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ANNALES MATHÉMATIQUES
BLAISE PASCAL
Peter Danchev
Fully Invariant Subgroups of n-Summable Primary Abelian
Groups
Volume 18, no 2 (2011), p. 245-250.
<http://ambp.cedram.org/item?id=AMBP_2011__18_2_245_0>
© Annales mathématiques Blaise Pascal, 2011, tous droits réservés.
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cedram
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Annales mathématiques Blaise Pascal 18, 245-250 (2011)
Fully Invariant Subgroups of n-Summable
Primary Abelian Groups
Peter Danchev
Abstract
We present a number of results concerning fully invariant subgroups of n-
summable groups.
1. Introduction
Throughout this paper suppose all groups are Abelian, p-primary for some
arbitrary but a fixed prime, written additively as is the custom when
exploring Abelian groups. All notions and notations are standard and
follow those from [8] and [9]. For instance, for any element g from a group
G, the letter |g|G will always denote the height of g in G, that is, the
maximal ordinal δ such that g ∈ pδG; whence g ∈ pδG \ pδ+1G.
A problem of interest is to find suitable properties of groups that are in-
herited by special subgroups called fully invariant. Recall that a subgroup
F of a group G is said to be fully invariant if each endomorphism of G
sends F into itself. For some classes of groups (e.g., fully transitive groups
or, in particular, totally projective groups) these subgroups are completely
described - see, for instance, [9] and [11]. Specifically, they have the form
G(α) = {x ∈ G : |pix|G ≥ αi, i < ω}, where α = {αi}i<ω is an increasing
sequence of ordinals and symbols ∞. In addition, for any ordinal γ, pγG
is fully invariant in G.
Moreover, it was shown in [10] and [12] that a group G is totally pro-
jective if, and only if, G(α) and G/G(α) are totally projective; however
whether or not this remains true for some arbitrary fully invariant sub-
group F is still unknown. It is worthwhile noticing that this extends the
classical result of Nunke [13] which says that G is totally projective if, and
only if, both pαG and G/pαG are totally projective.
Keywords: n-summable groups, fully invariant subgroups, quotients, σ-summable
groups.
Math. classification: 20K10.
245
P. Danchev
The purpose that motivates the writing of the present article is to try
to modify to what extent the quoted above results for totally projec-
tive groups can be proved for summable groups and their generalizations,
named n-summable groups. This continues our recent results in this di-
rections - cf. [5, 2, 4, 3].
2. n-Summable groups and their fully invariant subgroups
Imitating [7], the valuated pn-socle V is said to be n-summable if it is
isometric to the valuated direct sum ⊕i∈IVi of a collection of countable
valuated pn-socles Vi where i ∈ I and I is an index set. A group G is said
to be n-summable if G[pn], as a valuated pn-socle under the valuation
induced by the height function, is n-summable. In particular, a group is
1-summable exactly when it is summable in the usual sense of the term.
For more detailed information about summable and n-summable groups,
we refer the interested reader to [8] and [7].
Due to the importance of Nunke-like theorems, referred to above [13],
we prove the following (see also [1]).
Proposition 2.1. Suppose G is a group and λ is an ordinal.
(a) If G is n-summable, then pλG and G/pλ+nG are n-summable.
(b) If λ is countable and both pλG and G/pλG are n-summable, then
G is n-summable.
Proof. (a): If G[pn] is the valuated direct sum of countable valuated
groups, then the same can clearly be said of (pλG)[pn]. Next, let H be
a pλ+n-high subgroup of G, and let H ′ = [H + pλ+nG]/pλ+nG, so that
H ∼= H ′ are isotype in G and G′ = G/pλ+nG, respectively. It follows that
there are valuated direct sum decompositions
G[pn] = H[pn]⊕X and G′[pn] = H ′[pn]⊕X ′
where pλ+nG ⊆ X ⊆ pλG and X ′ ⊆ pλG′. Note that X ′ is, in fact, n-
summable (see, e.g., [7]). Therefore, since G is n-summable, we can con-
clude H[pn] and X are n-summable. This, in turn, implies that H ′[pn] and
pλ+nG are n-summable, so that G′ and pλ+nG are n-summable. The fact
that pλ+nG is n-summable readily implies that pλG shares this property.
246
Fully Invariant Subgroups
(b): Let H and X be as in part (a). Since pλG is n-summable, we
can infer that X has this property. We need, therefore, to show that H[pn]
is also n-summable. Since G/pλG is n-summable and λ is countable, we
can conclude that (G/pλG)[pn] is n-Honda, i.e., it is the ascending union
of subgroups Ym, for m < ω, such that Sn = {|y|G/pλG : y ∈ Ym} is
finite. If we let Y ′m = {x ∈ H : x + pλG ∈ Ym}, then clearly H[pn] is the
ascending union of the Y ′m and {|x|G : x ∈ Y ′m} ⊆ Sn ∪ [λ, λ + n − 1] is
finite. Therefore, H[pn] is also n-Honda, and hence n-summable. 
There are lots of examples of summable groups G such that G/pωG is
not summable; in fact suppose B is an unbounded direct sum of cyclics
and B is its torsion-completion. Then G = B/B[pn] can easily be seen to
be n-summable, but on the other hand
G/pωG = B/B[pn]/B[pn]/B[pn] ∼= B/B[pn] ∼= pnB
is not even summable. So Proposition 2.1(a) does not hold if λ + n is
replaced by λ.
In addition, if G is a totally projective group of length λ, where ω1 <
λ < ω1 · 2, then pω1G and G/pω1G are both n-summable, but G is not; so
Proposition 2.1(b) does not hold for uncountable λ.
As an immediate consequence, we yield the following (see [1] too):
Corollary 2.2. A group G is summable iff pω+1G and G/pω+1G are sum-
mable.
The last statement can slightly be extended to the following.
Proposition 2.3. A group G is summable if, and only if, pω+kG and
G/pω+kG are summable for some k < ω.
Proof. First, observe that
G/pω+1G ∼= G/pω+kG/pω+1G/pω+kG = G/pω+kG/pω+1(G/pω+kG).
Moreover, since a group A is summable uniquely when so is pmA for some
natural m, we derive that pω+kG is summable precisely when the same
holds for pω+1G. Henceforth, we apply the above corollary to infer the
wanted claim. 
Lemma 2.4. If L = (pω+1G)[p] and G/L is summable, then G is sum-
mable.
247
P. Danchev
Proof. First note that
pω+2G = p(pω+1G) ∼= (pω+1G)/(pω+1G)[p] = pω+1G/L = pω+1(G/L)
and, hence, pω+2G is summable. But this plainly implies that pω+1G is
summable, and in view of Corollary 2.2 it remains only to show that
G/pω+1G is also summable. Indeed, since L ⊆ pω+1G, we have
G/pω+1G ∼= G/L/pω+1G/L = G/L/pω+1(G/L)
and the latter group is really summable because G/L is so by hypothesis.

We will hereafter assume that all fully invariant subgroups of G are of
the type
G(α) = {x ∈ G : |pix|G ≥ αi, i < ω}
where α = {αi}i<ω is an increasing sequence of ordinals and symbols ∞.
Corollary 2.5. If L = G(α) and G/L is summable, then G/pL is sum-
mable.
Proof. Observe that L/pL = (pα(G/pL))[p] where α = α0 = ω+1. There-
fore, (G/pL)/(L/pL) ∼= G/L is summable and we need only employ the
previous lemma to G/pL. 
So, we come to the following sufficient condition for summability.
Theorem 2.6. Let pω·2G = 0. If α is an increasing sequence of ordinals
and symbols ∞ such that both G(α) and G/G(α) are summable, then G
itself is summable.
Proof. Let L = G(α). If the sequence α contains any symbol∞, then there
is a positive integer j such that pjL = 0. But then repeated applications of
Corollary 2.5 yield the desired conclusion that G/pjL ∼= G is summable.
Thus we may assume that α is an increasing sequence of ordinals that
does not contain symbols of the type ∞, and take λ = sup(α) where
α = {αi}i<ω. It is not hard to see that pωL = pλG where λ ≤ ω · 2.
Foremost, suppose for a moment λ = ω · 2, whence pωL = 0 so that
L is separable summable and thus a direct sum of cyclic groups. In view
of [12], we obtain that G is σ-summable. Furthermore, appealing to [6,
Proposition 6.5], we deduce that pωG is σ-summable, i.e., a direct sum of
cyclic groups and hence summable. Therefore pω+iG is summable for any
natural number i.
248
Fully Invariant Subgroups
Next, suppose that λ < ω · 2. Consequently, pλG is summable. But
λ = ω+m for some m < ω, so that pω+mG being summable immediately
forces that pω+iG is summable for all i < ω.
Furthermore, in virtue of Proposition 2.3, we need only verify in the
both cases for λ alluded to above that G/pαiG is summable for some
i ≥ 1, where αi = ω + i. But since piL ⊆ pαiG, we have again with this
proposition at hand that
G/pαiG ∼= (G/piL)/pαi(G/piL)
is summable since each G/piL is summable by the subsequent application
i times of Corollary 2.5. 
Acknowledgment. The author owes his deep thanks to Prof. Pat Keef
for their valuable communication in this subject.
References
[1] P. Danchev – “Commutative group algebras of summable p-
groups”, Commun. Algebra 35 (2007), p. 1275–1289.
[2] , “Notes on λ-large subgroups of primary abelian groups and
free valuated vector spaces”, Bull. Allahabad Math. Soc. 23 (2008),
p. 149–154.
[3] , “On λ-large subgroups of n-summable Cω1-groups”, SUT J.
Math. 44 (2008), p. 33–37.
[4] , “On some fully invariant subgroups of summable groups”,
Ann. Math. Blaise Pascal 15 (2008), p. 147–152.
[5] , “On λ-large subgroups of summable CΩ-groups”, Algebra
Colloq. 16 (2009), p. 649–652.
[6] P. Danchev & P. Keef – “Generalized Wallace theorems”, Math.
Scand. 104 (2009), p. 33–50.
[7] , “n-Summable valuated pn-socles and primary abelian
groups”, Commun. Algebra 38 (2010), p. 3137–3153.
[8] L. Fucs – Infinite abelian groups I and II, Academic Press, New York
and London, 1970 and 1973.
249
P. Danchev
[9] I. Kaplansky – Infinite abelian groups, University of Michigan
Press, Ann Arbor, 1954 and 1969.
[10] R. Linton – “On fully invariant subgroups of primary abelian
groups”, Mich. Math. J. 22 (1975), p. 281–284.
[11] , “λ-large subgroups of Cλ-groups”, Pac. J. Math. 75 (1978),
p. 477–485.
[12] R. Linton & C. Megibben – “Extensions of totally projective
groups”, Proc. Amer. Math. Soc. 64 (1977), p. 35–38.
[13] R. Nunke – “Homology and direct sums of countable abelian
groups”, Math. Z. 101 (1967), p. 182–212.
Peter Danchev
Department of Mathematics
Plovdiv State University
24 Tzar Assen St.
Plovdiv 4000
BGR
pvdanchev@yahoo.com
250


Fully Invariant Subgroups of $n$-Summable Primary Abelian Groups

David Mascré

Inégalités pour l’opérateur intégral fractionnaire sur différents espaces métriques mesurés

Anne-Laure Thiel

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Anne-Laure Thiel
Categorification of the virtual braid groups
Volume 18, no 2 (2011), p. 231-243.
<http://ambp.cedram.org/item?id=AMBP_2011__18_2_231_0>
© Annales mathématiques Blaise Pascal, 2011, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pas-
cal » (http://ambp.cedram.org/), implique l’accord avec les condi-
tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute
utilisation commerciale ou impression systématique est constitutive
d’une infraction pénale. Toute copie ou impression de ce fichier doit
contenir la présente mention de copyright.
Publication éditée par le laboratoire de mathématiques
de l’université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales mathématiques Blaise Pascal 18, 231-243 (2011)
Categorification of the virtual braid groups
Anne-Laure Thiel
Abstract
We extend Rouquier’s categorification of the braid groups by complexes of
Soergel bimodules to the virtual braid groups.
Catégorification des groupes de tresses virtuelles
Résumé
Nous généralisons la catégorification des groupes de tresses par complexes de
bimodules de Soergel due à Rouquier aux groupes de tresses virtuelles.
Introduction
Virtual links have been introduced by Kauffman in [5] as a geometric
counterpart of Gauss codes. A virtual knot diagram is a generic oriented
immersion of circles into the plane, with the usual positive and negative
crossings plus a new kind of crossings called virtual. Such crossings appear
for instance when one projects a generic link in a thickened surface onto a
plane (see [2] or [7]). Many invariants for classical links can be extended to
virtual links. Classical oriented links can be represented by closed braids;
likewise virtual links can be represented by the closures of virtual braids.
There is also an analogue of Markov theorem [3]. Now virtual braids with n
strands form a group, denoted VBn, which can be described by generators
and relations, generalizing the generators and relations of the usual braid
group with n strands Bn. The aim of this note is, using this presentation, to
categorify VBn in the weak sense of Rouquier. Rouquier actually proves
in [10] a stronger version of the result that we want to extend and in
particular he states the faithfulness of his categorification. More precisely,
to any word ω in the generators of VBn we associate a bounded cochain
complex F (ω) of bimodules such that if two words ω and ω′ represent the
Keywords: braid group, virtual braid, categorification.
Math. classification: 20F36, 05E10, 05E18, 13D99, 18G35.
231
A.-L. Thiel
same element of VBn, then the corresponding cochain complexes F (ω)
and F (ω′) are homotopy equivalent. So we obtain a morphism from the
group VBn to the group of isomorphim classes of invertible complexes
up to homotopy. This morphism, contrary to the case of Bn studied by
Rouquier, is not injective. We describe a part of its kernel in Remarks 3.5.
Note that, using Mazorchuk and Stroppel’s study of Arkhipov’s twisting
functor in [9], one can give a representation theoretic approach to the
categorification of virtual braids.
1. Virtual braids
Following [13] and [8], we recall the definition of the virtual braid group
VBn with n strands.
Definition 1.1. The virtual braid group VBn is the group generated by
2(n−1) generators σ1, . . . , σn−1 and ζ1, . . . , ζn−1 satisfying the braid group
relations
σiσj = σjσi, if |i− j| > 1, (1.1)
σiσi+1σi = σi+1σiσi+1, if 1 ≤ i ≤ n− 2, (1.2)
the permutation group relations
ζiζj = ζjζi, if |i− j| > 1, (1.3)
ζiζi+1ζi = ζi+1ζiζi+1, if 1 ≤ i ≤ n− 2, (1.4)
ζ2i = 1, if 1 ≤ i ≤ n− 1, (1.5)
and the mixed relations
σiζj = ζjσi, if |i− j| > 1, (1.6)
σiζi+1ζi = ζi+1ζiσi+1, if 1 ≤ i ≤ n− 2. (1.7)
The classical braid group Bn (see [4] for a definition) naturally embeds
in VBn as a subgroup generated by σ1, . . . , σn−1.
The braid group VBn can be depicted diagrammatically. To each gener-
ator σi (resp. σ−1i ) we associate the elementary braid diagram consisting
of a single positive (resp. negative) crossing between the ith and i + 1st
strand as shown in Figure 1.1 (resp. Figure 1.2). To each generator ζi
we associate the elementary virtual braid diagram with a single virtual
crossing between the ith and i+ 1st strand of Figure 1.3.
232
Categorification of the virtual braid groups
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
Anne-Laure Thiel
Institut de Recherche Mathématique
Avancée, Université de Strasbourg et
CNRS
7 rue René Descartes, F–67084
Strasbourg Cedex, France
thiel@math.unistra.fr
1
Figure 1.1. The positive braid σi
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
Anne-Laure Thiel
Institut de Recherche Mathématique
Avancée, Université de Strasbourg et
CNRS
7 rue René Descartes, F–67084
Strasbourg Cedex, France
thiel@math.unistra.fr
1
Figure 1.2. The negative braid σ−1i
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
1 i!1 i i+1 i+2 n
Anne-Laure Thiel
Institut de Recherche Mathématique
Avancée, Université de Strasbourg et
CNRS
7 rue René Descartes, F–67084
Strasbourg Cedex, France
thiel@math.unistra.fr
1
Figure 1.3. The virtual braid ζi
The multiplication law of the group VBn consists in concatenating these
elementary braids. We use the convention that braids multiply from bot-
tom to top: if D (resp. D′) is a virtual braid d agram representing an
element β (resp. β′) of VBn, then the product ββ′ is represented by the
diagram obtained by putting D′ on top of D and gluing the lower end-
points of D′ to the upper endpoints of D.
233
A.-L. Thiel
The braid group relations, the permutation group relations and the
mixed relations have a diagrammatical interpretation. They correspond
to planar isotopies and the generalized Reidemeister moves depicted in
Figures 1.4, 1.5 and 1.6.
Figure 1.4. Classical Reidemeister II–III moves
Figure 1.5. Virtual Reidemeister moves
Figure 1.6. The mixed Reidemeister move
2. Rouquier’s categorification of the braid groups
In this section we recall how Rouquier [10] categorified the braid group Bn.
234
Categorification of the virtual braid groups
2.1. Soergel bimodules
We first discuss some bimodules introduced by Soergel [11], [12] in his
work on representation theory.
Let R be the subalgebra of Q[x1, . . . , xn] defined by
R = Q[x1 − x2, x2 − x3, . . . , xn−1 − xn] = Q[x1 − x2, x1 − x3, . . . , x1 − xn].
The symmetric group Sn acts on Q[x1, . . . , xn] by ω(xi) = xω(i) for all
xi ∈ R and ω ∈ Sn. This action preserves R. Let Rω be the subalgebra
of elements of R fixed by ω. In particular Rτi is the subalgebra of R of
elements fixed by the transposition τi = (i, i+ 1). As an algebra,
Rτi = Q[x1 − x2, . . . , (x1 − xi) + (x1 − xi+1),
(x1 − xi)(x1 − xi+1), x1 − xi+2, . . . , x1 − xn].
Let us also consider the R–bimodules Bω = R ⊗Rω R for any ω ∈ Sn.
The R–bimodules Bτi will be denoted by Bi for simplicity of notation.
We introduce a grading on R, Rτi and Bi by setting deg(xk) = 2 for all
k = 1, . . . , n.
Two R-bimodule morphisms between these objects will be relevant to
us, namely bri : Bi → R and rbi : R{2} → Bi defined by
bri(1⊗ 1) = 1 and rbi(1) = (xi − xi+1)⊗ 1 + 1⊗ (xi − xi+1).
The curly brackets indicate a shift of the grading: if M =
⊕
i∈Z
Mi is a
Z-graded bimodule and p an integer, then the Z–graded bimodule M{p}
is defined by M{p}i = Mi−p for all i ∈ Z. The maps bri and rbi are
degree-preserving morphisms of graded R-bimodules.
2.2. Categorification of the braid groups
Following [6] and [10], to each braid generator σi ∈ Bn we assign the
cochain complex F (σi) of graded R–bimodules
F (σi) : 0 −→ R{2}
rbi−−→ Bi −→ 0, (2.1)
where Bi sits in cohomological degree 0. To σ−1i we assign the cochain
complex F (σ−1i ) of graded R–bimodules
F (σ−1i ) : 0 −→ Bi{−2}
bri−−→ R{−2} −→ 0, (2.2)
235
A.-L. Thiel
where Bi{−2} sits in cohomological degree 0. To the unit element 1 ∈ Bn
we assign the complex of graded R-bimodules
F (1) : 0 −→ R −→ 0, (2.3)
where R sits in cohomological degree 0; the complex F (1) is a unit for
the tensor product of complexes so tensoring any complex of graded R-
bimodules with F (1) leaves the complex unchanged. Finally to any word
σ = σε1i1 . . . σ
εk
ik
where ε1, . . . , εk = ±1, we assign the complex of graded
R-bimodules F (σ) = F (σε1i1 )⊗R · · · ⊗R F (σ
εk
ik
).
Rouquier proved the following result, which can be called a categorifi-
cation of the braid group Bn.
Theorem 2.1. [10] If ω and ω′ are words representing the same element
of Bn, then F (ω) and F (ω′) are homotopy equivalent complexes of R–
bimodules.
3. Categorification of the virtual braid groups
Our aim is to extend Rouquier’s categorification to the virtual braid group
VBn. The cochain complexes associated to the generators σi of VBn coming
from Bn will be the same as Rouquier’s complexes above. We have to assign
complexes to the generators ζi of VBn corresponding to virtual crossings
such that all these complexes satisfy the same relations as the generators
of VBn up to homotopy equivalence.
3.1. Twisted bimodules
In order to achieve this categorification, we consider the R-bimodule Rω
for each permutation ω ∈ Sn. As a left R-module, Rω is equal to R while
the right action of a ∈ R is the multiplication by ω(a). Note that Rid = R
as an R-bimodule. The following lemma is obvious.
Lemma 3.1. For all ω, ω′ ∈ Sn the map ψ : Rω ⊗R Rω′ → Rωω′ defined
for all a, b ∈ R by ψ(a⊗ b) = aω(b) is an isomorphism of R-bimodules.
The bimodules we will mostly use are the bimodules Ri = Rτi . The
reason why we consider these bimodules is that, by Lemma 3.1, they
possess the following interesting property:
Ri ⊗R Ri ∼= R
236
Categorification of the virtual braid groups
for all i = 1, . . . , n− 1.
Lemma 3.2. For all permutations ω, ω′ ∈ Sn the R-bimodules Rω⊗RBω′
and Bωω′ω−1 ⊗R Rω are isomorphic.
Proof. First note that there are natural isomorphisms of R-bimodules:
Rω ⊗R Bω′ ∼= Rω ⊗Rω′ R and Bωω′ω−1 ⊗R Rω ∼= R⊗Rωω′ω−1 Rω.
Now consider the map ψ : Rω ⊗Rω′ R → R ⊗Rωω′ω−1 Rω defined for all
a, b ∈ R by
ψ(a⊗ b) = a⊗ ω(b).
This map is well defined: for any c ∈ Rω′ , we just have to check that a⊗cb
and aω(c)⊗ b have the same image under ψ. This is true because c ∈ Rω′
implies that ω(c) ∈ Rωω′ω−1 . Moreover the map ψ is obviously a morphism
of R-bimodules.
Similarly, the map ϕ : R ⊗
Rωω′ω−1 Rω → Rω ⊗Rω′ R defined for all
a, b ∈ R by
ϕ(a⊗ b) = a⊗ ω−1(b)
is a well defined morphism of R-bimodules as well.
Finally, ψ and ϕ are easily seen to be inverse of each other. 
Now let us assign a complex of graded R-bimodules to each generator
of the virtual braid group. To the elements σi and σ−1i we assign the
complexes F (σi) and F (σ−1i ) defined by (2.1) and (2.2). To the element
ζi we assign the complex concentrated in degree 0:
F (ζi) : 0 −→ Ri −→ 0. (3.1)
Just as in Section 2.2 we assign to the unit element 1 of VBn the complex
F (1) of (2.3), and to a virtual braid word we assign the tensor product over
R of the complexes associated to the generators involved in the expression
of the word.
Remark 3.3. Consider ω = ζi1 . . . ζik a word in {ζ1, . . . , ζn−1} and let
ω̃ = τi1 . . . τik be the corresponding element of Sn. It follows from Lemma
3.1 that the complex F (ω) is isomorphic to 0 −→ Rω̃ −→ 0.
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A.-L. Thiel
3.2. Categorification of VBn
We now state our main result.
Theorem 3.4. If ω and ω′ are words representing the same element
of VBn, then F (ω) and F (ω′) are homotopy equivalent complexes of R–
bimodules.
Proof. By definition of VBn and in view of Theorem 2.1, it is enough to
check that there are homotopy equivalences between the complexes asso-
ciated to the braid words appearing in both sides of Relations (1.3)-(1.7).
Actually we will prove a stronger result: these complexes are isomorphic.
Permutation group relations. Let ω = ω′ be a permutation group relation.
Since the bimodules Rω̃ and Rω̃′ are equal, the complexes F (ω) and F (ω′)
are isomorphic in view of Remark 3.3.
Mixed relations. Let us first deal with Relation (1.6). We have to prove
that for |i− j| > 1 the complexes F (ζjσi) and F (σiζj) are isomorphic. We
have
F (ζjσi) : 0 −→ Rj ⊗R R{2}
id⊗ rbi−−−−→ Rj ⊗R Bi −→ 0
and
F (σiζj) : 0 −→ R⊗R Rj{2}
rbi⊗ id−−−−→ Bi ⊗R Rj −→ 0.
First observe that F (ζjσi) is naturally isomorphic to the following complex
of bimodules
0 −→ Rj{2}
d−−→ Rj ⊗Rτi R −→ 0
whose differential d sends each a ∈ Rj{2} to
a
(
τj(xi−xi+1)⊗1 + 1⊗ (xi−xi+1)
)
= a
(
(xi−xi+1)⊗1 + 1⊗ (xi−xi+1)
)
.
Similarly, F (σiζj) is isomorphic to
0 −→ Rj{2}
d′−−−→ R⊗Rτi Rj −→ 0
whose differential d′ sends each a ∈ Rj{2} to
a
(
(xi − xi+1)⊗ 1 + 1⊗ (xi − xi+1)
)
.
Since the transpositions τi and τj commute, the proof of Lemma 3.2 pro-
vides us the isomorphism of R-bimodules ψ : Rj⊗RτiR→ R⊗RτiRj . Using
238
Categorification of the virtual braid groups
the invariance of (xi−xi+1) under the action of τj , we easily check that the
following vertical maps and their inverse commute with the differentials.
0 // Rj{2} d //
id

Rj ⊗Rτi R
ψ

// 0
0 // Rj{2} d
′
// R⊗Rτi Rj // 0
Thus the complexes F (ζjσi) and F (σiζj) are isomorphic for all |i− j| > 1.
We finally deal with Relation (1.7). We have to show that the complexes
F (ζi+1ζiσi+1) and F (σiζi+1ζi) are isomorphic for i = 1, . . . , n − 2. The
complex F (ζi+1ζiσi+1) is equal to
0 −→ Ri+1 ⊗R Ri ⊗R R{2}
id⊗ id⊗ rbi+1−−−−−−−−−→ Ri+1 ⊗R Ri ⊗R Bi+1 −→ 0
and the complex F (σiζi+1ζi) is equal to
0 −→ R⊗R Ri+1 ⊗R Ri{2}
rbi⊗ id⊗ id−−−−−−−→ Bi ⊗R Ri+1 ⊗R Ri −→ 0.
There exist a natural isomorphism between F (ζi+1ζiσi+1) and the com-
plex
0 −→ Rτi+1τi{2}
d−−→ Rτi+1τi ⊗Rτi+1 R −→ 0
whose differential d sends each a ∈ Rτi+1τi{2} to
a
(
τi+1τi(xi+1 − xi+2)⊗ 1 + 1⊗ (xi+1 − xi+2)
)
=
a
(
(xi − xi+1)⊗ 1 + 1⊗ (xi+1 − xi+2)
)
.
Similarly, F (σiζi+1ζi) is isomorphic to
0 −→ Rτi+1τi{2}
d′−−−→ R⊗Rτi Rτi+1τi −→ 0
whose differential d′ sends each a ∈ Rτi+1τi{2} to
a
(
(xi − xi+1)⊗ 1 + 1⊗ (xi − xi+1)
)
.
Applying Lemma 3.2 and using the relation τiτi+1τi = τi+1τiτi+1 and
the involutivity of τi+1 yields the isomorphism
ψ : Rτi+1τi ⊗Rτi+1 R→ R⊗Rτi Rτi+1τi
given for all a ∈ Rτi+1τi , b ∈ R by
ψ(a⊗ b) = a⊗ τi+1τi(b).
239
A.-L. Thiel
Let us complete the proof by checking that the following vertical maps
commute with the differentials (and similarly for their inverse).
0 // Rτi+1τi{2}
d //
id

Rτi+1τi ⊗Rτi+1 R
ψ

// 0
0 // Rτi+1τi{2}
d′ // R⊗Rτi Rτi+1τi // 0
For any element a in Rτi+1τi{2} we compute both its image under d′ ◦ id
and ψ ◦ d. We obtain
d′ ◦ id(a) = a
(
(xi − xi+1)⊗ 1 + 1⊗ (xi − xi+1)
)
and
ψ ◦ d(a) = ψ
(
a(xi − xi+1)⊗ 1 + a⊗ (xi+1 − xi+2)
)
= a(xi − xi+1)⊗ 1 + a⊗ τi+1τi(xi+1 − xi+2)
= a
(
(xi − xi+1)⊗ 1 + 1⊗ (xi − xi+1)
)
This shows that the complexes F (ζi+1ζiσi+1) and F (σiζi+1ζi) are isomor-
phic for all i = 1, . . . , n− 2. 
Remarks 3.5.
• Let us call virtualisation moves the moves consisting in squeezing a
classical crossing between two virtual crossings, as shown in Figure 3.1.
We observe that the complexes F (ζiσεi ζi) and F (σεi ) are isomorphic for all
i = 1, . . . , n−1 and ε ∈ {−1, 1}. This is essentially due to the involutivity
of τi, which implies (cf. Lemma 3.1 and Lemma 3.2) that the bimodules
Ri ⊗R Bi ⊗R Ri and Bi are isomomorphic. Thus our categorification of
VBn does not detect the virtualisation moves.
• Adding the relation ζiσi+1σi = σi+1σiζi+1 to the presentation of VBn,
one obtains a presentation of the group of welded braids with n strands
defined in [1]. Noting that the only morphism between Rω and Rω′ is the
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Categorification of the virtual braid groups
Figure 3.1. Virtualisation moves
trivial one if ω 6= ω′, one can check that the complexes F (ζiσi+1σi)
Ri ⊗R Bi+1{2}
id⊗ id⊗ rbi
**VVV
VVVV
VVVV
VVVV
VVV
Ri ⊗R R{4}
id⊗ rbi+1
44jjjjjjjjjjjjjjjj
− id⊗ rbi **TTT
TTTT
TTTT
TTTT
T
Ri ⊗R Bi+1 ⊗R Bi
Ri ⊗R Bi{2}
id⊗ rbi+1⊗ id
44iiiiiiiiiiiiiiiiii
and F (σi+1σiζi+1)
Bi+1 ⊗R Ri+1{2}
id⊗ rbi⊗ id
++VVVV
VVVVV
VVVVV
VVVVV
R⊗R Ri+1{4}
rbi+1⊗ id
44iiiiiiiiiiiiiiiii
− rbi⊗ id **UUU
UUUU
UUUU
UUUU
UU
Bi+1 ⊗R Bi ⊗R Ri+1
Bi ⊗R Ri+1{2}
rbi+1⊗ id⊗ id
33hhhhhhhhhhhhhhhhhhh
are not equivalent up to homotopy.
241
A.-L. Thiel
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Categorification of the virtual braid groups
Anne-Laure Thiel
Institut de Recherche Mathématique
Avancée, Université de Strasbourg et
CNRS
7 rue René Descartes, F–67084
Strasbourg Cedex, France
thiel@math.unistra.fr
243


Categorification of the virtual braid groups

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