aa r X i v : . [ m a t h . QA ] S e p NOT EVEN KHOVANOV HOMOLOGY
PEDRO VAZA
BSTRACT . We construct a supercategory that can be seen as a skew version of (thickened) KLRalgebras for the type A quiver. We use our supercategory to construct homological invariants oftangles and show that for every link our invariant gives a link homology theory supercategorifyingthe Jones polynomial. Our homology is distinct from even Khovanov homology and we presentevidence supporting the conjecture that it is isomorphic to odd Khovanov homology. We alsoshow that cyclotomic quotients of our supercategory give supercategorifications of irreduciblefinite-dimensional representations of gl n of level 2.
1. I
NTRODUCTION
After the appearance of odd Khovanov homology in [15] there has been a certain interest inodd categorified structures and supercategorification (see for example [2, 3, 4, 5, 6, 7, 11, 14]).In contrast to (even) Khovanov homology, odd Khovanov homology has an anticommutativefeature. Both theories categorify the Jones polynomial and both agree modulo 2, but they areintrinsically distinct (see [20] for a study of the properties of odd Khovanov homology and acomparison with even Khovanov homology).A construction of odd Khovanov homology using higher representation theory is still missing.In the case of even Khovanov homology this question was solved in [24] using categorificationof tensor products and the WRT invariant and in [12] using categorical Howe duality.In this paper we construct a supercategorification of the Jones invariant for tangles using higherrepresentation theory. In particular, we define a supercategory in the spirit of Khovanov andLauda’s diagrammatics that can be seen as a superalgebra version of KLR algebras [8, 19] oflevel 2 for the A n quiver. We present our supercategory in the form of a graphical calculusreminiscent of the thick calculus for categorified sl [10] and sl n [22] (see also [3] for a thickcalculus for the odd nilHecke algebra). Our supercategory admits cyclotomic quotients thatsupercategorify irreducibles of U q p gl k q of level 2.We use cyclotomic quotients of our supercategories as input to Tubbenhauer’s [23] approachto Khovanov-Rozansky homologies. It is based in q -Howe duality and uses only the lower halfof the quantum group U q p gl k q to produce an invariant of tangles. In our case we obtain aninvariant that shares several similarities with odd Khovanov homology when restricted to links.For example, it decomposes as a direct sum of two copies of a reduced homology and it produceschronological Frobenius algebras, analogous to the ones that can be extracted from [15] (see [17]for explanations). Both theories coincide over Z { Z . We also give computational evidence thatour invariant is distinct from even Khovanov homology and we conjecture that for every link L it coincides with the odd Khovanov homology of L . Pedro Vaz
Acknowledgements.
We thank Daniel Tubbenhauer, Kris Putyra and Gr´egoire Naisse for inter-esting discussions. The author was supported by the Fonds de la Recherche Scientifique - FNRSunder Grant no. MIS-F.4536.19. 2. T
HE SUPERCATEGORY R The supercategory R p ν q . We follow [1] regarding supercategories. For objects
X, Y ina supercategory C we write Hom C p X, Y q (resp. Hom C p X, Y q ) for its space of even (resp. odd)morphisms and we write p p f q for the parity of f P Hom i C p X, Y q . If C has additionally a Z -grading we denote by q s X a grading shift up of X by s units and we consider only morphismsthat preserve the Z -grading. In this case we write Hom C p X, Y q “ ‘ s P Z Hom C p X, q s Y q . Wefollow the grading conventions in [12], which are aligned with the tradition in link homology.This means that a map of degree s from X to Y yields a degree zero map from X to q s Y .Fix a unital ring k . Let α , . . . , α n denote the simple roots of sl n and x´ , ´y their innerproduct: x α i , α i y “ , x α i , α i ˘ y “ ´ , and x α i , α j y “ otherwise. Fix also a choice of scalars Q consisting of r i , t ij P k ˆ for all i, j P I : “ t , . . . , n u , such that t ii “ and t ij “ t ji when | i ´ j | ‰ . Let also p ij be defined by p ii “ p i ` ,i “ and otherwise p ij “ .For each ν “ ř i P I ν i .i P N r I s , we consider the set of (colored) sequences of ν , CSeq p ν q : “ i p ε q ¨ ¨ ¨ i p ε r q r ˇˇ ε s P t , u , ÿ s ε s i s “ ν ( . By convention we write simply i s for i p q s . Two sequences i P CSeq p ν q and j P CSeq p ν q can beconcatenated into a sequence ij in CSeq p ν ` ν q . Definition 2.1.
The supercategory R p ν q is defined by the following data:(a) The objects of R p ν q are finite formal sums of grading shifts of elements of CSeq p ν q .(b) The morphism space Hom R p ν q p i , j q from i to j is the Z -graded k -supervector space gener-ated by vertical juxtaposition and horizontal juxtaposition of the diagrams below. Composi-tion consists of vertical concatenation of diagrams. By convention we read diagrams frombottom to top and so, ab consists of stacking the diagram for a atop the one for b . Diagramsare equipped with a Morse function that keeps trace of the relative height of the genera-tors. We consider isotopy classes of such diagrams that do not change the relative height ofgenerators. Generators. ‚ Simple and double identities i P Hom R p ν q p i, i q , i P Hom R p ν q p i p q , i p q q , ot even Khovanov homology 3 ‚ dots i P Hom R p ν q p i, q i q , ‚ splitters i P Hom R p ν q p i p q , q ´ ii q , i P Hom R p ν q p ii, q ´ i p q q , ‚ and crossings i j P Hom p ij R p ν q p ij, q ´x α i ,α j y ji q , i j P Hom R p ν q p i p q j, q ´ x α i ,α j y ji p q q , i j P Hom R p ν q p ij p q , q ´ x α i ,α j y j p q i q , i j P Hom R p ν q p i p q j p q , q ´ x α i ,α j y j p q i p q q . Relations.
Morphisms are subject to the local relations (1) to (14) below. ‚ For all f, g :(1) f i ¨ ¨ ¨ i k ¨ ¨ ¨ g i ¨ ¨ ¨ i k ¨ ¨ ¨ “ f i ¨ ¨ ¨ i k ¨ ¨ ¨ g i ¨ ¨ ¨ i k ¨ ¨ ¨ “ p´ q p p f q p p g q f i ¨ ¨ ¨ i k ¨ ¨ ¨ g i ¨ ¨ ¨ i k ¨ ¨ ¨‚ For all i, j, k P I :(2) i “ . Pedro Vaz i j “ $’’’’’’’’’’’’’’&’’’’’’’’’’’’’’% if i “ j,t ij i j if | i ´ j | ą ,t ij i j ` t ji i j if | i ´ j | “ , (3) i j “ p´ q p ij i j i j “ p´ q p ij i j for i ‰ j ,(4) t i,i ` i ` i ` t i ` ,ii ` i “ (5) i i ` i i “ r i i i “ i i ` i i (6) i kj “ i kj unless i “ k and | i ´ j | “ , (7) i ij ` i ij “ r i t ij i j i if | i ´ j | “ , (8) ot even Khovanov homology 5 (9) j j “ j j (10) j “ j “ j j “ (11) j “ “ jjkj “ kj kj “ kj (12) k j “ k j k j “ k j (13) kj “ kj kj “ kj (14)This ends the definition of R p ν q .In Subsection 2.5 below we show that R p ν q acts on a supercommutative ring. Definition 2.2.
We define the monoidal supercategory R “ à ν P N r I s R p ν q , the monoidal structure given by horizontal composition of diagrams. Pedro Vaz
Further relations in R p ν q . We have several consequences of the defining relations.
Lemma 2.3.
For all i P I , i i ´ i i “ , (15) i i i “ , (16) ii “ i i “ ii “ . (17) Proof.
By (2) and (6), r ´ i i i ´ r ´ i i i “ i i ´ i i “ , which proves (15).Also, i i i “ i i i ` i i i “ i i i ` i i i “ i i i ` i i i “ , and this proves (16). Relations (17) are an easy consequence of (10) together with (16). (cid:3) Lemma 2.4.
For all i, j P I with | i ´ j | “ , ii j “ ii j Proof.
Start from the equality ji i “ ji i ot even Khovanov homology 7 Sliding up the dot on the left-hand side using (4) and (1), followed by (8) to pass the ii -crossingto the left, and simplifying using (3) and (10) gives ´ r i t ij t ji ii j Proceeding similarly on the right-hand side, but sliding the ii -crossing to the right gives ´ r i t ij t ji ii j and the claim follows. (cid:3) Lemma 2.5.
For all i, j P I with | i ´ j | “ , i j “ . Proof.
We compute: i j (10) “ i j (14) “ i j (13) “ i j which is zero if i “ j ˘ by (4), (5) and (2). (cid:3) The following are easy consequences of the defining relations of R p ν q . Lemma 2.6.
For all i , j P I , i j “ i j i j “ i j Lemma 2.7.
For all i , j P I , i j “ $’’’’’&’’’’’% t ij i j if | i ´ j | ą , otherwise , Pedro Vaz i j “ $’’’’’&’’’’’% t ij i j if | i ´ j | ą , otherwise , i j “ $’’’’’&’’’’’% t ij i j if | i ´ j | ą , otherwise . Lemma 2.8. If | i ´ j | “ , i ij ´ i ij “ r i t ij i j i ´ r i t ij i j i If i ‰ j ‰ k , then relation (7) is true for all types of strands. Let
Seq p ν q : “ i p ε q ¨ ¨ ¨ i p ε r q r P CSeq p ν q ˇˇ ε s “ ( Ă CSeq p ν q . The superalgebra R p ν q “ à i , j P Seq p ν q Hom R p ν q p i , j q , is the sub-superalgebra of the Hom -superalgebra of R p ν q consisting of all diagrams having onlysimple strands. If we interpret R p ν q as a superalgebra version of a level 2 cyclotomic KLR alge-bra for sl n then R p ν q can be seen as version of the thick calculus [10, 22] for this superalgebra.It is not hard to see that both the center and the supercenter of R p ν q are zero.2.3. Cyclotomic quotients.
Fix a sl n -weight Λ and denote by R Λ p ν q , R Λ p ν q and R Λ p ν q thecyclotomic quotients of R p ν q , R p ν q and R p ν q . The following is immediate. Lemma 2.9. If Λ is of level 2 then the algebras R Λ p ν q b Z p Z { Z q and R Λ p ν q b Z p Z { Z q areisomorphic (after collapsing the Z { Z grading of R Λ p ν q ). We depict a morphism of R λ p ν q by decorating the rightmost region of each diagram D withthe weight Λ . This defines weights for all regions of D .The supercategory R Λ : “ ‘ ν P N r I s R Λ p ν q is not monoidal anymore, but it is a (left) modulecategory over R , where R acts by adding diagrams of R to the left of diagrams from R Λ . This ot even Khovanov homology 9 is expressed by a bifunctor(18) Φ : R ˆ R λ Ñ R λ . A super 2-category.
There is a super 2-category around R p ν q , paralleling the case ofKhovanov–Lauda and Rouquier. An element i “ i p ε q ¨ ¨ ¨ i p ε r q r in CSeq p ν q corresponds to aroot α i : “ ř s ε s α s . Let Λ p n, d q : “ t µ P t , , u n | µ ` ¨ ¨ ¨ ` µ n “ d u .Define R p n, d q as the super 2-category with objects the elements of Λ p n, d q and with mor-phism supercategories HOM R p n,d q p µ, µ q the various R p ν q . In other words, a 1-morphisms µ Ñ µ is a sequence i such that µ ´ µ “ α i and the 2-morphism espace i Ñ j is Hom R p ν q p i , j q .Similarly we define the super 2-category R Λ p n, d q by using the cyclotomic quotient with re-spect with the integral dominant weight Λ . Both super 2-categories R Λ p n, d q have diagrammaticpresentations with regions labeled by objects Λ . The 2-morphisms in R λ p n, d q are presented asa collection of 2-morphisms in R p n, d q with rightmost region decorated with Λ , subjected to thesame relations together with the cyclotomic condition. This defines a label for every region of adiagram of R Λ p n, d q .For later use, we denote F i λ : “ F i p ε q ¨¨¨ i p εr q r λ : “ F p ε q i ¨ ¨ ¨ F p ε r q i r λ the 1-morphisms of R Λ p n, d q and, by abbuse of notation, the objects of R Λ .2.5. Action on a supercommutative ring.
We now construct an action of R p ν q on exteriorspaces.2.5.1. Demazure operators on an exterior algebra.
Let V “ Ź p y , . . . , y d q be the exterior alge-bra in d variables. This algebra is naturally graded by word length. Denote by | z | the degree ofthe homogeneous element z .The symmetric group S d acts on V by the permutation action, wy i “ y w p i q for all w P S d .Define operators B i for i “ , . . . , d ´ on V by the following rules: B i p y k q “ i “ k, k ` , otherwise , and B i p f g q “ B i p f q g ` p´ q | f | f B i p g q , for all f , g P V such that f g ‰ .The following can be checked through a simple computation. Lemma 2.10.
The operators B i satisfy the relations B i “ , B i B j ` B j B i “ if | i ´ j | ą , and B i B i ` B i “ B i ` B i B i ` . Pedro Vaz
An action of R p ν q on supercommutative rings. For i P CSeq p ν q let P i “ Ź p x , , x ,ε , . . . , x d, , x d,ε d q i , be an exterior algebra in ř i ν i generators, and set P p ν q “ à i P CSeq p ν q P i . We extend the action of S d from V to P p ν q by declaring that wx r, “ x w p r q , , wx r,ε r “ x w p r q ,ε r ` , or w P S d .Below we denote by B u,z the Demazure operator with respect to the variables u and z .To the object i P R p ν q we associate the idempotent i P P i . The defining generators of R p ν q act on P as follows. A diagram D acts as zero on P i unless the sequence of labels in the bottomof D is i . ‚ Dots i r : p i ÞÑ x r, p i , ‚ Splitters(19) i r : p i ÞÑ B x r, ,x r, p p q i , i r : p i ÞÑ x r, B x r, ,x r, p p q i , ‚ Crossings i r i r ` : p i ÞÑ $’’&’’% r i r B x r, ,x r ` , p p q i if i r “ i r ` , p t i r ` i r x r, ` t i r i r ` x r ` , q s r p p i q if i s “ i s ` ` ,s r p p i q else , i r i r ` : p i ÞÑ if i r “ i r ` , or i s “ i s ` ` ,s r p p i q else , (20) i r i r ` : p i ÞÑ $’’&’’% if i r “ i r ` ,f , p x r, , x r, , x r ` , q s r p p i q if i s “ i s ` ` ,s r p p i q else , (21) ot even Khovanov homology 11 i r i r ` : p i ÞÑ $’’&’’% if i r “ i r ` ,f , p x r, , x r ` , , x r ` , q s r p p i q if i s “ i s ` ` ,s r p p i q else , (22)where f , p x r, , x r, , x r ` , q “ t i r i r ` t i r ` i r x r, x r ` , ` t i r i r ` t i r ` i r x r, x r, ` t i r ` i r x r, x r ` , .f , p x r, , x r ` , , x r ` , q “ ´ t i r i r ` x r, x r, ` t i r i r ` t i r ` i r x r, x r ` , ´ t i r ` i r t i r ` i r x r, x r ` , . Proposition 2.11.
The assignment above defines an action of R p ν q on P p ν q .Proof. By a long and rather tedious computation one can check that the operators above satisfythe defining relations of R p ν q .The relations involving the action of the generators of R p ν q are easy to check by direct com-putation. For example, for ν “ i ` j , with j “ i ` we have i ij p f q “ p t ij x ` t ji x q s r i B s p f q , and i ij p f q “ s r i B p t ij x ` t ji x q s p f q “ r i t ij f ´ p t ij x ` t ji x q s r i B s p f q , and so, for any f p x , x , x q P P iji , i ij p f q ` i ij p f q “ r i t ij i j i p f q . Setting as in [10], i : “ i i i : “ i i i i : “ i i Pedro Vaz and i j : “ i j ji : “ ji ji : “ i j then it follows that the action of the generators of R p ν q on P p ν q is given by the operators (19),(20), (21) and (22) and satisfy the defining relations of R p ν q . (cid:3)
3. A
TOPOLOGICAL INVARIANT
In [23] q -skew Howe duality is used to show how to write as a web in a form that uses onlythe lower part of U q p gl k q . In this language, the formula for the sl -comutator becomes oneof Luzstig’s higher quantum Serre relations from [13, § F i ’s in U ´ : “ U ´ q p gl k q .This allows a categorification of webs using only (cyclotomic) KLR algebras [8, 19] insteadof the whole 2-quantum group U p gl k q [9, 19]. In this context, the unit and co-unit maps ofthe several adjunctions in U p gl k q that are used as differentials in the Khovanov–Rozansky chaincomplex can be written as composition with elements of the KLR algebra. Taking cyclotomicKLR algebras of level 2 gives Khovanov homology. The approach in [23] is easily adapted totangles, which we do in in this section for level 2 in the context of the supercategories introducedin Section 2.3.1. Supercategorification of gl -webs and flat tangles. Our webs have strands labeled from t , , u which we depict as “invisible”, “simple”, and “double”, as in the example below. Allthe strands point either up or to the right and sometimes we omit the orientations in the pictures.
10 21 02
For λ “ p λ , ¨ ¨ ¨ , λ k q P t , , u k and ǫ P t , u with | λ | “ ℓ ` ǫ , we put Λ “ p q ℓ ǫ “p , . . . , , ǫ, , . . . , q and we define W p λ q “ HOM R Λ p k, | λ |q p Λ , λ q . Let W be a gl -web with all ladders pointing to the right. Suppose that W has the bottomboundary labelled λ and the top boundary labelled µ , whith λ, µ P t , , u k and | λ | “ | µ | . We ot even Khovanov homology 13 write W as a word in the F i ’s in U ´ q p gl k q applied to a vector v λ of gl k -weight λ . W λ ¨ ¨ ¨ λ k µ ¨ ¨ ¨ µ k “ F i ¨ ¨ ¨ F i r p v λ q . This gives a 1-morphism F p W q in R p k, | λ |q . Composition of 1-morphisms in R p k, | λ |q definesa superfunctor F p W q : W p λ q Ñ W p µ q . If λ is dominant and µ is antidominant then F p W q is a superfunctor from k -smod to k -smod that is, a direct sum of grading shifts of the identity superfunctor. In this case, there is a canonical1-morphism F can p W q in Hom R Λ p k, | λ |q p λ, µ q (23) F can “ F p k ´ ℓ ´ q p q ¨¨¨p q p q ¨ ¨ ¨ F p k ´ q p q ¨¨¨p ℓ ´ q p q F p k ´ q p q ¨¨¨ ℓ p q F p k ´ q p ǫ q ¨¨¨p ℓ ` q p ǫ q p q ℓ ǫ, which in terms of webs takes the form of the following example: ¨ ¨ ¨ ¨ ¨ ¨ We have that F p W q “ Hom R λ p k, | λ |q p λ, µ q is isomorphic to the graded k -supervector space Hom R Λ p F can p W q , F p W qq .3.2. The chain complex.
As explained in [23] any oriented tangle diagram T can be written inthe form of a web W T with all horizontal strands pointing to the right. In this case we say that T is in F -form . Pedro Vaz
Example 3.1.
For the Hopf link we have the following web diagram.
Suppose the bottom boundary of W T is p λ , ¨ ¨ ¨ , λ k q and the top boundary is p µ , ¨ ¨ ¨ , µ k q .Let Kom p λ, µ q be the category of complexes of HOM R p k, | λ |q p W p λ q , W p µ qq and Kom { h p λ, µ q itshomotopy category. To each tangle in F -form as above we associate an object in Kom { h p λ, µ q as follows.We first chop the diagram vertically in such way that each slice contains either a web withoutcrossings, or a single crossing together with vertical pieces (as in Example 3.1). Each slice thengives either a superfunctor or a complex of superfunctors, as explained below. By compositionwe get a complex F p W T q of superfunctors from W p λ q to W p µ q .3.2.1. Basic tangles. ‚ If T is a flat tangle, then we’re done by Subsection 3.1. ‚ To the positive crossing we associate the chain complex(24) ÞÑ q ´ F ¨˚˚˚˚˚˚˝
10 11 01 ˛‹‹‹‹‹‹‚
ÝÝÝÝÝÝÝÑ F ¨˚˚˚˚˚˚˝
10 11 01 ˛‹‹‹‹‹‹‚ with the leftmost term in homological degree zero. Algebraically this can be written β ` ÞÑ q ´ F F p , , q τ ÝÝÑ F F p , , q , where τ is the diagram above. ‚ To the negative crossing we associate the chain complex(25) ÞÑ F ¨˚˚˚˚˚˚˝
10 11 01 ˛‹‹‹‹‹‹‚
ÝÝÝÝÝÝÝÑ q F ¨˚˚˚˚˚˚˝
10 11 01 ˛‹‹‹‹‹‹‚ ot even Khovanov homology 15 with the righmost term in homological degree zero. Algebraically β ´ ÞÑ F F p , , q τ ÝÝÑ qF F p , , q . The normalized complex.
Let n ˘ be the number of positive/negative crossings in W T andlet w “ n ` ´ n ´ be the writhe of W T . We define the normalized complex(26) F p W T q : “ q w F p W T q . Topological invariance.Theorem 3.2.
For every tangle diagram T the homotopy type of F p W T q is invariant under theReidemeister moves. Theorem 3.3.
For every link L the homology of F p L q is a Z -graded supermodule over Z whosegraded Euler caracteristic equals the Jones polynomial.Proof of Theorem 3.2. The following is immediate.
Lemma 3.4.
For β ˘ a positive/negative crossing let W t and W b be the following tangles in F -form: W t “ β ˘ and W b “ β ˘
110 0110 0
Then the complexes F p W t q and F p W b q are isomorphic. Lemma 3.5 (Reidemeister I) . Consider diagrams D ` and D that differ as below. D ` “ D “ Then F p D ` q and F p D q are isomorphic in Kom { h ` p , , q , p , , q ˘ .Proof. We have F p D ` q “ q ´ F F F p , , q F F F p , , q . Pedro Vaz
The first term is isomorphic to F F p q p , , q ‘ q ´ F F p q p , , q via the map F F p q p , , q ‘ q ´ F F p q p , , q q ´ F F p , , q , » ˜ , ¸ while for the second term there is an isomorphism F F F p , , q F F p q p , , q , »
12 2 so that F p D ` q is isomorphic to the complex ¨˝ F F p q p , , q q ´ F F p q p , , q ˛‚ F F p q p , , q . ˆ t , , t , ¯ By Gaussian elimination one gets that the complex F p D ` q is homotopy equivalent to the oneterm complex q ´ F F p q p , , q concentrated in homological degree zero, which after normal-ization is F p D q . (cid:3) The other types of Reidemeister I move can be verified similarly. For example, replacingthe positive crossing by a negative crossing in Lemma 3.5 and using the inverses of the variousisomorphisms above results in a complex isomorphic to F p D ´ q that is homotopy equivalent tothe 1-term complex q F F p q p , , q concentrated in homological degree zero. Lemma 3.6 (Reidemeister IIa) . Consider diagrams D and D that differ as below. D “ D “ Then F p D q and F p D q are isomorphic in Kom { h ` p , , , q , p , , , q ˘ . ot even Khovanov homology 17 Proof.
In the following we write µ instead of p , , , q . The complex F p D q is q ´ F F F F µ F F F F µF F F F µ qF F F F µ, À ´ From the isomorphisms F F F F µ F F p q F µ F F F F µ, » » F F F F µ F F p q F µ F F F F µ, » »
32 2 1 3 2 1 and F F F F µ qF F p q F µq ´ F F p q F µ F F F F µ, À Pedro Vaz and simplifying the maps using the relations in R p ν q one gets that F p D q is isomorphic to thecomplex q ´ F F p q F µ qF F p q F µ À q ´ F F p q F µ À F F p q F µ qF F p q F µ, t
13 2 13 2 t Id ´ t
13 2 ´ t Id 132 2
By Gaussian elimination of the acylic two-term complexes q ´ F F p q F µ t Id ÝÝÝÑ q ´ F F p q F µ and qF F p q F µ ´ t Id ÝÝÝÝÑ qF F p q F µ one obtains that F p D q is homotopy equivalent to the com-plex F F p q F µ , with the middle-term in homological degree zero. (cid:3) Lemma 3.7 (Reidemeister III) . Consider diagrams D L and D R that differ as below. D L “ D R “ Then F p D L q and F p D R q are isomorphic in Kom { h ` p , , , , , q , p , , , , , q ˘ .Proof. The proof is inspired by [17, Lemma 7.9] (see also [18, § D L is the maping cone of the map q ´ F ¨˚˚˚˚˚˚˚˚˚˚˝ ˛‹‹‹‹‹‹‹‹‹‹‚ ¨¨¨ ¨¨¨ ÝÝÝÝÝÝÝÝÑ F ¨˚˚˚˚˚˚˚˚˚˚˚˝ ˛‹‹‹‹‹‹‹‹‹‹‹‚ ot even Khovanov homology 19 An easy exercice shows that the second complex is isomorphic to the complex F ¨˚˚˚˚˚˚˚˚˚˚˝ ˛‹‹‹‹‹‹‹‹‹‹‚ In [18, § F p D L q is homotopyequivalent to F p D R q . (cid:3) This finishes the proof of Theorem 3.2. (cid:3)
Not even Khovanov homology.
We now show that for links the invariant H p L q is distictfrom even Khovanov homology and shares common properties with odd Khovanov homology.3.4.1. Reduced homology.
Theorem 3.8.
For every link L there is an invariant H reduced p L q with the property H p L q » qH reduced p L q ‘ q ´ H reduced p L q . The proof of Theorem 3.8 follows a reasoning analogous to the proof of Theorem 3.2.A.in [21], for the analogous decomposition for Khovanov homology over Z { Z in terms of re-duced Khovanov homology.Before proving the theorem we do some preparation. Recall that for D a diagram of L thechain groups of F p D q are the various k -supervector spaces Hom R Λ p F can , F p W qq , where W runsover all the resolutions of D .If we write F can “ F i p q i p q ¨¨¨ i p q k then Hom R Λ p F can , F i i i i ¨¨¨ i k i k q is spanned by i δ i δ ¨ ¨ ¨ i k δ k , δ , . . . , δ k P t , u + . Introduce linear maps X and ∆ on Hom R Λ p F can , F i i i i ¨¨¨ i k i k q as follows. Map ∆ is defined onthe factors as ∆ ˜ ¨ ¨ ¨ ¨ ¨ ¨ ¸ “ , ∆ ˜ ¨ ¨ ¨ ¨ ¨ ¨ ¸ “ ¨ ¨ ¨ ¨ ¨ ¨ , Pedro Vaz and extended to
Hom R Λ p F can , F i i i i ¨¨¨ i k i k q using the Leibiz rule. The map X is defined by X ˜ i δ i δ ¨ ¨ ¨ i k δ k ¸ “ $’’’&’’’% i i δ ¨ ¨ ¨ i k δ k if δ “ , otherwise . Since
Hom R Λ p F can , F p W qq » Hom R Λ p F can , F i i i i ¨¨¨ i k i k q ˆ Hom R Λ p F i i i i ¨¨¨ i k i k , F p W qq the maps ∆ and X induce maps on Hom R Λ p F can , F p W qq , denoted by the same symbols. Lemma 3.9.
Both maps X and ∆ commute with the differential of F p D q , ∆ “ , and moreover X ∆ ` ∆ X “ Id F p D q .Proof. Straightforward. (cid:3)
Proof of Theorem 3.8.
We have that ∆ is acyclic and therefore F p D q » ker p ∆ q ‘ q ker p ∆ q , and so the claim follows by setting F reduced p D q “ q ker p ∆ q . (cid:3) A chronological Frobenius algebra.
We now examine the behaviour of the functor F un-der merge and splitting of circles. First define maps ı and ε , F ¨˚˚˝ ˛‹‹‚ εı F ¨˚˚˝ ˛‹‹‚ as ı : F p q p , q ÝÝÝÝÝÑ F p , q ε : F p , q ÝÝÝÝÝÑ F p q p , q . Note that, contrary to [15], p p ı q “ and p p ε q “ .We now consider the following two cases ( a ) and ( b ) below.( a ) F ¨˚˚˚˚˚˝ ˛‹‹‹‹‹‚ µδ F ¨˚˚˚˚˚˝ ˛‹‹‹‹‹‚ The maps µ and δ are given by µ : F F p , , q
21 1 2
ÝÝÝÝÝÝÝÝÝÝÝÝÝÑ F F F F p , , q , ot even Khovanov homology 21 and δ : F F F F p , , q
21 2 1
ÝÝÝÝÝÝÝÝÝÝÝÝÝÑ F F p , , q . We have p p µ q “ and p p δ q “ . Decomposing F F p , , q and F F F F p , , q into a directsum of several copies of F p q F p q p , , q with the appropriate grading shifts we fix bases C p “ , p “ , p “ , p “ G of F F p , , q , and C p “ , p “ G of F F F F p , , q . Then we compute δ ˜ ¸ “ ´ t
12 1 2 ` t
21 1 2 δ ˜ ¸ “ t
21 1 2 and µ ˜ ¸ “ µ ˜ ¸ “ µ ˆ ¸ “ µ ˜ ¸ “ t t ´
121 1 2
Using this one sees that easily that µδ “ , as in the case of the odd Khovanov homologyof [15].Setting to 1 all t ij ’s and renaming x , a , a , a ^ a y the basis vectors of F F p , , q and x , a “ a y the basis vectors of F F F F p , , q one can give the maps δ, µ, ı and ε a formthat coincides with the corresponding maps in [15, § δ and µ coincide with the corresponding maps in [15], the parities of ı and ε are reversed withrespect to [15]. Pedro Vaz ( b ) F ¨˚˚˚˚˚˝ ˛‹‹‹‹‹‚ µ δ F ¨˚˚˚˚˚˝ ˛‹‹‹‹‹‚ The maps µ and δ are given by µ : F F p , , q
12 2 1
ÝÝÝÝÝÝÝÝÝÝÝÝÝÑ F F F F p , , q , and δ : F F F F p , , q
12 1 2
ÝÝÝÝÝÝÝÝÝÝÝÝÝÑ F F p , , q . Proceeding as above we fix a basis C p “ , p “ G of F F F F p , , q and C p “ , p “ , p “ , p “ G of F F p , , q , to get δ ˜ ¸ “ ´ t
21 2 1 ` t
12 2 1 δ ˜ ¸ “ t
12 2 1 and µ ˜ ¸ “ µ ˜ ¸ “ ot even Khovanov homology 23 µ ˆ ¸ “ µ ˜ ¸ “ t t ´
112 2 1
In this case we also have µ δ “ .Contrary to the previous case, we have p p µ q “ and p p δ q “ . The maps µ and δ can alsobe made to agree with [15], but the parity is reversed (as with ı and ε above).3.4.3. A sample computation.
We now compute the homology of the left-handed trefoil T in itslowest and highest homological degrees. Consider the following presentation of T , The computation of H p T q is fairly simple: up to an overall degree shift it is the homology indegree 1 of the complex(27) q F t F F b µq F t F F b µ À q F t F F b µ À q F t F F b µ The three terms in homological degree zero are isomorphic to F p q p q . Composing theisomorphisms from F p q p q to F , F and to F with the corresponding mapsabove gives three maps that differ by a sign.By inspection, one sees that up to a sign, these theree maps are equal to the map δ from thecase ( a ) in the previous subsection. The cokernel map in (27) is therefore two-dimensional.Adding the degree shifts one obtains H p T q “ q ´ k ‘ q ´ k . Pedro Vaz
We now compute H ´ p H q . Up to an overall degree shift it is computed as the homology indegree zero of the complex F F F µ qF F F µ À qF F F µ À qF F F µ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ Here µ “ p , , , , q and the factors F and F are the upper and lower closures of thediagram. We write F t for F and F b for F and sometimes we write F t F F b µ instead of F F F µ , etc..., and we only depict the pertinent part of the morphisms.In the following we will use the identities(28) ¨ ¨ ¨ µ “ ¨ ¨ ¨ µ “ ´ t t t t t t ¨ ¨ ¨ µ The first equality follows from Lemma 2.4 after using (3) on the second strand labelled 4 to pullit to the left. The second equality can be checked by a applying (3) three times.Coming back to H ´ p T q we apply the isomorphisms F » qF p q ‘ q ´ F p q ,F » qF p q ‘ q ´ F p q ,F » F p q , to obtain the isomorphic complex ˜ qF t F p q F b µq ´ F t F p q F b µ ¸ ¨˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˝ ´ t
21 4 3 3 2 1 t
12 4 3 3 2 1 ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‚
ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ ¨˚˚˚˚˝ q F t F p q F b µF t F p q F b µqF t F F b µqF t F p q F b µ ˛‹‹‹‹‚ . ot even Khovanov homology 25 By Gaussian elimination of the acyclic complex qF t F p q F b µ t ÝÝÝÝÝÝÝÝÝÑ qF t F p q F b µ. we obtain the homotopy equivalent complex q ´ F t F p q F b µ ¨˚˚˚˚˚˚˚˚˚˚˚˚˚˝ ´ t t ´ ´ t t ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‚ ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ ¨˚˝ q F t F p q F b µF t F p q F b µqF t F F b µ ˛‹‚ . Applying the isomorphisms(29) F p q » qF p q p q ‘ q ´ F p q p q and F p q » F p q p q gives the isomorphic complex ˜ F t F p q p q F b µq ´ F t F p q p q F b µ ¸ ¨˚˚˚˚˚˚˚˚˚˚˝ t t t ´ t
34 4 3 2 1 ´ t
43 4 3 2 1 f g ˛‹‹‹‹‹‹‹‹‹‹‚
ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ ¨˚˝ q F t F p q p q F b µF t F p q p q F b µqF t F F b µ ˛‹‚ , or ˜ F t F p q p q F b µq ´ F t F p q p q F b µ ¸ ¨˚˚˚˚˚˚˚˚˚˚˝ t t t ´ t
34 4 3 2 1 t t t t t f g ˛‹‹‹‹‹‹‹‹‹‹‚ ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ ¨˚˝ q F t F p q p q F b µF t F p q p q F b µqF t F F b µ ˛‹‚ , where f (resp. g ) is the composite of the map from F p q p q (resp. q ´ F p q p q ) to q ´ F p q in (29) and ´ t t Pedro Vaz
Gaussian elimination of the acyclic complex F t F p q p q F b µ ´ t ÝÝÝÝÝÝÝÝÑ F t F p q p q F b µ, yields the homotopy equivalent complex q ´ F t F p q p q F b µ ¨˚˝ h ˛‹‚ ÝÝÝÑ ˜ q F t F p q p q F b µqF t F F b µ ¸ , where h “ ´ t t ` t t t t Since we are only interested in the lowest homological degree we restrict to considering thecomplex q ´ F t F p q p q F b µ h ÝÝÑ qF t F F b µ. Finally, applying the isomorphism F t F F b » F t F p q F b results in the isomorphic com-plex q ´ F t F p q p q F b µ ÝÝÑ qF t F F b µ. Adding the shift corresponding to the normalization (26), and using the fact that F t F p q p q F b µ is a k -supervector space of graded dimension q ` q ´ , yields H ´ p T q “ q ´ k ‘ q ´ k , which agrees with the odd Khovanov homology of T .4. F URTHER PROPERTIES OF R In this section we sketch several of its higher representation theory properties of R , some ofthem we have used in the previous section.4.1. Supercategorical action on R Λ p k, d q . Given a gl n -weight Λ “ p Λ , . . . , Λ n q we write Λ “ p Λ ´ Λ , . . . , Λ n ´ ´ Λ n q for the corresponding sl n -weight. The super algebra R Λ p ν q for gl k is defined to be the same as the superalgebra R Λ p ν q for sl k .We now explain how the bifunctor Φ : R ˆ R Λ Ñ R Λ in (18). gives rise to an action of gl k on R Λ p k, d q for Λ a dominant integrable gl k -weight of level 2 with Λ ` ¨ ¨ ¨ Λ n “ d . A diagram D in R Λ p k, d q with leftmost region labelled µ defines a web W D with bottom boundary labelled Λ and with top boundary labelled µ . We denote f i , e i P U q p gl k q the Chevalley generators. ot even Khovanov homology 27 Behind Tubbenhauer’s construction in [23] there is the observation that the transformation(30) baba ` bb ´ bab ba ` b ´ turns any web into a web with all horizontal edges pointing to the right. This goes through theobvious embedding of gl k into gl k ` . ‚ The generator f i acts by stacking the web(31) ¨ ¨ ¨ µ i µ i ` ¨ ¨ ¨ on the top of W D . This means that f i acts on R Λ p n, d q as the functor that adds a strand labelled i to the left of D . ‚ To define the action of e i we stack the web ¨ ¨ ¨ µ i µ i ` ¨ ¨ ¨ on the top of W D , then we use Tubbenhauer’s trick (30) to put in a form that uses only F ’s. Thetransformation in (30) is not local and in order to be well defined one needs to keep trace of theindices before and after acting with an e i . Tubbenhauer’s trick gives ¨ ¨ ¨ µ i ´ µ i µ i ` µ i ` µ i ` µ i ` ´ ¨ ¨ ¨ Everytime we act with an e i we embed U q p gl k q ã Ñ U q p gl k ` q and set e i p W D q “ f p µ q ¨¨¨ i ´ p µi ´ q f p µ i q i f p µ i ` ´ q i ` f i ` p µi ` q ¨¨¨ k p µk q p µ, qp W D q . After being acted with an e j , f i acts on W D through the web corresponding to f i ` p µ, q . Pedro Vaz
We define the action of e i on R Λ p k, d q as the superfunctor that adds p µ q ¨ ¨ ¨ i p µ i q i ` p µ i ` ´ q ¨ ¨ ¨ k p µ k q to the left of D (here p µ q , etc..., are the thicknesses) that is, we act with the identity 2-morphismof F p µ q ¨¨¨ i ´ p µi ´ q F p µ i q i F p µ i ` ´ q i ` F i ` p µi ` q ¨¨¨ k p µk q p µ, q .Denote Φ p e i q and Φ p f i q the morphisms in R Λ that act as endofunctors of R Λ p n, d q throughthe action above. It is clear that Φ p uv q “ Φ p u q Φ p v q for u , v P U q p gl k q . Note that Φ p qp µ q is acanonical element F can p µ q as introduced in (23). Lemma 4.1.
We have natural isomorphisms Φ p e i q Φ p f i qp λ q » Φ p f i q Φ p e i qp λ q ‘ Φ p q ‘r λ i s p λ q if λ i ě , Φ p f i q Φ p e i qp λ q » Φ p e i q Φ p f i qp λ q ‘ Φ p q ‘r´ λ i s p λ q if λ i ď . Proof.
These are instances of the categorified higher Serre relations. Denote F u “ F p λ q ¨¨¨ i ´ p λi ´ q and F d “ F i ` p λi ` q ¨¨¨ k p λk q . We have Φ p e i q Φ p f i qp λ q “ F u F p λ i ´ q i F p λ i ` q i ` F d F i p λ, q» F u F p λ i ´ q i F p λ i ` q i ` F i p . . . , λ i , λ i ` , , λ i ` , . . . q F d , p λ, q , and Φ p f i q Φ p e i qp λ q “ F t F i ` F p λ i q i F p λ i ` ´ q i ` F b p λ, q , and therefore, it is enough to check that the relations above are satisfied by the superfunctors F p λ i ´ q i F p λ i ` q i ` F i p λ i , λ i ` , q and F i ` F p λ i q i F p λ i ` ´ q i ` p λ i , λ i ` , q . Suppose λ i ě λ i ` . Then wehave λ i P t , u and λ i ` P t , u . The computations involved are rather simple and we cancheck the four cases separately.(1) p λ i , λ i ` q “ p , q : Φ p e i q Φ p f i qp λ q “ F p λ i ´ q i F p λ i ` q i ` F i p λ i , λ i ` q “ F i p , q “ ‘ F can p , q , “ Φ p f i q Φ p e i qp λ q ‘ Φ p qp λ q . (2) p λ i , λ i ` q “ p , q : Φ p e i q Φ p f i qp λ q “ F i ` F i p , , q “ Φ p f i q Φ p e i qp λ q . (3) p λ i , λ i ` q “ p , q : Φ p e i q Φ p f i qp λ q “ F i F i p , , q » qF p q i p , , q ` q ´ F p q i p , , q “ Φ p q ‘r s p λ q . (4) p λ i , λ i ` q “ p , q : Φ p e i q Φ p f i qp λ q “ F i F i ` F i p , , q » ‘ F p q i F i ` p , , q “ Φ p f i q Φ p e i qp λ q ‘ Φ p qp λ q . ot even Khovanov homology 29 An this proves the first isomorphism in the statement. The second isomorphism can be checkedusing the same method. (cid:3)
The proof of Lemma 4.1 uses several supernatural transformations between the various com-positions of Φ p f i qp λ q and Φ p e i qp λ q and Φ p qp λ q that can be given a presentation in terms of thediagrams from R . We act with such diagrams by stacking them on the top of the diagrams forthe image of Φ . On the weight space p , q these maps coincide with the maps used to define thechain complex for a tangle diagram in the previous section. In the general case these maps areunits and co-units of adjunctions in the following. Lemma 4.2.
Up to degree shifts, the functor Φ p e i q is left and right adjoint to Φ p f i q . Lemma 4.3.
We have the following natural isomorphisms: Φ p e j q Φ p f i qp λ q » Φ p f i q Φ p e j qp λ q for i ‰ j, Φ p f i q Φ p f i ˘ q Φ p f i qp λ q » Φ p f p q i q Φ p f i ˘ qp λ q ‘ Φ p f i ˘ q Φ p f p q i qp λ q , Φ p e i q Φ p e i ˘ qp λ q Φ p e i q » Φ p e p q i q Φ p e i ˘ qp λ q ‘ Φ p e i ˘ q Φ p e p q i qp λ q . Proof.
The proof consists of a case-by-case computation. We ilustrate the proof with the case of Φ p e i q Φ p f i ` qp λ q » Φ p f i ` q Φ p e i qp λ q and leave the rest to the reader. We have Φ p e i q Φ p f i ` qp λ q “ F p λ i q i F p λ i ` ´ q i ` F p λ i ` ` q i ` F i ` p λ q , and Φ p f i ` q Φ p e i qp λ q “ F p λ i q i F i ` F p λ i ` ´ q i ` F p λ i ` q i ` p λ q , which are zero unless λ i ` “ and λ i ` P t , u . If λ i ` “ these can be written Φ p e i q Φ p f i ` qp λ q “ F p λ i q i F p λ i ` ` q i ` F i ` p λ q , and Φ p f i ` q Φ p e i qp λ q “ F p λ i q i F i ` F i ` F p λ i ` q i ` p λ q . The case λ i ` “ is immmediate and the case λ i ` “ follows from the Serre relation (8)-(9). (cid:3) As explained in [1, Sections 1.5 and 6] the Grothendieck group of a ( Z -graded) monoidalsupercategory is a Z r q ˘ , π s{p π ´ q -algebra. Nontrivial parity shifts will occur when applyingTubbenhauer’s trick. All the above can be used to prove the following. Theorem 4.4.
The assignement above defines an action of U q p gl k q on R Λ p k, d q . With this ac-tion we have an isomorphism of K p R Λ p k, d qq with the irreducible, finite-dimensional, U q p gl k q -representation of highest weight Λ at π “ . Pedro Vaz R EFERENCES [1] J. Brundan and A. Ellis, Monoidal supercategories.
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