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arXiv:1604.05246v2 [math.QA] 9 May 2016
Odd Khovanov’s arc algebra
Gr´egoire Naisse and Pedro Vaz
Universit´e catholique de Louvain
Louvain-la-Neuve, Belgium
May 10, 2016
Abstract
We construct an odd version of Khovanov’s arc algebra Hn. Extending the center to
elements that anticommute, we get a subalgebra that is isomorphic to the oddification
of the cohomology of the (n, n)-Springer variety. We also prove that the odd arc algebra
can be twisted into an associative algebra.
Contents
1 Introduction 2
1.1 Sketch of the construction and main results . . . . . . . . . . . . . . . . . . 2
2 Reminders 3
2.1 Khovanov’s arc algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 OddKhovanovhomoloy ............................. 5
2.3 Odd cohomology of the Springer varieties . . . . . . . . . . . . . . . . . . . . 7
3 Odd arc algebra 9
3.1 An example : OH2
C................................ 12
3.2 The odd center of OH n
C.............................. 13
4 The odd center of OH nand the (n, n)-Springer variety 14
5 Turning OHn
Cinto an associative algebra 22
5.1 The Putyra-Shumakovitch associator . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Twisting OHn
C................................... 26
5.3 Classification results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Perspectives 29
1
1 Introduction
Arc algebras were first introduced by Khovanov in [20] to extend his categorification of
Jones’ link invariant [19] to tangles. One of his main ingredients is a certain Frobenius
algebra of rank 2, which coincides with the cohomology ring of complex projective space.
In a follow-up paper [21], Khovanov showed that the arc algebra Hnfrom [20] is closely
related to the geometry of Springer varieties. Indeed, he proved that the center of Hnis
isomorphic to the cohomology ring of the (n, n)-Springer variety. Later on, Chen and Kho-
vanov defined in [11] subquotients of Hnwith the aim of giving an explicit categorification of
the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
To do so, they categorified the n-folded tensor power of the fundamental representation of
Uq(sl(2)) together with its weight space decomposition. Additionally, Khovanov’s arc alge-
bra was further studied in the sequence of papers [5, 6, 7, 8, 9], where most of its interesting
representation-theoretic properties were revealed.
Khovanov’s arc algebra was later generalized in several directions by several authors. It
was first generalized to sl3-web algebras in [26] and then to sln-web algebras in [25] and
further studied in [35, 36]. In [33], a version of the arc algebra associated with gl(1|1)
was constructed, motivated by a representation-theoretic categorification of the Alexander
polynomial. In [14, 15], a Khovanov algebra of type Dwas introduced, in connection with
orthosymplectic Lie algebras. More recently, a variant of Khovanov’s arc algebra based on
Blanchet’s version of Khovanov homology [3] was constructed in [16]. This was extended
in [17] to gl2-arc and web algebras associated with the variants of Khovanov homology
from [10] and [12]. One of the main properties of the arc/web algebras above is that, except
the ones from [14, 15] for which it is not know, they all admit topological constructions using
cobordisms or foams.
In [28], Ozsvath, Rasmussen and Szabo used an exterior version of the Frobenius algebra
originally used by Khovanov to give an odd version of Khovanov homology. Odd Khovanov
homology agrees with the even (usual) Khovanov homology from [19] modulo 2, but they
differ over fields of characteristic other than 2. Moreover, both categorify the Jones poly-
nomial (see for example [34] for further properties). Odd Khovanov homology was given
a (Bar-Natan style [2]) topological set-up by Putyra in [30]. He introduced the so-called
chronological cobordisms, which are cobordisms together with some extra structure related
to a height function.
In this paper we use the set-up from [28, 30] and construct an odd version of Khovanov
arc algebra from [20].
1.1 Sketch of the construction and main results
The first step in the construction of an odd version of Khovanov’s arc algebra is to replace the
TQFT obtained by the Frobenius algebra from [19] by the chronological TQFT from [28, 30].
As explained in [30], in order to get a well-defined category of cobordisms one has to choose
an orientation for each of the local Morse moves. It was proved in [30] (and in [28] in an
algebraic set-up) that any consistent choice of orientations would result in the same link
homology. This is no longer the case if one tries to extend odd Khovanov homology to
2
tangles, since the context is different from [28, 30]. In particular, a priori there is no reason
for two different choices of orientations to result in isomorphic odd arc algebras.
Our first result is that we get a family of odd arc algebras indexed by all possible choices.
We denote OHn
Cthe odd arc algebra associated with the choice Cof chronological cobor-
disms. As a second result, we get that for all Cand all n≥2, the odd arc algebra OHn
Cis
nonassociative. This is done in Section 3.
In Section 4, we prove an odd version of Khovanv’s results from [21]. Namely, we prove
that the odd center of OH n
Cis isomorphic to the odd cohomology of the (n, n)-Springer
variety as given by Lauda and Russell in [24]. In this paper they constructed an oddification
of the cohomology of the Springer variety associated to any partition, by replacing polynomial
rings and symmetric functions by their odd counterparts.
As mentioned above, the algebra OHn
Cis not associative. But this is not too big a
problem, since it is a quasialgebra in the sense of Albuquerque and Majid. They defined in [1]
the notion of quasialgebra, which is a nonassociative graded algebra with an associator given
by a 3-cocycle coming from a higher structure, that is, a monoidal category. In Section 5,
we introduce a grading on OHnby a groupoid and prove the quasi-associativity of OHn
C,
the associator depending only on C. The idea of looking at an odd version of Khovanov’s
arc algebra as a quasialgebra goes back to the attempts of Putyra and Shumakovitch to
extend the odd Khovanov homology to tangles. An extended discussion over generalised
quasialgebras can be found in the unpublished work of Putyra [29]. We prove that the
associator admits a primitive which gives a twist τ. Twisting the multiplication of OHn
Cby
this τdefines an associative algebra which keeps the odd flavor of OHn
C. In addition, we
prove that all choices of Cand of twist lead to isomorphic algebras.
Acknowledgments. We would like the thank Krzysztof Putyra for the discussions and
ideas leading to the results of Section 5. We thank also Daniel Tubbenhauer for comments
on a previous version of this paper. G.N. is a Research Fellow of the Fonds de la Recherche
Scientifique - FNRS, under Grant no. 1.A310.16. P.V. was supported by the Fonds de la
Recherche Scientifique - FNRS under Grant no. J.0135.16.
2 Reminders
To begin, we recall the three main constructions we will use: the Khovanov arc algebra, the
TQFT from odd Khovanov homology and the oddification of the cohomology of the Springer
varieties.
2.1 Khovanov’s arc algebra
As the construction in this paper follows Khovanov’s original setup from [20], we give below
a sketch of the construction of the arc algebra Hn.
Crossingless matchings. Let Bnbe the set of crossingless matchings of 2npoints, that
is, all ways one can pair 2npoints on a horizontal line by arcs placed below this line. For
3
b∈Bn, we denote by W(b) the reflection of bacross the horizontal line and by W(b)athe
gluing of W(b) on the top of a∈Bn. It is clear that W(b)ais a disjoint union of circles. For
example, we have in B2:
a=❴❴❴❴❴❴ , b =❴❴❴❴❴❴ ,
W(b) = ❴❴❴❴❴❴ , W (b)a=❴❴❴❴❴❴ ❴❴❴❴❴❴ .
We also write W(d)cW (b)afor the concatenation of W(d)con top of W(b)a, which is the
disjoint union of W(d)cand W(b)a, see for example (2).
Contraction cobordisms. Given a diagram W(c)bW (b)a, we construct a cobordism
Scba :W(c)bW (b)a→W(c)a(1)
by contracting the arcs of bwith their symmetric counterparts in W(b) by saddles:
❴❴❴❴❴
❴❴❴❴❴
arcs contraction
−−−−−−−−−−−−→
by a saddle
❴❴❴❴❴
❴❴❴❴❴
rescaling
≃❴❴❴❴❴
This gives a surface with one saddle point for each arc in b. Therefore, Scba has (minimal)
Euler characteristic −n, is embedded in R2×[0,1] and is unique up to isotopy. Indeed,
contracting the symmetric arcs in two different orders gives rise to homeomorphic surfaces
and thus, the construction does not depend on any choice. Moreover, Scba can be given a
canonical orientation. The picture to keep in mind is:
❴❴❴❴❴❴❴❴❴❴❴❴
W(c)
bW (b)
❴❴❴❴❴❴❴❴❴❴❴❴
a
Scba
−−−−−−→
❴❴❴❴❴❴❴❴❴❴
❴❴❴❴❴❴❴❴❴❴
≃
W(c)
❴❴❴❴❴❴❴❴❴❴❴
a
(2)
Frobenius algebra. Let A:= Z[X]/(X2) be the Z-graded abelian group with grading
given by deg 1 = −1 and deg X= 1. This group possesses the structure of a Z-algebra when
equipped with the polynomial multiplication. However, notice that this multiplication has
degree 1 and does not gives a graded algebra structure. We turn Ainto a Frobenius algebra
by defining a trace,
tr : A→Z,tr(1) = 0,tr(X) = 1.
As the trace is non-degenerate, this defines a TQFT,
F: 2Cob →Z-grmod,
where Z-grmod is the category of finite-dimensional Z-graded Z-modules and 2Cob the cat-
egory of oriented cobordisms between 1-manifolds, see [23]. From now on, unless stated
otherwise, we will always assume that graded means Z-graded.
4
Thus, we get F(W(b)a)≃A⊗|W(b)a|for |W(b)a|the number of circle components in
W(b)a. Moreover, the comultiplication map is explicitly given by
∆ : A→A⊗A, ∆(1) = X⊗1 + X⊗1,∆(X) = X⊗X.
Applying this TQFT on the cobordism (1), we get a morphism,
F(W(c)b)⊗ZF(W(b)a)≃F(W(c)bW (b)a)F(Scba)
−−−−→ F(W(c)a).(3)
This morphism has degree nsince the multiplication and comultiplication maps in Ahave
degree 1 and Scba possesses nsaddle points.
Arc algebra. Define the graded abelian groups
Hn:= M
a,b∈Bn
b(Hn)a, b(Hn)a:= F(W(b)a){n},
where the notation {n}means that we shift the degree up by n. Therefore, as the maximal
number of components in W(b)ais n, every element x∈F(W(b)a) has degree deg x≥ −n
and thus, Hnis a Z+-graded group. In order to define a multiplication in Hn, we first let
the product d(Hn)c⊗Zb(Hn)a→Hnbe zero whenever c6=b. Then, for the other cases, we
define the multiplication such that the diagram
c(Hn)b⊗Zb(Hn)a
≃
//c(Hn)a
F(W(c)b)⊗ZF(W(b)a){2n}(3)
//F(W(c)a){n}
≃
OO
commutes. The associativity of the multiplication follows from the fact that Fis a TQFT.
Moreover, the sum Pa∈Bn1a, with 1athe unit in a(Hn)a≃A⊗n{n}, is a unit for Hn. All
of this sum up to:
Proposition 2.1. (Khovanov, [20, Proposition 1]) Structures, described above, make Hn
into a Z+-graded associative unital Z-algebra.
2.2 Odd Khovanov homoloy
Ozsvath, Rasmussen and Szabo constructed in [28] an odd version of Khovanov homology
using some “projective TQFT” replacing F(projective meaning here that it is well-defined
only up to sign). Putyra extended in [30] the work of Bar-Natan for Khovanov homology [2]
by giving a topological framework for the odd homology: the chronological cobordisms. In
addition, Putyra’s work allows the construction of the odd Khovanov homology using a
well-defined functor. In this subsection, we mainly follow the exposition in [30].
5
Chronological cobordims. Recall that a chronological 2-cobordism is a 2-cobordism
equipped with a chronology, that is, a Morse function with one critical point at each critical
level. Moreover, at each critical point, we choose an orientation of the space of unstable
directions in the gradient flow induced by the chronology. We write that choice by an ar-
row. These chronological 2-cobordisms, taken up to isotopy which preserves the orientations
and the chronology, form a category with composition given by gluing. We denote it by
2ChCob. Every chronological 2-cobordism can be built from the six elementary chronologi-
cal 2-cobordisms:
,
GGA
,
GGA
,
GA
,
DG
,(4)
which are called respectively a birth, a merge, a split, a positive death, a negative death and
a twist. As we are only interested in chronological 2-cobordisms, we will forget the prefix 2-.
The odd functor. We describe the functor OF : 2C hCob →Z-grmod from [28]. Morally,
objects of 2ChCob are disjoint unions of circles. For Ssuch an union we denote by V(S) the
free abelian group generated by the components of Swith a grading such that each generator
has degree 2. We define
OF (S) := V∗V(S){−|S|},
with V∗V(S) being the exterior algebra generated by the elemens of V(S) and |S|the number
of components.
We now define the functor on each of the elementary cobordisms (4). Let S1and S2be
objects of 2C hCob with S2containing one circle more than S1. For a birth of a circle from
S1to S2, there is a canonical inclusion V(S1)⊂V(S2) (the new generator being the circle
cupped by the birth cobordism). This induces a morphism
OF !:V∗V(S1)⊂
−→ V∗V(S2), v 7→ 1∧v.
Consider a merge of two circles a1, a2in S2to a single one in S1with an arrow a1Aa2.
The arrow represent one of the two possible choices of orientation of the merge, the other
being denoted a2Aa1. There is an isomorphism of groups V(S1)≃V(S2)/{a1−a2}and
thus the canonical projection V(S2)→V(S2)/{a1−a2}induces a morphism
OF
GGA
a1a2
:V∗V(S2)→V∗(V(S2)/{a1−a2})≃V∗V(S1).
It is not hard to see that the choice of orientation does not change the result in this case
and we get
OF
GGA
a1a2
=OF
D GG
a1a2
.
6
Now say we have split sending a∈S1to b1Ab2in S2. Again, there is a natural identifi-
cation V(S2)≃V(S1)/{b1−b2}, but now we also use the isomorphism
V∗(V(S1)/{b1−b2})≃(b1−b2)∧V∗V(S1)
to get a morphism
OF
GGA
b1b2
:V∗V(S2)≃(b1−b2)∧V∗V(S1)⊂
−→ V∗V(S1).
As a matter of fact, this morphism is easily computable by replacing the occurences of aby
b1(or b2) and multiplying by (b1−b2). For example, 1 is sent to b1−b2and ais sent to
(b1−b2)∧b1=b1∧b2. Notice also that reversing the orientation changes the sign of the
morphism
OF
GGA
b1b2
=−OF
D GG
b1b2
.
Suppose we have a positive (in other words, anticlockwise oriented) death of a∈S2. We
associate to it the morphism given by contraction with the dual of a1
OF
GA
a1
:V∗V(S2)→V∗V(S1), v 7→ a∗
1(v).
The negative one is given by the opposite.
Finally, the twist is given by a the permutation of the corresponding terms
OF
a1a2
a1a2
:V∗V(S2)→V∗(V(S2)) ,
a17→ a2,
a27→ a1,
a1∧a27→ a2∧a1.
Remark 2.2. Since the orientations of the merges does not change the result of the functor,
we will ignore them in our discussion.
2.3 Odd cohomology of the Springer varieties
First, let us recall the definition of a Springer variety.
Definition 2.3. Let λ= (λ1,...,λm) be a partition of m,Embe a complex vector space
of dimension mand zλ:Em→Embe a nilpotent linear endomorphism with |λ|nilpotent
Jordan blocks of size λ1, . . . , λm. The Springer variety for the partition λis
Bλ:= {complete flags in Emstabilized by zλ}.
7
The cohomology ring of Bλcan be computed by quotienting the polynomial ring in m
variables by the ideal of partial symmetric functions (see [13] for more details). Write (n, n)
for the partition λ= (n, n) of 2n.
Theorem 2.4. (Khovanov, [21, Theorem 1.1]) There is an isomorphism of graded algebras
Z(Hn)≃H(Bn,n,Z).
Lauda and Russell constructed in [24] an oddification of the cohomology of the Springer
varieties, denoted OH(Bλ,Z). Like the usual cohomology is obtained as a quotient of the
polynomials by the partial symmetric functions, they constructed OH(Bλ,Z) as a quotient
of the ring OP olmof odd polynomials
OP olm:= Zhx1,...,xmi
hxixj+xjxi= 0 for all i6=ji,deg(xi) = 2,
by some ideal. Since we only need the case m= 2nand λ= (n, n) for our discussion, we
restrict to this case from now on.
Definition 2.5. (Lauda & Russell, [24]) The odd cohomology of the (n, n)-Springer variety
is the quotient
OH(Bn,n,Z) := O P ol2n/OIn
where OInis the left ideal generated by the set of odd partial symmetric functions
OCn:= (ǫI
r:= X
1≤i1<···<ir≤2n
xI
i1...xI
irk∈ {1,2,...,n},|I|=n+k,
r∈ {n−k+ 1, n −k,...,n+k}),
for all Iordered subset of {1,...,2n}of cardinality n+kand
xI
ij:= (0,if ij/∈I,
(−1)I(ij)−1xij,otherwise,
with I(ij) the position of ijin I.
In general, the odd cohomology of a Springer variety is only a module over the odd
polynomials. However, in case λ= (n, n), it is a graded algebra. This is due to the fact that,
thanks to [24, Lemma 3.6], x2
i∈OInfor all i. Thus, OInis a 2-sided ideal. As a matter of
fact, OH(Bn,n,Z) also possesses the structure of a superalgebra with superdegree given by
dividing the degree by 2. Finally, by construction, the algebra OH(Bn,n,Z) is isomorphic
modulo 2 to H(Bn,n,Z).
Example 2.6. OH(B2,2,Z) is given by the odd polynomials in 4 variables x1, x2, x3, x4
quotiented by the (not minimal) relations:
x1−x2+x3−x4= 0,
−xixj+xixk−xjxk= 0,∀i < j < k ∈[1,4],
−x1x2+x1x3−x1x4−x2x3+x2x4−x3x4= 0,
xixjxk= 0,∀i < j < k ∈[1,4],
−x1x2x3+x1x2x4−x1x3x4+x2x3x4= 0,
x1x2x3x4= 0.
8
3 Odd arc algebra
In this section, we construct an odd version of the Khovanov arc algebra Hn. Therefore, we
will closely follow the construction from above, replacing the TQFT Fby the odd functor
OF from Section 2.2. First, we define for all n≥0 the following graded abelian groups,
OHn:= M
a,b∈Bn
b(OHn)a, b(OHn)a:= OF (W(b)a){n},
such that OHnis Z+-graded.
The first difficulty we encounter when we try to define a multiplication as in Section 2.1 is
that we have to choose a chronology and signs for the splits. In the odd Khovanov homology
from [28], the signs are forced by the requirement that the cube of resolutions anticommutes
(all possible choices leading to isomorphic cubes). However, in our case, there is no condition
other than that the cobordisms must be embedded in R2×[0,1]. This means we have to
consider all possible choices.
Contraction cobordisms. For each a, b, c ∈Bn, there is a canonical cobordism with
minimal number of critical points (up to homeomorphism and embedded in R2×[0,1]) from
the diagram W(c)bW (b)ato W(c)a. This corbordism is given by contracting the arcs of b
with their symmetric counterparts in W(b), as in the definition of Hnin Section 2.1. To be
able to apply OF , we need to define a chronological cobordism and there are several ways
to do so:
•We have to choose a chronology, or in other words, we have to choose an order for
contracting the symmetric arcs of bW (b), with taking care of never contracting two
arcs before having contracted the one surrounding them. This is required to get an
embedded surface.
•We have to give an orientation for the critical points, especially for the splits (we do
not need to orient the merges by Remark 2.2). We express the two possibilities by an
arrow:
a
a❴❴❴❴❴
❴❴❴❴❴
−→
b1b2
D G
❴❴❴❴❴
❴❴❴❴❴
or
a
a❴❴❴❴❴
❴❴❴❴❴
−→
b1b2
GA
❴❴❴❴❴
❴❴❴❴❴
meaning that we split the component ain two components b1, b2with the orientation
b2Ab1in the first case and b1Ab2in the second one.
Remark 3.1. There is always at least one possible choice: it suffices to go through the end
points of bfrom left to right and contracting whenever we encounter an arc which was not
already contracted, then orienting the splits from left to right (i.e. putting an arrow from
the component passing through the left point to the one passing through the right point).
We now assume that for each triplet a, b, c ∈Bnwe have chosen a chronological cobor-
dism. We write it Ccba and denote the collection all of them by C:= {Ccba|a, b, c ∈Bn}. In
addition, we write Cnfor the set of all possible choices of such a set C.
9
Multiplication. Like in the even case, we let the multiplication
d(OHn)c⊗Zb(OHn)a→ {0} ⊂ OHn
be zero for c6=b. We define the multiplication c(OHn)b⊗Zb(OHn)a→c(OHn)ausing the
morphism OF (Ccba). More precisely, there is a morphism
OF (W(c)b)⊗ZOF (W(b)a)→O F (W(c)bW (b)a) : (x, y)7→ x∧y(5)
induced by the inclusions W(c)b⊂W(c)bW (b)aand W(b)a⊂W(c)bW (b)a. We compose it
with OF (Ccba) to obtain the multiplication by making the following diagram commutes:
c(OHn)b⊗Zb(OHn)a
≃
//c(OHn)a
OF (W(c)b)⊗ZOF (W(b)a)(5)
//OF (W(c)bW (b)a)OF (Ccba )//OF (W(c)a).
≃
OO
This map is grading preserving thanks to the minimality hypothesis on the Euler Charac-
teristic of Ccba which guarantees that the degree of OF (Ccba) is n.
Unit. We write 1afor the unit in the exterior algebra V∗V(W(a)a) and we easily check that
the sum Pa∈Bn1ais a unit for the multiplication defined in the paragraph above. We also
write b1afor the unit in the exterior algebra V∗V(W(b)a) (notice b1ais not an idempotent).
Proposition 3.2. For n≥2and any choice C∈ Cn, the multiplication defined above is not
associative.
Proof. First, suppose that n= 2 and let C∈ C2be an arbitrary choice of cobordisms.
Consider a, b ∈B2such that
a=❴❴❴❴❴❴ , b =❴❴❴❴❴❴ .
Take x=b1∈b(OHn)b, y =b1a∈b(OHn)aand z=a1b∈a(OHn)b, where b1is the element
in the exterior algebra coming from the outer circle in the diagram W(b)b. We write b2for
the element generated by the inner circle. Then we compute x(yz) as follows. The cobordism
Cbab :→
is given by a merge followed by a split such that the element yz is
yz =α(b1−b2),
where α∈ {±1}depends on the chosen orientation for the unique split in Cbab. Then, x(yz)
is given by
Cbbb :→, x(yz) = −αb1∧b2
10
since Cbbb is composed by two merges. Now, we compute (xy)zby
Cbba :→, xy =c1,
with c1the element coming from the unique circle in W(b)aand then
Cbab :→,(xy)z=αb1∧b2.
This means that for every C∈ C2, (xy)z=−x(yz). To conclude the proof, we observe that
this example can be extended for all n≥2 by adding the same arcs at right of aand b,
an:= ❴❴❴❴❴❴ ❴❴❴ ... ❴❴❴ , bn:= ❴❴❴❴❴❴ ❴❴❴ ... ❴❴❴ ,
such that the exact same computation can be done for all n≥2.
Definition 3.3. We denote by OH n
Cthe Z+-graded nonassociative, unital Z-algebra given
by OHnwith the multiplication obtained from a C∈ Cn.
Remark 3.4. We sometimes write b(OHn
C)a. By this, we mean that we take the elements
of the group b(OH n)a, but viewed as elements in OHn
C.
As we get a family of algebras for all n≥0, it is legitimate to ask if is it possible to
classify them. We give some partial answer to this question in Section 5.
Remark 3.5. From now on, unless otherwise specified, all assertions are valid for all n∈N
and C∈ Cnand we forget the specify them.
To ensure that OHn
Cis an odd version of Hn, we have to check that the two algebras
agree modulo 2.
Proposition 3.6. There is an isomorphism of graded algebras
Hn⊗ZZ/2Z≃OHn⊗ZZ/2Z.
Proof. The result follows directly from the construction since the functor Fused to define
Hnagrees up to sign with OF .
It is interesting (and it will be useful) to notice that OHn
Ccontains a collection of exterior
algebras as subalgebras.
Proposition 3.7. There is an inclusion of graded algebras
M
a∈BnV∗Zn≃M
a∈Bn
a(OHn
C)a⊂OHn
C.
Proof. It is enough to notice that OF (W(a)a)≃V∗Znas W(a)ais a collection of ncircle,
and to remark that the multiplication
a(OHn
C)a⊗Za(OHn
C)a→a(OHn
C)a
is the usual product in an exterior algebra. Indeed, the cobordism Caaa consists of nmerges
and no split and thus, gives the exterior product.
11
Diagrammatic notation. To simplify the notation, we propose a way to write the gen-
erators of OH nusing diagrams. First, notice that there is an order on the components of
W(b)a, for a, b ∈Bn, given by reading the diagram from left to right. More precisely, for
a1, a2∈W(b)a, we say that a1< a2whenever a1passes through an end point of a(or
equivalently b) which is at the left of all end points contained in a2.
An element x1∧ · · · ∧ xkin b(OH n)ais written as the diagram W(b)awhere we draw
the components with a plain line for each xiand a dashed one for the others. Moreover, we
require that x1∧ · · · ∧ xkis in the order induced by reading the diagram from left to right,
otherwise we add a sign to recover this order. Thus, we get for example in OH2:
=b1∧1,−=b2∧b1,
=a1b,=c1,
with ai, biand cifollowing the conventions from the proof of Proposition 3.2.
3.1 An example : OH2
C
We construct explicit multiplication tables for OH2
C, with Cthe choice from Remark 3.1.
Since the multiplication maps for ∗2(OH 2)a⊗b(OH2)∗1→0 and ∗2(OH2)b⊗a(OH2)∗1→0
are zero for all ∗1,∗2∈ {a, b}, we give the tables only for ∗2(OH2)a⊗a(OH2)∗1→ ∗2(OH2)∗1
and ∗2(OH2)b⊗b(OH2)∗1→ ∗2(OH2)∗1. Moreover, these tables are written with the
convention:
y
x xy
By direct computation we get
OH2
C
0 0 0
−0 0 0
0 0 0 0 0
0− −
0 0 0 −0
and
12
OH2
C
0 0 0
−0 0 0
0 0 0 0 0
0−
0 0 0 0
3.2 The odd center of OHn
C
When talking about exterior algebras (or in general superalgebras), it is common to consider
the supercenter which is an extension of the center to the elements that anticommute. In
the same spirit, we define the odd center for OHn
C.
Definition 3.8. We define the parity of an homogeneous element z∈a(OHn)bby
p(z) := deg(z)−deg(a1b)
2=deg(z)−n+|W(b)a|
2mod 2,
with |W(b)a|the number of circle components in W(b)a.
One can easily see that this number counts the factors of z=a1∧... ∧am, i.e.
p(a1∧ · · · ∧ am) = mmod 2.
Definition 3.9. We call odd center of OHn
Cthe subset
OZ(OHn
C) := z∈OH n
C|zx = (−1)p(x)p(z)xz, ∀x∈OHn
C.
Remark 3.10. The parity does not give a grading on OHn
C, since there are elements x, y in
OHn
Csuch that
p(xy)6=p(x) + p(y) mod 2.
This means that OH n
Cis not a superalgebra with respect to p. As a matter of fact, the parity
descends to a degree on the antisymmetric subalgebra La∈Bna(OHn
C)a⊂OHn
C, in which
the odd center lives.
Proposition 3.11. There are inclusions Z(OHn
C)⊂OZ(OHn
C)⊂La∈Bna(OHn
C)a.
Proof. The second inclusion is immediate since for every z∈OZ(OHn
C) one can decompose
z=Pa,b∈Bnbzawith bza∈b(OHn
C)aand get bza= 1bz1a=z(1b1a) = 0, unless b=a. The
first inclusion is obtained by first noticing that Z(OHn
C)⊂La∈Bna(OHn
C)aby the same
argument as before, and then observing that every commutating element has even parity.
Moreover, one can check that the odd center is an associative superalgebra with superde-
gree given by the parity, and is characterized by the following property.
13
Proposition 3.12. An element z=Pa∈Bnzais in OZ(OHn
C)if and only if zb.b1a=b1a.za
for all a, b ∈Bn.
Proof. An element z=Pa∈Bnzacommutes with x=Pa,b∈Bnbxaif and only if zcommutes
with every bxa. Moreover, z.bxa= (zb.b1a)∧bxa= (−1)p(z)p(x)bxa∧(zb.b1a) and we have
bxa∧(zb.b1a) = bxa∧(b1a.za) = bxa.z if and only if zb.b1a=b1a.za.
The following result allows us to write OZ(OH n) with no ambiguity.
Proposition 3.13. For all C, C ′∈ Cn, there is an isomorphism of graded (super)algebras
OZ(OHn
C)≃OZ(OHn
C′).
Proof. The condition zb.b1a=b1a.zafrom Proposition 3.12 depends only on the multiplica-
tions maps
b(OHn)b⊗Zb(OHn)a→b(OH n)aand b(OHn)a⊗Za(OHn)a→b(OH n)a.
It is not hard to see that those are defined using only cobordisms without split such that
they do not depend on C. Moreover, by Proposition 3.7, La∈Bna(OH n
C)abehaves like a
direct sum of exterior algebras and thus, does not depend on C.
4 The odd center of OHnand the (n, n)-Springer variety
We are now ready to prove one of the main results of this paper, which is to construct an
explicit isomorphism between the odd cohomology of the (n, n)-Springer variety and the odd
center of the odd Khovanov arc algebra.
Theorem 4.1. There is an isomorphism of graded (super)algebras between OZ(OH n)and
OH(Bn,n,Z). Moreover, this isomorphism is given by
h:OH(Bn,n,Z)→OZ(OH n), xi7→ X
a∈Bn
ai,
where aiis generated by the circle component of W(a)apassing through the ith end point
of a, counting from left.
The proof of this theorem will occupy the rest of this section and is split in three steps.
First, we define a morphism h0:OP ol2n→OZ(OH n
C) and prove that OInfrom Definition 2.5
lies in the kernel of this map, inducing the map hon OH(Bn,n,Z). Secondly, we show that
his injective using the equivalence up to sign between the odd and the even case together
with Theorem 2.4. Finally, we show that the ranks of the two algebras are equal using the
cohomology of a geometric construction based on hypertori.
14
Existence of h
To construct h, we first define the algebra map
h0:OP ol2n→OHn
C, xi7→ X
a∈Bn
ai,
where aiis generated by the circle component of W(a)apassing through the ith end point
of a, counting from left. It is well defined since
h0(xixj) = X
a∈Bn
ai∧aj=−X
a∈Bn
aj∧ai=−h0(xjxi).
Lemma 4.2. The image of h0lies in the odd center of OHn
C
h0(OP ol2n)⊂OZ(OHn).
Proof. The proof is straightforward from Proposition 3.12 and the fact that for all a, b ∈Bn,
we have
(h0(xi))b1a=bi.b1a=b1a.ai=b1a(h0(xi)),
since aiand biare both sent to the component of W(b)apassing through the ith point.
Now, we want to show that ǫI
ris in the kernel of h0for all Iand ras in Definition 2.5.
This is equivalent to show that ǫI
rlies in the kernel of the morphism
ha:OP ol2n→a(OHn
C)a, xi7→ ai,
for all a∈Bn, since h=Pa∈Bnha.
For the sake of simplicity, we fix an element a∈Bn. We also denote by E2nthe ordered
set {1,...,2n}and we see it as the end points of a, from left to right. For I⊂E2n, we call
a pair of distinct points in Iwhich are linked by an arc in a, an arc of I. The other points
of Iare called free points. For R={i1,...,ir} ⊂ I, we also write
ǫI
R:= xI
i1...xI
ir
such that
ǫI
r=X
R⊂I,|R|=r
ǫI
R.
Lemma 4.3. Let I⊂E2nbe a subset with |I|=n+k. Then, Icontains at least karcs and
at most n−kfree points.
Proof. There is at most nfree points and we pick n+kpoints, thus we have to pick at least
karcs. Then, it remains n+k−2kpoints which can be free.
Lemma 4.4. If R⊂I⊂E2ncontains an arc of I, then ha(ǫI
R) = 0.
Proof. This assertion comes from the fact that if iis connected to i′in a, then ai=ai′and
thus, ha(xixi′) = ai∧ai′= 0.
15
Lemma 4.5. For all R⊂I⊂E2nwith |R|> n, one has ha(ǫI
R) = 0.
Proof. There is at most nfree points in I, but Rcontains at least n+ 1 points. So, R
contains an arc and the result follows from the preceding lemma.
Lemma 4.6. For all R⊂I⊂E2nwith |I|=n+kand |R| ≥ n−k+ 1, there exists an arc
(j, j′)in Iwith jor j′in R.
Proof. We have to choose n−k+ 1 points in I, but there are at most n−kfree points by
the Lemma 4.3.
For x∈I⊂E2n, we write
FI(x) := (1,if xis free in I,
0,otherwise,
and, for R⊂Iand (j, j′) an arc of Isuch that j∈Ror j′∈R, we define
pR,I ((j, j′)) := X
x∈I
j<x<j ′
FI(x)−X
y∈R
j<y<j′
FI(y) + X
z∈R
j<z<j′
(1 −FI(z)) mod 2.
We say that a point x(resp. an arc (k, k′)) is contained in an arc (j, j′) if j < x < j′(resp.
j < k < k′< j′). Therefore, pR,I ((j, j′)) counts the number of free points of Icontained in
the arc (j, j′) and which are not in Rplus the number of points of Ralso contained in (j, j′)
and which are not free. Denote the set of all maximal sub-arcs of (j, j′) with an extremity
in Rby ]j, j′[R. Also, write ]j, j [max
Rfor the set of all maximal sub-arcs of (j, j′) with an
extremity in R, that is, the sub-arcs not contained in any other arcs from ]j, j′[R.
Lemma 4.7. If R⊂I⊂E2ncontains no arc of I, then for any arc (j, j′)in Iwith jor j′
in Rwe have
pR,I ((j, j′)) = X
(k,k′)∈]j,j ′[max
R
(pR,I ((k, k′)) + 1) + X
x∈I\R
x/∈(k,k′)∈]j,j′[R
FI(x) mod 2,
with the left sum being on all maximal sub-arcs of (j, j′)with kor k′∈Rand the right sum
on all x∈I\Rthat are not contained in any of these sub-arcs.
Proof. All points contained in a maximal sub-arc (k, k′) from the left sum are contained in
(j, j′). Moreover, kor k′is a free point in Rand thus, has a contribution by +1.
Lemma 4.8. Let R⊂I⊂E2nbe subsets with |I|=n+kand n−k+ 1 ≤ |R| ≤ n. If R
does not contain any arc of I, then there exists an arc (j, j′)of Iwith jor j′in Rand such
that pR,I ((j, j′)) = 0 mod 2.
Proof. First, by Lemma 4.6 there is at least one arc of Iwith an extremity in R. We write
L6=∅for the set of all such arcs. Now, we suppose by contradiction that pR,I = 1 mod 2
on all those arcs. By the Lemma 4.7, we have for all (j, j′)∈L,
pR,I ((j, j′)) = X
x∈I\R
x/∈(k,k′)∈]j,j′[R
FI(x) mod 2,
16
with xnot contained in any sub-arc (k, k′) of (j, j′) such that kor k′is in R. Since by
contradiction hypothesis this sum must be equal to 1 mod 2, there is at least one such x
which is free and thus at least |L|free points in I\R. However, we know that there is at
least n−k− |L|+ 1 free points in Rand by Lemma 4.3 there is at most n−kfree points
in I, which is a contradiction.
Lemma 4.9. Let R⊂I⊂E2nbe subsets and (j, j′)be an arc of Iwith j∈R(resp. j′∈R).
We have
ha(ǫI
R) = (−1)(pR,I ((j,j′))+1) ha(ǫI
R′),
with R′obtained by taking Rwhere we remove jand add j′(resp. j′and j), respecting the
order of I.
Proof. The proof is an induction on the size of I∩]j, j′[ and R. If I∩]j, j′[= ∅, then clearly
pR((j, j′)) = 0 mod 2 and we get the result since I(j′) = I(j) + 1 and thus, xI
j=−xI
j′.
Then, to get the result in the general case, we check how the sign changes when we add free
points and arcs to Iand points to R. We leave the details to the reader.
Proposition 4.10. For all a∈Bn,k∈ {1,...,n},r≥n−k+ 1 and all I⊂E2nsuch that
|I|=n+k, we have
ha(ǫI
r) = 0.
Proof. This result directly follows from Lemmas 4.5, 4.4, 4.8, and 4.9.
Corollary 4.11. The map h0induces a homomorphism of graded (super)algebras given by
h:OH(Bn,n,Z)→OZ(OH n), xi7→ X
a∈Bn
ai.
Injectivity of h
We will need the following result.
Lemma 4.12. The algebra homomorphism induced by h
h:OH(Bn,n,Z)⊗ZZ/2Z→OZ(OH n)⊗ZZ/2Z
is an isomorphism.
Proof. The proof results from the commutativity of the diagram
OH(Bn,n,Z)⊗ZZ/2Zh//
≃
OZ(OHn)⊗ZZ/2Z
≃
H(Bn,n,Z)⊗ZZ/2Z≃//Z(Hn)⊗ZZ/2Z
which comes from the equivalence modulo 2 between the odd and the even cases, knowing
that H(Bn,n,Z) is isomorphic to Z(Hn) by a morphism similar to h, see [21, Section 5.3] for
more details.
17
Proposition 4.13. The homomorphism his injective.
Proof. From [24, Theorem 3.8] we know that there exists a basis OB for OH(Bn,n,Z) and
that it gives rise to a basis OB ⊗1 in the tensor product OH(Bn,n,Z)⊗ZZ/2Z. Hence,
if we suppose by contradiction that there is a relation in the image of OB by h, we would
have the same relation in the image of OB ⊗1 by h. But, by the preceding lemma his an
isomorphism, so we have a relation in the basis OB ⊗1 which is a contradiction.
Computing the rank of OZ(OHn)
To show the existence of an isomorphism between Z(Hn) and H(Bn,n), Khovanov con-
structed in [21] a manifold e
Susing products of 2-spheres. We make a similar construction,
but using circles instead of spheres. The inspiration from Khovanov work should be clear.
All cohomology groups and rings in this section are supposed to be taken on Z.
Definition 4.14. For an a∈Bn, let Ta⊂T2n:= S1× · · · × S1
|{z }
2n
be the set of all points
(x1,...,x2n)∈T2nsuch that if iis linked to jby an arc of a, then xi=xj. We also define
e
T:= [
a∈Bn
Ta⊂T2n.
One can notice that Ta≃Tnas we equalize npairs of coordinates. In the same spirit,
we have that Tb∩Ta≃T|W(b)a|, with xk=xlwhenever the kth and the lth end points are in
the same component of W(b)a. As a result, e
Tis a collection of hypertori identified to each
other on certain subtori.
It is well-know that the cohomology ring of an n-torus is the exterior algebra generated
by nelements of degree 1. If we forget the grading, then we get an isomorphism of superrings
a(OHn
C)a≃H(Ta) and an isomorphism of abelian groups b(OHn
C)a≃ab H(Tb∩Ta) for all
a, b ∈Bn. Lifting the (super)ring structure from H(Tb∩Ta), we get a (super)ring structure
on b(OHn
C)aand (super)ring morphisms
γa;b,a :a(OHn
C)a→b(OHn
C)a, x 7→ b1ax,
γb;b,a :b(OHn
C)b→b(OHn
C)a, x 7→ xb1a.
The inclusions Tb∩Ta⊂Taand Tb∩Ta⊂Tbinduce ring morphisms on the cohomology
ψa;b,a :H(Ta)→H(Tb∩Ta),
ψb;b,a :H(Tb)→H(Tb∩Ta).
Lemma 4.15. The morphisms defined above are such that the following diagram of ring
morphisms commutes:
H(Tb)ψb;b,a //
≃
H(Tb∩Ta)
≃
H(Ta)
ψa;b,a
oo
≃
b(OHn
C)bγb;b,a //b(OHn
C)a a(OHn
C)a.
γa;b,a
oo
18
Proof. Say H(Tb)≃V∗{t1,...,tn}. Then the map ψb;a,b identify tkwith tlwhenever the kth
and the lth end points of bare in the same component of W(b)a. The map γb;b,a does exactly
the same on the generators of b(OHn
C)bas Cbba merges the corresponding components.
Definition 4.16. Let Iand Jbe finite sets and Ai, Bjbe rings for all i∈Iand j∈J.
Moreover, let βi,j :Ai→Bjbe ring morphisms for some pairs (i, j)∈I×Jwith
β:= Xβi,j :Y
i∈I
Ai→Y
j∈J
Bj.
We define the equalizer Eq(β) of βas the subring of QAisuch that for (ai)i∈I∈Eq(β) we
have
βi,j (ai) = βk,j (ak)
whenever βi,j and βk,j are defined.
By Proposition 3.12, we get that OZ(OH n) = Eq(γ) for γ:= Pa6=bγa;b,a +γb;b,a. Thus,
if we define ψ:= Pa6=bψa;b,a +ψb;b,a, then by Lemma 4.15, we get a commutative diagram
H(e
T)κ//Eq(ψ)//
≃
L
a∈Bn
H(Ta)ψ//
≃
L
a6=b∈Bn
H(Tb∩Ta)
≃
OZ(OHn)≃//Eq(γ)//L
a∈Bn
a(OHn
C)aγ//L
a6=b∈Bn
b(OHn
C)a,
(6)
with κcoming from the factorization by Eq(ψ) of the map φ:H(e
T)→La∈BnH(Ta) induced
by the inclusions Ta֒→e
T. This factorization exists since im φ⊂Eq(ψ). Our goal now is to
prove that κis an epimorphism such that rk(OZ(OH n)) ≤rk(H(e
T)).
Definition 4.17. We say that there is an arrow a→bfor a, b ∈Bnif there exists a
quadruplet 1 ≤i < j < k < l ≤2nsuch that (i, j),(k, l)∈aand (i, l),(j, k)∈b. Visually,
we have
❴❴❴❴❴❴ijk l −→ ijk l❴❴❴❴❴❴ .
This leads to a partial order a≺bif there exists a chain a→a1→ · · · → ak→b. We
extend (arbitrarily) this partial order to a total order <on Bn.
Lemma 4.18. For all a∈Bn, we have
T<a ∩Ta=[
b→a
(Tb∩Ta)
with T<a := Sb<a Tb.
Proof. We use similar arguments as in [21, Lemma 3.4], replacing Sby T.
19
Lemma 4.19. There exists a cellular decomposition of Tawhich restrains to a decomposition
of T<a ∩Ta, which itself restrains to the decomposition of Tb∩Tafor all b→a. This
decomposition is such that there is n
kcells of dimension kin Ta.
Proof. We construct a similar decomposition as in [21, Lemma 3.5]. We stress the fact that
the cells are not in even degree only for our case.
Corollary 4.20. The morphism
H(T<a ∩Ta)→M
b<a
H(Tb∩Ta),
induced by the inclusions (Tb∩Ta)⊂(T<a ∩Ta)is injective.
We remark that T≤a=T<a ∪Tasuch that there is a Mayer-Vietoris sequence:
... //Hm−1(Ta∩T<a)
δ
rr❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡
Hm(Ta∪T<a)//Hm(Ta)⊕Hm(T<a)//Hm(Ta∩T<a)
δ
rr❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡
Hm+1(Ta∪T<a )//...
(7)
Proposition 4.21. The following sequence is exact:
H(T≤a)φ//Lb≤aH(Tb)ψ−
//Lb<c≤aH(Tb∩Tc),
where φis induced by the morphisms Tb֒→T≤a, and where we define
ψ−:= X
b<c≤a
(ψb,c −ψc,b),
with
ψb,c =ψb;b,c :H(Tb)→H(Tb∩Tc),
induced by the inclusion (Tb∩Tc)֒→Tb.
Proof. The proof is an induction on ausing Corollary 4.20 like in [21, Proposition 3.8] with
the only difference that we lose the left part 0 →in our sequence.
Proposition 4.22. There is an epimorphism of superrings
k:H(e
T)→OZ(OHn).
Proof. We take amaximal in the Proposition 4.21 giving an exact sequence
H(e
T)φ//LbH(Tb)ψ−
//Lb<c H(Tb∩Tc)
and we observe that by definition
Eq(ψ) = ker(ψ−) = im(φ)
such that κ:H(e
T)→Eq(ψ) from diagram (6) is surjective.
20
Now we show that the rank of H(e
T) is the same as the rank of OH(Bn,n,Z). By [24,
Corollary 3.9], we already know that rk(OH(Bn,n,Z)) = 2n
n.
Lemma 4.23. For all k≥0, the cohomology groups
Hk(Ta∪T<a), H k(Ta), Hk(T<a )and Hk(Ta∩T<a)
are free of ranks satisfying the relation
rk(Hk(Ta∪T<a)) = rk(Hk(Ta)) + rk(Hk(T<a )) −rk(Hk(Ta∩T<a)).
Proof. The proof is an induction on Bn. We consider the Mayer-Vietoris sequence (7) and we
claim that the morphisms Hk(Ta)→Hk(Ta∩T<a) are surjective and thus, that the boundary
operators δare zero. Indeed, the decomposition from Lemma 4.19 has the same number of
k-cells as the rank of Hk(Ta) so they are independent generators for the cohomology. Seeing
that the cell decomposition restricts to Ta∩T<a , the cohomology groups Hk(Ta∩T<a) are
free of rank given by the cell decomposition and the morphisms are surjective. This claim
gives us an isomorphism
Hk(Ta∩T<a)≃Hk(Ta)⊕Hk(T<a)
Hk(Ta∪T<a).
If ais minimal, then the lemma is trivial. For the general case, we get the result by induction
since Hk(Ta), Hk(T<a) and Hk(Ta∩T<a ) are free and so is Hk(Ta∪T<a ).
Proposition 4.24. H(e
T)is a free abelian group of rank
rk(H(e
T)) = 2n
n.
Proof. We obtain a cellular partition of e
Tby first taking the cellular decomposition of Ta0
from Lemma 4.19, with a0∈Bnthe minimal element, and then by adding the cells Tam\T<am
for all am∈Bnfollowing the total order. We claim that the rank of H(e
T) is given by the
number of cells of the partition. Indeed, all cohomology groups are free and the relation from
the Lemma 4.23 gives us the claim since rk(Hk(Tam)) −rk(Hk(Tam∩T<am)) counts exactly
the number of cells of Tam\T<am. Finally, like in [21] (and proved in [27, Lemma 3.64]), the
number of cells is 2n
nand this concludes the proof.
Corollary 4.25. We have
rk(OZ(OHn)) = rk(OH(Bn,n,Z)).
Proof. By Proposition 4.13, we have
rk(OZ(OHn)) ≥rk(OH (Bn,n,Z))
and by Propositions 4.22 and 4.24, we get
rk(OZ(OHn)) ≤rk(H(e
T)) = 2n
n= rk(OH (Bn,n,Z)).
The two inequalities together conclude the proof.
Corollary 4.11, Proposition 4.13 and Corollary 4.25 together prove Theorem 4.1. Notice
that this also prove that κis in fact an isomorphism of superrings.
21
5 Turning OHn
Cinto an associative algebra
In this section, we show that we can twist the multiplication of OHn
C, turning it into an
associative Z[i]-algebra. To do so, we begin by proving that OH n
Cis a quasialgebra in the
sense of Albuquerque-Majid [1], extended to grading by groupoid like in [29]. Finally, we
give some classification results on all those algebras.
5.1 The Putyra-Shumakovitch associator
The material in this subsection is due to Putyra and Shumakovitch [31] 1.
Grading by a groupoid. A groupoid is a small category with every morphism admitting
an inverse. We say that a ring Ris graded by a groupoid Gif
R=M
g∈Hom(G)
Rgand Rg1Rg2⊂Rg1◦g2,
whenever g1and g2are composable, and Rg1Rg2= 0 otherwise.
Arc grading. Let Gnbe the groupoid with objects given by the elements of Bnand with a
unique morphism a→bfor all a, b ∈Bn. By uniqueness of the morphisms, the composition
is such that a→b→cis equal to a→c. We can view the morphism a→bas the diagram
W(b)awith the composition defined for all a, b, c ∈Bnby W(c)b◦W(b)a=W(c)a. It is
clear that Gnis a groupoid as every morphism a→bpossesses an inverse b→aand a→a
is the identity.
Example 5.1. We can put Gnin form of a diagram. For example, G2can be pictured as
❴❴❴❴❴❴ ❴❴❴❴❴❴ ""❴❴❴❴❴❴
❴❴❴❴❴❴ ❴❴❴❴❴❴
++
❴❴❴❴❴❴
❴❴❴❴❴❴ ❴❴❴❴❴❴
kk
❴❴❴❴❴❴ ❴❴❴❴❴❴
bb.
The decomposition
OHn=M
a,b∈Bn
b(OHn)a=M
W(b)a∈Hom(Gn)
b(OHn)a
gives a grading of OH n
Cby Gn. We call by the quantum degree the grading by Zcoming
from the gradation on V∗V(S) , written ||q. We call by the arc degree the grading from Gn,
denoted | |B. We get a bigrading, written | |, by the groupoid Gn× Z , with Zbeing the
abelian group Zviewed as a category with one abstract object ⋆, morphisms given by the
integers and composition obtained by taking the sum, i.e.
⋆z1
−→ ⋆z2
−→ ⋆=⋆z1+z2
−−−→ ⋆.
1And we would like to thank Krzysztof Putyra for explaining it to us.
22
Notice that Hom(Gn× Z)≃Hom(Gn)×Z.
As a matter of fact, OHn
Cis graded by a subgroupoid with arrows given by diagrams
W(b)aand a quantum degree in 2Zor 2Z+ 1 depending on whether |W(b)a| ≡ nmod 2 or
not.
Definition 5.2. We denote by Gn× Z2the groupoid given by the same objects as Gn× Z
but with arrows given by
(a, ⋆)(W(b)a,2k)
−−−−−−→ (b, ⋆),for all k∈Zif |W(b)a| ≡ nmod 2,
(a, ⋆)(W(b)a,2k+1)
−−−−−−−−→ (b, ⋆),for all k∈Zif |W(b)a| 6≡ nmod 2.
Quasialgebra. Recall that we proved in Proposition 3.2 that OHn
Cis not an associative
algebra. We claim that it is almost one: it is a quasialgebra in the sense of [1] extended to
grading by groupoids [29]. Before defining quasialgebras, recall that the nerve of a category
Cis the simplicial set generated by its morphisms. We write it N(C) and we denote by Nn(C)
the set of compositions of nmorphisms in C.
Definition 5.3. Aquasialgebra Ais a (nonassociative) R-algebra graded by a groupoid G
with a 3-cocycle
φ:N3(G)→R∗
where R∗⊂Rare the invertible elements, such that
(xy)z=φ(|x|,|y|,|z|)x(yz) (8)
for all (homogeneous with compatible degrees) x, y, z ∈A. We call φthe associator of A.
Remark 5.4. The condition for φbeing a 3-cocycle means that
φ(h, k, l)φ(g, hk, l)φ(g, h, k) = φ(gh, k, l)φ(g, h, kl)
for all sequences
eg
←− dh
←− ck
←− bl
←− a∈ G.
The condition (8) forces φto be a 3-cocycle on deg A:= {deg(x)∈Hom(G)|x∈A}since,
the multiplication being well defined, the following diagram must commute:
((A⊗A)⊗A)⊗Aφ⊗id //
φ
(A⊗(A⊗A)) ⊗A
φ
(A⊗A)⊗(A⊗A)
φ**
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
A⊗((A⊗A)⊗A)
id ⊗φ
tt✐
✐
✐
✐✐
✐
✐
✐
✐✐
✐
✐
✐
✐✐
✐
A⊗(A⊗(A⊗A)).
(9)
We can view Z∗={±1}as the group Z/2Zby the isomorphism x∈Z/2Z7→ (−1)x.
Because of that, we will write the associator of OHn
Cas a map with codomain Z/2Z.
23
Lemma 5.5. There exists a unique map
ϕch
C:N3(Gn)→Z/2Z
such that for all a, b, c, d ∈Bnwe have
OF (Cdba ◦Cdcb IdW(b)a) = (−1)ϕch
C(W(d)c,W (c)b,W (b)a)OF (Cdca ◦IdW(d)cCcba).
Proof. The two cobordisms Cdba ◦Cdcb IdW(b)aand Cdca ◦IdW(d)cCcba have the same Euler
characteristic and the same source and target, and thus are homeomorphic. This means they
are related by changes of chronology and orientations, which induce only potential changes
of sign.
Lemma 5.6. For all choices C∈ Cn, the cobordism Ccba is composed by the same number
of splits.
Proof. This is immediate as all Ccba have the same Euler characteristic.
We define
ϕcom :N3(Gn× Z)⊂(Hom(Gn)×Z)3→Z/4Z,
((W(d)c, k),(W(c)b, l),(W(b)a, m)) 7→ (k−n+|W(d)c|)s(W(c)b, W (b)a) mod 4,
where s(W(c)b, W (b)a) is the number of splits coming from the cobordism Ccba , which does
not depend on Cby Lemma 5.6. If we take x, y, z ∈OH n
C, then
ϕcom(|x|,|y|,|z|) = 2p(x)s(|y|B,|z|B).(10)
In this spirit, we define the parity p((W(d)c, k)) := k−n+|W(d)c|
2∈ {0,1/2,1,3/2}for any
element in Hom(Gn× Z). Notice that the parity gives an integer for every element in the
subgroupoid Gn× Z2from Definition 5.2.
Lemma 5.7. The map
ϕC:= ϕch
C+ϕcom/2 : N3(Gn× Z2)→Z/2Z
is such that
(xy)z= (−1)ϕC(|x|,|y|,|z|)x(yz)
for all x, y, z ∈OHn
C.
Proof. Suppose we have x∈d(OHn
C)c,y∈c(OHn
C)band z∈b(OHn
C)a. We compute
xy =S(d, c, b)∧x∧y,
(xy)z=S(d, b, a)∧S(d, c, b)∧x∧y∧z,
yz =S(c, b, a)∧y∧z,
x(yz) = S(d, c, a)∧x∧S(c, b, a)∧y∧z,
where S(d, c, b) are the terms coming from the splits of the cobordism Cdcb. Notice that we
make an abuse of notation by identifying x, y with their images in d(OHn
C)bin the first line,
and so on.
This computation means that the non-associativity comes from two phenomena:
24
•The commutation between the elements coming from the splits of the product yz and
the left term x, that is
S(d, c, a)∧x∧S(c, b, a)∧y∧z= (−1)p(x)p(S(c,b,a))S(d, c, a)∧S(c, b, a)∧x∧y∧z.
By (10), we have p(x)p(S(c, b, a)) = p(x)s(W(c)b, W (b)a) = ϕcom(|x|,|y|,|z|)/2.
•The change of chronology and orientations between the cobordisms Cdba ◦Cdcb IdW(b)a
and Cdca ◦IdW(d)cCcba, meaning that
S(d, b, a)∧S(d, c, b) = (−1)ϕch
C(|x|B,|y|B,|z|B)S(d, c, a)∧S(c, b, a)
by Lemma 5.5.
In conclusion, we have
(xy)z= (−1)ϕch
C(|x|B,|y|B,|z|B)+ϕcom(|x|,|y|,|z|)/2x(yz)
and this finishes the proof.
Lemma 5.8. The map ϕCis a 3-cocycle. More generally, the map
ψC:= 2ϕch
C+ϕcom :N3(Gn× Z)→Z/4Z,
is a 3-cocycle.
Proof. We mainly use Remark 5.4. Take a, b, c, d, e ∈Bn. Injecting e1d,d1c,c1band b1a
in (9), we get
chedcb +chedba +chdcba =sedcscba +checba +chedca (11)
where scba =s(W(c)b, W (b)a), chdcba =φch
C(W(d)c, W (c)b, W (b)a), and so on. Now suppose
we have a sequence (e, ⋆)g
←− (d, ⋆)h
←− (c, ⋆)k
←− (b, ⋆)l
←− (a, ⋆)∈ Gn× Z . We compute
ψC(h, k, l) + ψC(g, hk, l) + ψC(g, h, k) =
2(p(h)scba +chdcba +p(g)sdba +chedba +p(g)sdcb +chedcb),
and
ψC(gh, k, l) + ψC(g, h, kl) = 2 (p(gh)scba +checba +p(g)sdca +chedca).
We see that p(gh) = p(g) + p(h) + sedc and
sdba +sdcb =scba +sdca = #splits in the cobordism W(d)cW (c)bW (b)a→W(d)a,
such that by (11) we get
ψC(h, k, l) + ψC(g, hk, l) + ψC(g, h, k) = ψC(gh, k, l) + ψC(g, h, kl),
which concludes the proof for ψC. We get the claim for ϕCby seeing that ϕC=ψC|G×Z2/2.
Theorem 5.9. The nonassociative ring OHn
Cis a quasialgebra with associator ϕC.
Proof. This is an immediate consequence from Lemmas 5.7 and 5.8.
We call ϕCthe Putyra-Shumakovitch associator.
25
5.2 Twisting OHn
C
Twisted multiplication. The idea of twisting a G-graded R-algebra Aby a map
τ:N2(G)→R∗
is to define a new algebra Aτby the same elements as A, but with a multiplication given by
Aτ⊗RAτ→Aτ,(x, y)7→ x∗τy:= τ(|x|,|y|)xy
for all x, y ∈Aτ, and where xy is the product in A.
Proposition 5.10. Let Abe a quasialgebra graded by Gwith associator φ. If φis a cobound-
ary, then there exist a twist τsuch that Aτis associative.
Proof. By definition of coboundary, there exists a map
τ:N2(G)→R∗
such that
φ(g, h, k) = τ(g, h)τ(g, hk)−1τ(gh, k)τ(h, k)−1(12)
for all sequence dg
←− ch
←− bk
←− a∈ G. Let Aτbe the twisting of Aby this τ. Then we have
(x∗τy)∗τz=τ(|x|,|y|)τ(|xy|,|z|)(xy)z,
x∗τ(y∗τz) = τ(|x|,|yz|)τ(|y|,|z|)x(yz),
for all x, y, z ∈Aτand thus, by (8) and (12), we conclude that Aτis associative.
The geometric realization of the nerve of a category C, denoted |N(C)|, is a topological
space constructed by gluing simplexes respecting the simplicial structure of the nerve.
Lemma 5.11. The geometric realization of N(Gn)is a simplex of dimension Cn−1, for Cn
the nth Catalan number:
|N(Gn)| ≃ ∆(Cn−1).
Proof. The proof is immediate from the fact that Bnhas cardinality Cnand Gnpossesses
one unique morphism between each pair of objects.
Lemma 5.12. The cohomology groups of Gn× Z are
H0(N(Gn× Z),Z/4Z)≃Z/4Z,
H1(N(Gn× Z),Z/4Z)≃Z/4Z,
H≥2(N(Gn× Z),Z/4Z)≃0.
Proof. First, by Lemma 5.11, we get that |N(Gn× Z)| ≃ ∆m×S1for m=Cn−1. By the
K¨unneth formula, we get
Hk(N(Gn× Z),Z/4Z)≃M
i+j=k
Hi(∆m,Z/4Z)⊗Hj(S1,Z/4Z),
which gives the claim.
26
It still remains some technicalities to resolve before being able to apply Proposition 5.10
to OHn
C: we do not know the cohomology of Gn× Z2and thus, we are not able to show that
the Putyra-Shumakovitch associator is a coboundary. Therefore, we work with ψC, which
has an image in Z/4Zand gives square roots of −1. Hence, we must consider the extended
version OHn
C⊗ZZ[i] to the Gaussian integers, with Z[i]∗={1, i, −1,−i} ≃ Z/4Z. By
Lemmas 5.7 and 5.8, OH n
C⊗ZZ[i] is a quasialgebra graded by Gn× Z with ψCas associator.
Theorem 5.13. For all C∈ Cnthere exists a map τC:N2(G2× Z)→Z/4Zsuch that the
twisted algebra (OHn
C⊗ZZ[i])τCis associative.
Proof. By Lemma 5.12, we have
H3(N(Gn× Z),Z/4Z)≃0,
and thus every 3-cocycle is a coboundary. In particular, the associator ψCis a coboundary
and we can apply Proposition 5.10.
Remark 5.14. Notice that the twist is not necessarily unique and thus we get potentially
a family of associative algebras for each C∈ Cn.
Example 5.15. We construct an explicit (OH2
C)τCbased on the choice Cfrom Remark 3.1.
We twist this algebra with
τC(( ,1 + 4k),(,∗)) = i, τC(( ,2 + 4k),(,∗)) =i,
τC(( ,2 + 4k),(,∗)) = −1, τC(( ,3 + 4k),(,∗)) = −1,
τC(( ,3 + 4k),(,∗)) = −i, τC(( ,0 + 4k),(,∗)) = −i,
for every k∈Zand τC= 1 everywhere else. In short, we get the multiplication table:
(OH2
C)τC
0 0 −0
−0 0 −0
0 0 0 0 0
0− −
0 0 0 0
for aand the one for bstays the same. An exhaustive computation (which can easily be
done by computer) confirms that dτCgives the associator in this case.
Remark 5.16. For this example, the twisting in OHn
Cresults in integer coefficients which
is not surprising since the geometric realization of G2× Z2has dimension 2 and thus there
exists a twist for the associator ϕCin this case. In general, this is not true and we lose the
property that the algebra agrees modulo 2 with Hn⊗ZZ[i].
27
5.3 Classification results
For now, we have a family of quasialgebras {OHn
C}indexed by Cnand a family of associative
algebras indexed by Cnand the twists. In this section, we partially classify these families.
Proposition 5.17. Let C, C ′be two choices in Cnand ϕC, ϕC′be respectively the associators
of OHn
Cand OHn
C′. If ϕC=ϕC′, then the two quasialgebras are isomorphic, OHn
C≃OHn
C′.
Proof. Seeing that Ccba and C′
cba are related by a change of chronology and orientations,
there is a map ηC,C ′:N2(Gn)→Z/2Zsuch that
OF (Ccba) = (−1)ηC,C′(W(c)b,W (b)a)OF (C′
cba),
for all a, b, c ∈Bn. Writing ∗Cfor the product in OHn
Cand ∗C′for the one in OHn
C′, this
means that x∗Cy= (−1)ηC,C′(|x|B,|y|B)x∗C′yfor all x, y ∈OHn. We compute
OF (Cdba ◦Cdcb IdW(b)a) = (−1)ηC,C′(W(d)b,W (b)a)+ηC,C′(W(d)c,W (c)b)OF (C′
dba ◦C′
dcb IdW(b)a),
OF (Cdca ◦IdW(d)cCcba) = (−1)ηC,C′(W(d)c,W (c)a)+ηC,C′(W(c)b,W (b)a)OF (C′
dca ◦IdW(d)cC′
cba),
such that by definition of the associators, we get
dηC,C′=φch
C′−φch
C,(13)
and thus, as φch
C=φch
C′by hypothesis, ηC,C′is a 2-cocycle. By Lemma 5.11, we know that
H2(N(Gn),Z/2Z)≃0,
and consequently, ηC,C ′is a coboundary. Hence, ηC,C′=dλC,C′for a λC,C′:N1(Gn)→Z/2Z.
This means that the morphism
x7→ λC,C′(|x|B)x:OHn
C→OHn
C′,
is an isomorphism of quasialgebras.
For the associative twisted algebra, the case is much simpler and all algebras are isomor-
phic. This means that the choice of Cand of twist τCdoes not matter in the end.
Proposition 5.18. For all choices C, C′∈ Cnand all choices of twists τC, τC′, there is an
isomorphism (OH n
C⊗ZZ[i])τC≃(OHn
C′⊗ZZ[i])τC′.
Proof. For all x, y ∈OHn, we have
x∗τCy=iτC(|x|,|y|)+2ηC,C′(|x|B,|y|B)−τC′(|x|,|y|)x∗τC′y,
where ∗τCis the product in (OHn
C⊗ZZ[i])τC, and we write
θC,C′:= τC+ 2ηC,C′−τC′:N2(Gn× Z)→Z/4Z.
We compute dθC,C ′=ψC+ 2dηC,C ′−ψC′
(13)
= 0, and by using similar arguments as in the
proof of Proposition 5.17, we conclude that the two algebras are isomorphic.
28
Corollary 5.19. The associative twisted algebra is uniquely determined up to isomorphism.
We write it OHn
τ.
Remark 5.20. Finding a twist is not an easy task, which can entail some serious difficulties
to construct OHn
τ.
Proposition 5.21. The three arc algebras are not isomorphic
Hn⊗Z[i]6≃ OHn
τ6≃ OHn
C⊗Z[i].
Proof. The first two are associative algebras in opposition with the last one. Let us begin by
the case n= 2. We know that the center of H2has a graded rank 1 + 3q2+ 2q4. Howewer,
by a similar argument as in Proposition 3.11, we have
Z(OH2
τ)⊂a(OH2)a⊕b(OH2)b.
However, OH2
τbehaves like an exterior algebra on this subset, implying that elements anti-
commute and thus the graded rank of the center is 0 in degree 2. Therefore, H2and OH2
τ
have non-isomorphic centers and thus, cannot be isomorphic as algebras.
This can be extended for all n≥2. Suppose x, y ∈a(OHn
τ)awith |x|q=|y|q= 1, their
products are given by
(x, y)7→ τ(|x|,|y|)x∧y,
(y, x)7→ τ(|y|,|x|)y∧x,
since OF (Caaa) is the product in the exterior algebra V∗V(W(a)a). However |x|=|y|,
implying τ(|x|,|y|) = τ(|y|,|x|) and thus, xy =−yx.
Proposition 5.22. The odd center of OH2
τis not isomorphic to OH(B2,2,Z[i]).
Proof. It is not hard to compute that the odd center of OH 2
τis generated by the elements
OZ(OH2
τ) = h+,−,−, , i
and thus has graded rank 1 + 2q2+ 2q4. However, we know that OH (B2,2,Z[i]) has graded
rank 1 + 3q2+ 2q4.
Despite this result, one can still show that OHn
τcontains a subalgebra which is isomorphic
to OH(Bn,n,Z[i]).
6 Perspectives
One natural application of the work in this paper is the construction of odd link homology
theories (Putyra-Shumakovitch’s work in progress using the structure of quasialgebras [31]).
The fact that the twist τis not explicit may bring several technical difficulties in defin-
ing (and working with) (OH n
τ, OH n
τ)-bimodules. Another possibility consists to work with
quasibimodules, that is bimodules with the associativity axiom given by an associator, as
29
in [29]. With such a theory at hand, it is not hard to guess that the braid group action
on the category of complexes of (OHn
C, OH n
C)-quasibimodules up to homotopy descends to
an action of the (−1)-Hecke algebra from [24, Section 4] on its (odd) center, paralleling the
even case (see [20, Section 5.3]).
The fact that the twisted odd arc algebra OHn
τis defined over the Gaussian integers was
forced by technical reasons. One question that we leave open is to find whether it is possible
to twist OHn
Cover the integers.
The construction in this paper shares several features with Ehrig-Stroppel’s Khovanov
arc algebra of type Dfrom [14]. It would be interesting to find a connection between these
two arc algebras.
In [7], an action of the 2-Kac-Moody of Rouquier [32] (and therefore of Khovanov-
Lauda’s [22]) on Khovanov’s arc algebras was constructed. The results of Rouquier on
strong categorical actions [32], together with the fact that our associative arc algebra is not
isomorphic to Khovanov’s, imply that the 2-Kac-Moody algebra does not act on it. It seems
plausible to expect that the odd arc algebra admits an action of an algebra akin to Brun-
dan and Kleshchev’s Hecke-Clifford superalgebra from [4], which could be seen as a super
counterpart of the cyclotomic KLR algebra.
Another challenging problem we would like to mention is to find the representation-
theoretic context (category O) for the odd arc algebras. The analogy with [14] and [15],
taken together with [18] and the results in [17, Section 5.1] might suggest a connection to
category Ofor Lie superalgebras.
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