 1 d ) 1n = 0. The other turns can be proved similarly.

4.
Definitions of r1 n , r −1 1 n and (rE1 n ) ′ 4.1. Ribbon bicomplex. As it was already mentioned in Introduction, the bicomplex r1 n categorifying the ribbon element has "chain groups" with the total q-degree shift − n 2 2 − n and homological shift [n/2, n/2]. Here note that for an additive category C and a free abelian group G of finite rank one can construct a functor − ⊗ G : C → C.
The horizontal differential d H k,l : To define the vertical differential we proceed as follows. Consider the linear map where 1 denotes the identity 2-morphism of E (k) 1 n . This map induces an algebra homomorphism be its degree l part. Then the vertical differential is In what follows to simplify the notation we will often omit the tensor sign between the graphical part and Λ l W k . Also, we will sometimes use a bullet labeled with a symmetric function instead of a box.
Before giving the proof, let us show how the beginning of this bicomplex looks like.

Figure 1 The ribbon bicomplex
Proof. The first formula (d H ) 2 = 0 is immediate from the definition, since c ∧ c = 0. For the next equation we have to check that the following square anticommutes: After moving all c j down, by using their centrality, the anticommutativity reduces to where the lower index 2 indicates the stand where this endomorphism acts. This is easily seen to hold after accomplishing the second summand for j = 0 to zero by means of the formula The formula (4.2) follows from the fact that e k+1 ∈ Dot(E (k) 1 n ) (or e k+1 ∈ Dot(F (k) 1 n )) is zero and for any d, r and s, since Let us explain this in more details. The formula (4.1) contains one summand without dots, let us call it A, and the other summands called B. Putting A on top of d V k−1,l is zero by the first identity printed above. Now any square resulting from putting B on top of d V k−1,l gives rise to the second identity with t = k − i j and s = k − i j ′ or s = 0.
The first identity can be proved as follows. By using the associativity of the trivalent vertices in the thick calculus (Proposition 2.4 in [11]) we can attach the second horizontal line to the first one k+1 k+1 = 0 and then apply eq. (2.70) in [11]. The proof of the second identity is similar after sliding e t+1 and e s+1 down by using the comultiplication rule in the ring of symmetric polynomials (or eq. (2.67) in [11]).
The differentials are obtained by rotating the original differentials by π. We have with α k,l defined as before by (4.1).

Intermediate bicomplex.
Recall that rE1 n is the bicomplex obtained by composing r1 n+2 to the left of E1 n . Its "chain groups" are F (k) E (k) E1 n −kn − 3k Λ l W k with the total q-degree shift − n 2 2 − 3n − 4 and homological shift [n/2 + 1, n/2 + 1]. We will denote them by C k,l E. The differentials are those of r1 n+2 extended by identity on E1 n . The Euler characteristic of rE1 n is is an indecomposable summand of rE1 n defined as follows. The "chain groups" are . . , w k+1 } and the same total shifts as for rE1 n . The horizontal differential d ′H k,l sends x to c ′ ∧ x where The vertical differential d ′V k,l : C ′ k,l → C ′ k+1,l is defined as follows is defined similar to (4.1). For 1 < i 1 < i 2 < · · · < i l ≤ k + 1, we have It can be verified similarly to the previous case that (rE1 n ) ′ is a bicomplex. The identity which replaces (4.2) here is It can be proven by using the comultiplication of Schur functions implying that

5.
Chain maps between rE1 n and (rE1 n ) ′ In this section we will define the chain maps between rE1 n and the intermediate complexes.
. . , w k+1 }. Let us first define a linear map is the Kronecker delta-function. In the matrix form, this map can be represented as follows: where as before (e i ) 2 means that e i sits on the second strand.
The map β k extends to an algebra homomorphism which in the matrix form can be written as β k = ⊕ k l=0 β k,l . This defines the matrix β k,l as the matrix of (l, l)-minors of the matrix β k,1 = M (β k ) or, alternatively, We set β k,0 = 1.
For example, with the same notation as before where each entry of this matrix is a determinant of the corresponding 2 × 2 matrix of β 3,1 .
One can easily prove that The inverse mapf = ⊕ k,lfk,l : (rE1 n ) ′ → rE1 n is defined bȳ is given by the adjugate matrix, or the transpose of the cofactor matrix for β k,l . For example, Proposition 5.1. The map ff : (rE1 n ) ′ → (rE1 n ) ′ is equal to identity.
Proof. We have Proposition 5.2. The map f : rE1 n → (rE1 n ) ′ is a chain map between bicomplexes.
Proof. Let us first check the commutativity of the horizontal square: We have to show that for any 1 Let us first assume i 1 > 1. After the substitution of maps this identity can be rewritten as follows: After accomplishing the j = 0 summand of the last term to zero by using (4.3) and applying twice it is not difficult to verify that it holds. The case i 1 = 1 reduces to the same identities and is left to the reader.
To see that f k+1,l (d V k,l E) = d ′V k,l f k,l we need to verify that k+1 k+2 which can be easily seen after the substitution of maps. Hence, also the induced maps Λ l W k → Dot(F (k) E (k+1) )Λ l W ′ k+1 have to coincide.
Theorem 5.3. The bicomplex (rE1 n ) ′ is a strong deformation retract of rE1 n in Kom(U ).

Proof of Theorem 5.3
This section is devoted to the proof of Theorem 5.3. After defining the homotopies, Propositions 6.1, 5.1 and 5.2 establish all properties of the strong deformation retract.
6.1. Homotopies. The horizontal homotopy h H k,l : C k,l E → C k,l−1 E is defined as follows: ,l E is set to be zero. Now Theorem 5.3 reduces to the following proposition.
(1) The proof of the first identity requires to show that for any 1 ≤ i, j ≤ k, we have To prove the first statement we use the associativity of the trivalent vertices.
The second statement works similarly. We first move dots down, then cancel terms that coincide and finally use the same trick.
(2) Here we need to show that for all 0 ≤ s, u ≤ k − 1 which holds for all 0 ≤ s, u ≤ k (here e −1 are assumed to be zero). This identity is easy to prove by using the sliding rules from Appendix. (4) Let us compute all terms of this equation. The "dh"-part contains the diagonal term of the form The off-diagonal terms of "dh" and "hd" are Finally, from the chain maps we get Due to (5.1), the first term in (6.2) cancels with the last term in (6.1) and the remaining terms cancel with the off-diagonal contributions from "dh" and "hd". Further details are left to the reader.

7.
Chain maps between Er1 n and (rE1 n ) ′ Let us denote by EC k,l the "chain groups" of the complex Er1 n , obtained by composing E1 n with r1 n . These groups are EF (k) E (k) 1 n ⊗ Λ l W k −kn − k with the total degree shift − n 2 2 − 3n − 4. The differentials are those of r1 n extended by identity on E1 n . The Euler characteristic of this complex is given by (4.4).
Define an algebra homomorphism . We will also need another map a k : be the derivation along the homomorphism γ k induced by a k , i.e. γ k,0 (1) = 0 and for l ≥ 0 we have Then we define the map and set g = ⊕ k,l g k,l : Er1 n → (rE1 n ) ′ . The inverse map p = ⊕ k,l p k,l : (rE1 n ) ′ → Er1 n is defined as follows: The bicomplex (rE1 n ) ′ is a strong deformation retract of the bicomplex Er1 n in Kom(U).

Proof of Theorem 7.1
We will split the proof of Theorem 7.1 into lemmas and prove them separately. Lemma 8.1. g : Er1 n → (rE1 n ) ′ is a chain map between bicomplexes, i.e.
. Proof. Let us check the commutativity of a general horizontal square: We assume k ≥ 2, otherwise there is nothing to check. We start with EF (k) E (k) w i1 ∧ · · · ∧ w i l ∈ EC k,l and first apply the map g k,l followed by d ′H k−1,l−1 . Then we get Applying first the differential and then the map, we get We claim that these expressions are equal in Dot(F (k−1) E (k) 1 n )Λ l W ′ k−1 . Indeed, assume 1 = i 1 < i 2 < · · · < i l and t = i s for all 1 ≤ s ≤ l. Then collecting the coefficients in front of w t ∧ w i1 ∧ . . . w ij · · · ∧ w i l with j > 1 in the both formulas we get (−1) j+k−ij (e k−ij ) 3 times where the last term comes from setting i 0 = 1 and picking the t-th summand in γ k (w 1 ) = − k j=2 c j−1 w j . Allowing i 1 = 1, but i j = 1, leads to the same identity. Let us consider the case i j = 1. Collecting the coefficients of w t ∧ w i2 ∧ · · · ∧ w i l in (8.1) we get

Using (8.3) few times, we can reduce the claim to (4.3).
Let us consider the following vertical square Similar considerations as before lead in all cases to the following true identity: Proof. Putting g on the top of p we get for any 1 < i 2 < · · · < i l After cancellation we get Using the Reidemeister move listed in Appendix we can see that the only non-zero term without bubbles is the desired identity, and all the bubble terms cancel since where the last identity is equivalent to (4.2). Here we are again using centrality of c i 's.
We set h H k,l := k k k k q k,l : EC k,l → EC k,l−1 .
Proof. Equation (8.5) reduces to Indeed, moving the line starting at the lowest left corner up through the 3-valent vertex (note that the bubble terms cancel) and then moving the e j and e i to the middle of the strand, we get that the left hand side is equal to where y := e j−1 e i − e i−1 e j . Now using the associativity and the invariance under the 3. Reidemeister move with all strands going in the same direction we can see that both summands vanish.
Equation (8.6) reduces to the following identity for 1 ≤ i < j ≤ k − 1, which can be proved similarly. Finally, equation (8.7) follows from which holds for any 0 ≤ i < j ≤ k.

Vertical homotopy. We set
c i−1 w i and (X) j means that we replace w ij with X. As before the lower indices indicate the strands on which the morphism is acting.
Let us illustrate this definition with few examples.
Hence, we get Proof. The proof of the last equality is based on the two identities given in Lemmas 5,6 in Appendix. The rest is similar to the previous computations and hence left to the reader.
8.3. Proof of Lemma 4. Thus we proved, that the map κ E : Er1 n g (rE1 n ) ′f rE1 n has a homotopy inverseκ To construct κ F we apply the symmetry σω to κ E assuming that Theorem 1 (Symmetry) holds. We getκ together with its homotopy inverse
Since any 1-morphism inU is a direct sum of compositions of E t and F t ′ with t, t ′ ∈ , it is enough to check the statement for the generators. Lemma 4 defines the maps κ E and κ F as well as their homotopy inverses. Applying symmetry, we can define η F = ψ(κ F ) and η E = ψ(κ E ). The details are left to the reader.

8.5.
Comments on the naturality of maps κ X . To prove Conjecture 2 (Naturality) we need to show that for any chain map f : X → Y the squares below commute up to chain homotopy. The commutativity of similar diagrams for η X will follow then by applying symmetry functors. It is enough to check the commutativity for short chain complexes f : X → Y , where X, Y are E1 n , E 2 1 n , F E1 n or 1 n and the differential is one of the generating 2-morphisms: dot, crossing, cup or cap. We leave this problem for future investigations.

Symmetry 2-functors
The 2-category U has the symmetry group G = ( /2 ) 3 generated by the involutive 2-functors ω, σ, ψ described below. 9.1. 2-functor ω. Consider the operation on the diagrammatic calculus that rescales the crossing → − for all n ∈ , inverts the orientation of each strand and sends n → −n. This gives a strict invertible 2-functor ω : U → U given by Finally, this operation extends toU. The images of the idempotents e a , e ′ a under the 2-functor ω are new idempotents which are equivalent to the old ones and leave thick calculus invariant. 9.2. 2-functor σ. The operation on diagrams that rescales the crossing → − for all n ∈ , reflects a diagram across the vertical axis, and sends n to −n leaves invariant the relations on the 2-morphisms of U.
This operation is contravariant for composition of 1-morphisms, covariant for composition of 2morphisms, and preserves the degree of a diagram. This symmetry gives an invertible 2-functor that acts on 2-morphisms via the symmetry described above. This 2-functor extends to a 2-functor Note that σ acts contravariantly on 1-morphisms in Kom(U).
Furthermore, σ extends toU . The images of the idempotents e a , e ′ a under σ are equivalent idempotents, leaving thick calculus invariant. 9.3. 2-functor ψ. This operation reflects across the horizontal axis and invert orientation. This gives an invertible 2-functor defined by and on 2-morphisms ψ reflects the diagrams representing summands across the x-axis and inverts the orientation.
Again, this 2-functor extends straightforward to the Karoubi envelopeU. The images of the idempotents e a , e ′ a under symmetry functors are equivalent idempotents with the same properties as before.

Symmetries of the ribbon bicomplex
In this section we describe the behavior of the ribbon bicomplex under the symmetry 2-functors.
10.1. The image under ω. The "chain groups" of the bicomplex ω(r1 −n ) are with the total shifts − n 2 2 + n and [n/2, n/2]. The horizontal differential ω(d H k,l ) : Similarly, the vertical differential is Since all the relations inU are invariant under symmetries, ω(r1 −n ) is a bicomplex.

Isomorphic bicomplex
This section provides a construction of the bicomplex r1 n , which is isomorphic to the ribbon bicomplex and invariant under σω.
Then there exists an isomorphism H : Its inverse is defined by replacing h i with its antipode (−1) i e i , hence we have We can use this map to define a non-trivial transformation H k,l : C k,l → C k,l of the "chain groups" C k,l = F (k) E (k) 1 n ⊗ Λ l W k of the ribbon bicomplex as follows.
For l = 1 we set where the matrix is written in the basis w 1 , w 2 , . . . , w k and all symmetric polynomials are sitting on the second strand.
This map obviously extends to Λ l W k by setting Inserting (11.1), we get Hence, for 1 ≤ i 1 < i 2 · · · < i l ≤ k and 1 ≤ j 1 < j 2 · · · < j l ≤ k The inverse map is defined in a similar way by using For 1 ≤ i 1 < i 2 · · · < i l ≤ k and 1 ≤ j 1 < j 2 · · · < j l ≤ k we have 11.2. The bicomplex r1 n . Let us denote by r1 n the image of r1 n under applying the isomorphism H k,l to each "chain group" C k,l . The horizontal and vertical differentials of r1 n are given by Let us compute them.
It will be convenient in what follows to introduce a simplified notation in Flag N . Corresponding to a fixed value of N , we set n = 2k − N and write n−2 F½ N n := Γ N (F 1 n ) as a shorthand for the various bimodules. Juxtaposition of these symbols represents the tensor product of the corresponding bimodules. For example, where E + := E and E − := F. The 2-functor Γ N maps a composite E ε 1 n of 1-morphisms inU to the tensor product E ε 1 N n in Flag N . Note that because tensor product of bimodules is only associative up to coherent isomorphism our notation is ambiguous unless we choose a parenthesization of the bimodules in question. We employ the convention that all parenthesis are on the far left. Hence, Γ N preserves composition of 1-morphisms only up to coherent 2-isomorphism.
It is sometimes convenient to use the following isomorphisms from [2].
We also define bimodules Let us denote by r½ N n the image under Γ N of the ribbon complex. 12.2. Proof of Theorem 5. We first prove that tensoring on the left (or on the right) with r½ N n acts as a left (resp. right) multiplication with the identity on any left (resp. right) H k;N -module up to degree shift.
We will consider the left action only, the right action can be proved similarly. Note that it is enough to compute the left action of r½ N n on H k;N . Let us first ignore the homological shift for simplicity. Set n = N . Then r½ N N = ½ N N −N 2 /2 − N in Kom(Flag N ), simply because E½ N N = 0, and the result holds.
Assume n = N − 2k, then where the first homotopy equivalence holds due to centrality of r½ N n . Now observe that in Flag N . Hence, r½ N n acts as a left multiplication with ½ N n −N 2 /2 − N on F (k) ½ N N in Com(Flag N ).
But F (k) ½ N N is isomorphic to H k;N = ½ N n as a left module over itself. Hence we have the first statement.
Similarly, tensoring with r −1 ½ N n on the left and on the right is homotopic to the identity functor shifted by N 2 /2 + N , which is inverse to r½ N n . Since homological shifts for r1 n and r −1 1 n are inverse to each other, we have the result.
12.3. The inverse limit of Schur quotients. The Schur quotientU N ofU is defined by setting 1 N +2 = 0 (see [15] for a more general definition). Applying sl(2) relations, an easy induction argument shows that 1 n = 0 inU N for all n < −N and n > N . Moreover, this quotient is not empty, since the functor Γ N :U → Flag N factorises throughU N by its very definition.
For any N ′ > N there is a natural projection Ψ N ′ ,N :U N ′ →U N defined by setting 1 N +2 = 0. Taking all together, these maps define an inverse system of 2-categories whose inverse limit isU (compare [2]). Now let us consider Com b (U N ). The induced functor Com b (U ) → Com b (Flag N ) factorises again through Com b (U N ). Using the natural projections for all N ′ > N we can construct l Com(U) = lim ← − Com b (U N ) .
Observe that r ±1 r ∓1 1 n belongs to l Com(U ), since its projections to Com b (U N ) are well-defined for all N and compatible with each other.

Appendix
Let us collect the identities we need in the proofs. In what follows an x labeled bullet on a thick line will mean h x inserted.
13.2. Reidemeister moves. From Corollary 5.8 in [11] (for b = k, a = 1) and the bubble slide rule (4.11), we get Similar, to the proof of Proposition 5.8 in [13] we can show that for all colors n ∈ of the right most region the following equation holds  to simplify the resulting terms. The last equality follows from (13.1) and (13.1). It is easy to check that all terms with bubbles sum to zero. Proof. We first apply (13.2) to the first two summands on the right hand side, then all diagrams without smoothings will look as shown below.
k k The strategy of the proof will be to move all dots into the position shown by the dashed line. Doing so for the last two summands we get c i−1 (e 1 ) 3 (e j−1 ) 4 − c i−1 (e 1 ) 1 (e j−1 ) 4 ∈ Dot(EF (k) EE (k−1) ) minus the second term on the left hand side of the identity (obtained after sliding the dot though the down pointed k-line), and in addition various terms with bubbles. Let us first compare the terms without smoothings. The second term on the right hand side will contribute −(c i ) 234 (e j−1 ) 4 ∈ Dot(EF (k) EE (k−1) ) .
Finally, in the first term on the right hand side we replace (c i ) 23 with c i − c i−1 (e 1 ) 1 and move the dot down. The contribution of this term to the part without smoothings will be c i (e j−1 ) 4 − c i−1 (e 1 ) 3 (e j−1 ) 4 ∈ Dot(EF (k) EE (k−1) ) and in addition from moving the dot, we get the first term on the left hand side. Collecting all nonsmoothed terms together we get zero. It remains to show that all bubble terms vanish. This easy check is left to the reader.