THE ADJOINT GROUP OF A COXETER QUANDLE

arXiv:1702.07104v2 [math.GT] 1 Mar 2017

TOSHIYUKI AKITA

A BSTRACT. We give explicit descriptions of the adjoint group of the Coxeter
quandle associated with a Coxeter group W . The adjoint group turns out to be an
intermediate group between W and the corresponding Artin group, and fits into a
central extension of W by a finitely generated free abelian group. We construct 2cocycles of W corresponding to the central extension. In addition, we prove that
the commutator subgroup of the adjoint group is isomorphic to the commutator
subgroup of W .

1. I NTRODUCTION
A nonempty set X equipped with a binary operation X × X → X, (x, y) 7→ x ∗ y
is called a quandle if it satisfies the following three conditions:
(1) x ∗ x = x (x ∈ X),
(2) (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) (x, y, z ∈ X),
(3) For all y ∈ X, the map X → X defined by x 7→ x ∗ y is bijective.
Quandles have been studied in low dimensional topology as well as in Hopf algebras. To any quandle X one can associate a group Ad(X) called the adjoint group
of X (also called the associated group or the enveloping group in the literature). It
is defined by the presentation
Ad(X) := hex (x ∈ X) | e−1
y ex ey = ex∗y (x, y ∈ X)i
Although adjoint groups play important roles in the study of quandles, not much
is known about the structure of them, partly because the definition of Ad(X) by a
possibly infinite presentation is difficult to work with in explicit calculations. See
Eisermann [9] and Nosaka [14] for generality of adjoint groups, and [6, 9, 10, 14]
for descriptions of adjoint groups of certain classes of quandles.
In this paper, we will study the adjoint group of a Coxeter quandle. Let (W, S) be
a Coxeter system, a pair of a Coxeter group W and the set S of Coxeter generators
of W . Following Nosaka [14], we define the Coxeter quandle XW associated with
(W, S) to be the set of all reflections of W :
XW :=

[

w−1 Sw.

w∈W

The quandle operation is given by the conjugation x ∗ y := y−1 xy = yxy. The symmetric group Sn of n letters is a Coxeter group (of type An−1 ), and the associated
Coxeter quandle XSn is nothing but the set of all transpositions. In their paper
Key words and phrases. quandle, Coxeter group, Artin group, pure Artin group.
1

2

T. AKITA

[2], Andruskiewitsch-Fantino-Garcı́a-Vendramin showed that Ad(XSn ) is an intermediate group between Sn and the braid group Bn of n strands, in the sense
that the canonical projection Bn  Sn splits into a sequence of surjections Bn 
Ad(XSn )  Sn , and that Ad(XSn ) fits into a central extension of the form
0 → Z → Ad(XSn ) → Sn → 1.
See [2, Proposition 3.2] and its proof. The primary purpose of this paper is to
generalize their results to arbitrary Coxeter quandles. We will show that Ad(XW )
is an intermediate group between W and the Artin group AW associated with W
(Proposition 3.3), and will show that Ad(XW ) fits into a central extension of the
form
(1.1)

φ

0 → Zc(W ) → Ad(XW ) → W → 1,

where c(W ) is the number of conjugacy classes of elements in XW (Theorem 3.1).
As is known, the central extension (1.1) corresponds to the cohomology class uφ ∈
H 2 (W, Zc(W ) ). We will construct 2-cocycles of W representing uφ (Proposition 4.2
and Theorem 4.7).
Finally, Eisermann [9] claimed that Ad(XSn ) is isomorphic to the semidirect
product An o Z where An is the alternating group on n letters, but he did not write
down the proof. We will generalize his result to Coxeter quandles (Theorem 5.1
and Corollary 5.2).
Notation. Let G be a group, g, h ∈ G elements of G and m ≥ 2 a natural number.
Define
(gh)m := ghg · · · ∈ G.
| {z }
m

For example, (gh)2 = gh, (gh)3 = ghg, (gh)4 = ghgh. Let GAb := G/[G, G] be
the abelianization of G, AbG : G → GAb the natural projection, and write [g] :=
g[G, G] ∈ GAb for g ∈ G. For a quandle X and elements xk ∈ X (k = 1, 2, 3, . . . ), we
denote
x1 ∗ x2 ∗ x3 ∗ · · · ∗ x` := (· · · ((x1 ∗ x2 ) ∗ x3 ) ∗ · · · ) ∗ x` ,
for simplicity.
2. C OXETER GROUPS AND C OXETER QUANDLES
2.1. Coxeter groups. Let S be a finite set and m : S × S → N ∪ {∞} a map satisfying the following conditions:
(1) m(s, s) = 1 for all s ∈ S
(2) 2 ≤ m(s,t) = m(t, s) ≤ ∞ for all distinct s,t ∈ S.
The map m is represented by the Coxeter graph Γ whose vertex set is S and whose
edges are the unordered pairs {s,t} ⊂ S such that m(s,t) ≥ 3. The edges with
m(s,t) ≥ 4 are labeled by the number m(s,t). The Coxeter system associated with
Γ is the pair (W, S) where W is the group generated by s ∈ S and the fundamental
relations (st)m(s,t) = 1 (s,t ∈ S, m(s,t) < ∞):
(2.1)

W := hs ∈ S | (st)m(s,t) = 1(s,t ∈ S, m(s,t) < ∞)i.

COXETER QUANDLE

3

The group W is called the Coxeter group (of type Γ), and elements of S are called
Coxeter generators of W (also called simple reflections in the literature). Note
that the order of the product st is precisely m(s,t). In particular, every Coxeter
generator s ∈ S has order 2. It is easy to check that the defining relations in (2.1)
are equivalent to the following relations
(2.2)

s2 = 1 (s ∈ S), (st)m(s,t) = (ts)m(s,t) (s,t ∈ S, s , t, m(s,t) < ∞).

Finally, the odd subgraph Γodd is a subgraph of Γ whose vertex set is S and whose
edges are the unordered pairs {s,t} ⊂ S such that m(s,t) is an odd integer. We refer
Bourbaki [3] and Humphreys [12] for further details of Coxeter groups.
2.2. Conjugacy classes of reflections. Let
XW :=

[

w−1 Sw

w∈W

be the set of reflections in W as in the introduction (the underlying set of the Coxeter quandle). Let OW be the set of conjugacy classes of elements of XW under W ,
and RW a complete set of representatives of conjugacy classes. We may assume
RW ⊆ S. Let c(W ) be the cardinality of OW . The following facts are well-known
and easy to prove.
Proposition 2.1. The elements of OW are in one-to-one correspondence with the
connected components of Γodd . Consequently, c(W ) equals to the number of connected components of Γodd .
To be precise, the conjugacy class of s ∈ S corresponds to the connected component
of Γodd containing s ∈ S. The proof of Proposition 2.1 can be found in [4, Lemma
3.6].
Proposition 2.2. WAb is the elementary abelian 2-group with a basis {[s] ∈ WAb |
s ∈ RW }. In particular, WAb  (Z/2)c(W ) .
2.3. Coxeter quandles. Now we turn our attention to Coxeter quandles. Let XW
be the Coxeter quandle associated with (W, S) and
Ad(XW ) := hex (x ∈ XW ) | e−1
y ex ey = ex∗y (x, y ∈ XW )i
the adjoint group of XW . Observe that
(2.3)

e−1
y ex ey = ex∗y = ey−1 xy

and
(2.4)

ey ex e−1
y = ex∗y ,

where (2.4) follows from e−1
y ex∗y ey = ex∗y∗y = ex .
Proposition 2.3. Ad(XW ) is generated by es (s ∈ S).
Proof. Given x ∈ XW , we can express x as x = (s1 s2 · · · s` )−1 s0 (s1 s2 · · · s` ) for some
si ∈ S (0 ≤ i ≤ `) by the definition of XW . Applying (2.3), we have
ex = e(s1 s2 ···s` )−1 s0 (s1 s2 ···s` ) = es0 ∗s1 ∗s2 ∗···∗s` = (es1 es2 · · · es` )−1 es0 (es1 es2 · · · es` ),
proving the proposition.



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T. AKITA

Proposition 2.4. Ad(XW )Ab is the free abelian group with a basis {[es ] | s ∈ RW }.
In particular, Ad(XW )Ab  Zc(W ) .
Proof. By the definition of Ad(XW ), the abelianization Ad(XW )Ab is generated by
[ex ] (x ∈ XW ) subject to the relations [ex ] = [ey−1 xy ], [ex ][ey ] = [ey ][ex ] (x, y ∈ XW ).
Consequently, [ex ] = [ey ] ∈ Ad(XW )Ab if and only if x, y ∈ XW are conjugate in
W . We conclude that Ad(XW )Ab is the free abelian group with a basis {[es ] | s ∈
RW }.

Let φ : Ad(XW ) → W be a surjective homomorphism defined by ex 7→ x, which
is well-defined by virtue of (2.3), and let CW := ker φ be its kernel.
Lemma 2.5. CW is a central subgroup of Ad(XW ).
Proof. Given g ∈ CW , it suffices to prove g−1 ex g = ex for all x ∈ XW . To do so, set
g = eεy11 eεy22 · · · eεy`` (εi ∈ {±1}). Applying (2.3) and (2.4), we have
g−1 ex g = (eεy11 eεy22 · · · eεy`` )−1 ex (eεy11 eεy22 · · · eεy`` ) = ex∗y1 ∗y2 ∗···∗y` .
Now x ∗ y1 ∗ y2 ∗ · · · ∗ y` = (y1 y2 · · · y` )−1 x(y1 y2 · · · y` ), and the proposition follows
from φ (g) = yε11 yε22 · · · yε`` = y1 y2 · · · y` = 1.

Lemma 2.6. If x, y ∈ XW are conjugate in W , then e2x = e2y ∈ CW .
Proof. It is obvious that e2x ∈ CW for all x ∈ XW . Since CW is a central subgroup,
2
2
e2x = e−1

y ex ey = ey−1 xy holds for all x, y ∈ XW , which implies the lemma.
3. A RTIN GROUPS AND THE PROOF OF THE MAIN RESULT
Now we state the main result of this paper:
Theorem 3.1. CW is the free abelian group with a basis {e2s | s ∈ RW }. In particular, CW  Zc(W ) .
As a consequence of Theorem 3.1, Ad(XW ) fits into a central extension of the
form 0 → Zc(W ) → Ad(XW ) → W → 1 as stated in the introduction. We begin with
the determination of the rank of CW .
Proposition 3.2. rank(CW ) = c(W ).
φ

Proof. The central extension 1 → CW → Ad(XW ) → W → 1 yields the following
exact sequence for the rational homology of groups:
H2 (W, Q) → H1 (CW , Q)W → H1 (Ad(XW ), Q) → H1 (W, Q)
(see Brown [5, Corollary VII.6.4]). Here the co-invariants H1 (CW , Q)W coincides
with H1 (CW , Q) because CW is a central subgroup of Ad(XW ). It is known that the
rational homology of a Coxeter group is trivial (see Akita [1, Proposition 5.2] or
Davis [7, Theorem 15.1.1]). As a result, we have an isomorphism H1 (CW , Q) 
H1 (Ad(XW ), Q) and hence we have
rank(CW ) = dimQ CW ⊗ Q = dimQ H1 (CW , Q) = dimQ H1 (Ad(XW ), Q) = c(W )
as desired.



COXETER QUANDLE

5

Given a Coxeter system (W, S), the Artin group AW associated with (W, S) is the
group defined by the presentation
AW := has (s ∈ S) | (as at )m(s,t) = (at as )m(s,t) (s,t ∈ S, s , t, m(s,t) < ∞)i.
In view of (2.2), there is an obvious surjective homomorphism π : AW → W defined
by as 7→ s (s ∈ S). The pure Artin group PW associated with (W, S) is defined to be
the kernel of π so that there is an extension
π

1 → PW ,→ AW → W → 1.
In case W is the symmetric group on n letters, AW is the braid group of n strands,
and PW is the pure braid group of n strands. Little is known about the structure of
general Artin groups. Among others, the following questions are still open.
(1) Are Artin groups torsion free?
(2) What is the center of Artin groups?
(3) Do Artin groups have solvable word problem?
(4) Are there finite K(π, 1)-complexes for Artin groups?
See survey articles by Paris [15–17] for further details of Artin groups.
Proposition 3.3. The assignment as 7→ es (s ∈ S) yields a well-defined surjective
homomorphism ψ : AW → Ad(XW ).
Proof. As for the well-definedness, it suffices to show (es et )m(s,t) = (et es )m(s,t) for
all distinct s,t ∈ S with m(s,t) < ∞. Applying the relation ex ey = ey ex∗y (x, y ∈ XW )
repeatedly as in
(et es )m(s,t) = et es et es · · · = es et∗s et es · · · = es et et∗s∗t es · · · = · · · ,
we obtain
(et es )m(s,t) = (es et )m(s,t)−1 ex ,
where
m(s,t)

z
}|
{
x = t ∗ s ∗ t ∗ s ∗ · · · = (st)−1
m(s,t)−1t(st)m(s,t)−1
= (st)−1
m(s,t)−1 (ts)m(s,t)
= (st)−1
m(s,t)−1 (st)m(s,t) .
Here we used the relation (st)m(s,t) = (ts)m(s,t) . It follows from the last equality
that x is the last letter in (st)m(s,t) , i.e. x = s or x = t according as m(s,t) is odd or
m(s,t) is even. We conclude that (es et )m(s,t)−1 ex = (es et )m(s,t) as desired. Finally,
the surjectivity follows from Proposition 2.3.

As a result, the adjoint group Ad(XW ) is an intermediate group between a Coxeter group W and the corresponding Artin group AW , in the sense that the surjection
π : AW  W splits into a sequence of surjections AW  Ad(XW )  W .
Proposition 3.4. PW is the normal closure of {a2s | s ∈ S} in AW . In other words,
PW is generated by g−1 a2s g (s ∈ S, g ∈ AW ).

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T. AKITA

Proof. Given a Coxeter system (W, S), let F(S) be the free group on S and put
2
R := {(st)m(s,t) (ts)−1
m(s,t) | s,t ∈ S, s , t, m(s,t) < ∞}, Q := {s | s ∈ S}.

Let N(R), N(Q) be the normal closure of R, Q in F(S), respectively. The third
isomorphism theorem yields a short exact sequence of groups
1→

F(S) p
F(S)
N(R)N(Q)
→
→
→ 1.
N(R)
N(R)
N(R)N(Q)

Observe that the left term N(R)N(Q)/N(R) is nothing but the normal closure of Q
in F(S)/N(R). Now F(S)/N(R)N(Q) = F(S)/N(R ∪ Q) = W by the definition of
W and F(S)/N(R) is identified with AW via s 7→ as (s ∈ S). Under this identification, the map p is the canonical surjection π : AW  W and hence the left term
N(R)N(Q)/N(R) coincides with PW . The proposition follows.

Remark 3.5. Digne-Gomi [8, Corollary 6] obtained a presentation of PW by using Reidemeister-Schreier method. Their presentation is infinite whenever W is
infinite. Proposition 3.4 can be read off from their presentation.
Now we prove Theorem 3.1. Consider the commutative diagram
1

PW
ψ

1

CW

π

AW

W

1

W

1

ψ
φ

Ad(XW )

whose rows are exact. Since ψ : AW → Ad(XW ) is surjetive, one can check that
its restriction ψ : PW → CW is also surjective. Since PW is generated by g−1 a2s g
(s ∈ S, g ∈ AW ) by Proposition 3.4, CW is generated by
ψ(g−1 a2s g) = ψ(g)−1 e2s ψ(g) = e2s ,
where the last equality follows from the fact that CW is central by Lemma 2.5.
Combining with Lemma 2.6, we see that CW is generated by {e2s | s ∈ RW }. Now
rank(CW ) = c(W ) by Proposition 3.2 and |RW | = c(W ) by the defintion of RW ,
CW must be a free abelian group of rank c(W ) and {e2s | s ∈ RW } must be a basis
of CW .
4. C ONSTRUCTION OF 2- COCYCLES
Throughout this section, we assume that the reader is familiar with group cohomology and Coxeter groups. The central extension
φ

1 → CW → Ad(XW ) → W → 1
corresponds to the cohomology class uφ ∈ H 2 (W,CW ) (see Brown [5, §IV.3] for
precise). In this section, we will construct 2-cocycles representing uφ . Before
doing so, we claim uφ , 0. For if uφ = 0 then Ad(XW )  CW × W . But this is
not the case because Ad(XW )Ab  Zc(W ) by Proposition 2.4 while (CW × W )Ab 
CW ×WAb  Zc(W ) × (Z/2)c(W ) by Proposition 2.2. Now we invoke the celebrated
Matsumoto’s theorem:

COXETER QUANDLE

7

Theorem 4.1 (Matsumoto [13]). Let (W, S) be a Coxeter system, M a monoid
and f : S → M a map such that ( f (s) f (t))m(s,t) = ( f (t) f (s))m(s,t) for all s,t ∈ S,
s , t, m(s,t) < ∞. Then there exists a unique map F : W → M such that F(w) =
f (s1 ) · · · f (sk ) whenever w = s1 · · · sk (si ∈ S) is a reduced expression.
The proof also can be found in [11, Theorem 1.2.2]. Define a map f : S →
Ad(XW ) by s 7→ es , then f satisfies the assumption of Theorem 4.1 as in the proof
of Proposition 3.3, and hence there exists a unique map F : W → Ad(XW ) such
that F(w) = f (s1 ) · · · f (sk ) = es1 · · · esk whenever w = s1 · · · sk (si ∈ S) is a reduced
expression. It is clear that F : W → Ad(XW ) is a set-theoretical section of φ :
Ad(XW ) → W . Define c : W × W → CW by c(w1 , w2 ) = F(w1 )F(w2 )F(w1 w2 )−1 .
The standard argument in group cohomology (see Brown [5, §IV.3]) implies the
following result:
Proposition 4.2. c is a normalized 2-cocycle and [c] = uφ ∈ H 2 (W,CW ).
Remark 4.3. In case W = Sn is the symmetric group of n letters, Proposition 4.2
was stated in [2, Remark 3.3].
Now we deal with the case c(W ) = 1 more closely. All finite irreducible Coxeter
groups of type other than Bn (n ≥ 2), F4 , I2 (m) (m even) satisfy c(W ) = 1. Among
en (n ≥ 2), D
e n (n ≥ 4), Ee6 , Ee7 and
affine irreducible Coxeter groups, those of type A
Ee8 fulfill c(W ) = 1. For simplifying the notation, we will identify CW with Z by
e2s 7→ 1 and denote our central extension by
φ

0 → Z → Ad(XW ) → W → 1.
Proposition 4.4. If c(W ) = 1 then H 2 (W, Z)  Z/2.
Proof. A short exact sequence 0 →√Z ,→ C → C× → 1 of abelian groups, where
C → C× is defined by z 7→ exp(2π −1z), induces the exact sequence
H 1 (W, C) → H 1 (W, C× ) → H 2 (W, Z) → H 2 (W, C)
δ

(see Brown [5, Proposition III.6.1]). It is known that H k (W, C) = 0 for k > 0
(see the proof of Proposition 3.2), which implies that the connecting homomorphism δ : H 1 (W, C× ) → H 2 (W, Z) is an isomorphism. We claim that H 1 (W, C× ) =
Hom(W, C× )  Z/2. Indeed, W is generated by S consisting of elements of order 2, and all elements of S are mutually conjugate by the assumption c(W ) = 1.
Thus Hom(W, C× ) consists of the trivial homomorphism and the homomorphism
ρ defined by ρ(s) = −1 (s ∈ S).

φ

Corollary 4.5. If c(W ) = 1 then 0 → Z → Ad(XW ) → W → 1 is the unique nontrivial central extension of W by Z.
In general, given a homomorphism of groups f : G → C× , the cohomology class
δ f ∈ H 2 (G, Z), where δ : Hom(G, C× ) → H 2 (G, Z) is the connecting homomorphism as above, can be described as follows. For each g ∈ G, choose a branch of
log( f (g)). We argue log( f (1)) = 0. Define τ f : G × G → Z by
1
√ {log( f (g)) + log( f (h)) − log( f (gh))} ∈ Z.
(4.1)
τ f (g, h) =
2π −1

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T. AKITA

By a diagram chase, one can prove the following:
Proposition 4.6. τ f is a normalized 2-cocycle and [τ f ] = δ f ∈ H 2 (G, Z).
Assuming c(W ) = 1, let ρ : W → C× be the homomorphism defined by ρ(s) =
−1 (s ∈ S) as in the proof of Proposition 4.4. Note that ρ(w) = (−1)`(w) (w ∈ W )
where `(w) is the length of w (see Humphreys [12, §5.2]). For each w ∈ W , choose
a branch of log(ρ(w)) as
(
0
if `(w) is even,
√
log(ρ(w)) =
π −1 if `(w) is odd.
Applying (4.1), the corresponding map τρ : W ×W → Z is given by
(
1 if `(w1 ) and `(w2 ) are odd,
(4.2)
τρ (w1 , w2 ) =
0 otherwise.
Now Corollary 4.5 and Proposition 4.6 imply the following theorem:
Theorem 4.7. If c(W ) = 1 then τρ : W ×W → Z defined by (4.2) is a normalized
2-cocycle and [τρ ] = uφ ∈ H 2 (W, Z).
5. C OMMUTATOR SUBGROUPS OF ADJOINT GROUPS
As was stated in the introduction, Eisermann [9, Example 1.18] claimed that
Ad(XSn ) is isomorphic to the semidirect product An o Z where An is the alternating
group on n letters. We will generalize his result to Coxeter quandles by showing
the following theorem:
Theorem 5.1. φ : Ad(XW ) → W induces an isomorphism


φ : [Ad(XW ), Ad(XW )] → [W,W ].
Proof. Consider the following commutative diagram with exact rows and columns.
1
CW

1

[Ad(XW ), Ad(XW )]
φ

1

[W,W ]

Ad(XW )

1
AbAd(XW )

AbAd(XW )

φ

W

1

ker φAb

Ad(XW )Ab

1

φAb
AbW

WAb

1

1

From Proposition 2.2 and Proposition 2.4, we see that ker φAb is the free abelian
group with a basis {[e2s ] | s ∈ RW }, which implies that AbAd(XW ) : CW → ker φAb

COXETER QUANDLE

9

is an isomorphism because it assigns [e2s ] to e2s (s ∈ RW ). Since φ is surjective, it is obvious that φ ([Ad(XW ), Ad(XW )]) = [W,W ]. We will show that φ :
[Ad(XW ), Ad(XW )] → [W,W ] is injective. To do so, let g ∈ [Ad(X), Ad(X)] be an
element with g ∈ ker φ = CW . Then [g] := AbAd(XW ) (g) = 1 by the exactness of the
middle row. But it implies g = 1 because AbAd(XW ) : CW → ker φAb is an isomorphism.

Corollary 5.2. There is a group extension of the form
AbAd(X

)

1 → [W,W ] → Ad(XW ) −→W Zc(W ) → 1.
If c(W ) = 1 then the extension splits and Ad(XW )  [W,W ] o Z.
Since c(Sn ) = 1 and [Sn , Sn ] = An , we recover the claim by Eisermann mentioned above.
Acknowledgement. The author thanks Daisuke Kishimoto and Takefumi Nosaka
for valuable comments and discussions with the author. He also thanks Ye Liu
for informing the paper Digne-Gomi [8] and careful reading of earlier drafts of
this paper. The author was partially supported by JSPS KAKENHI Grant Number
26400077.
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D EPARTMENT OF M ATHEMATICS , H OKKAIDO U NIVERSITY, S APPORO , 060-0810 JAPAN
E-mail address: akita@math.sci.hokudai.ac.jp

