The adjoint group of a Coxeter quandle

TL;DR

This paper explicitly describes the adjoint group of a Coxeter quandle, revealing its structure as a central extension of the Coxeter group and its relation to Artin groups and root systems.

Contribution

It provides a detailed description of the adjoint group of Coxeter quandles, including its construction, relation to Coxeter and Artin groups, and the connection to root systems.

Findings

The adjoint group is an intermediate group between W and A_W.

It fits into a central extension of W by a free abelian group.

The root system is a rack and its adjoint group matches that of the quandle.

Abstract

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $\Phi_W$ associated with a Coxeter group $W$ turns out to be a rack. We prove that the adjoint group of $\Phi_W$ is isomorphic to the adjoint group of $Q_W$.

Equations

$$\mathrm{Ad}(Q):=⟨e_x(x\in Q)\mid e_y^{-1}{e}_x{e}_y=e_{x\ast y}(x,y\in Q)⟩.$$

$$Q_W:=\bigcup _{w\in W}{w}^{-1}Sw.$$

$$0\to \mathbb{Z}\to \mathrm{Ad}(Q_{{\mathfrak{S}}_n})\to {\mathfrak{S}}_n\to 1.$$

$$0\to {\mathbb{Z}}^{c(W)}\to \mathrm{Ad}(Q_W)\stackrel{\varphi }{\to }W\to 1,$$

$$(gh{)}_m:=\underset{m}{\underbrace{ghghg\cdots }}\in G.$$

$$x_1\ast x_2\ast x_3\ast \cdots \ast x_{\ell }:=(\cdots ((x_1\ast x_2)\ast x_3)\ast \cdots )\ast x_{\ell },$$

$$W:=⟨s\in S|(st{)}^{m(s,t)}=1(s,t\in S,m(s,t)<\infty )⟩.$$

$$s^2=1(s\in S),(st{)}_{m(s,t)}=(ts{)}_{m(s,t)}(s,t\in S,s\ne t,m(s,t)<\infty ).$$

$$Q_W:=\bigcup _{w\in W}{w}^{-1}Sw$$

$$\mathrm{Ad}(Q_W):=⟨e_x(x\in Q_W)\mid e_y^{-1}{e}_x{e}_y=e_{x\ast y}(x,y\in Q_W)⟩$$

$$e_y^{-1}{e}_x{e}_y=e_{x\ast y}=e_{y^{-1}xy}$$

$$e_y{e}_x{e}_y^{-1}=e_{x\ast y},$$

$$e_x=e_{(s_1{s}_2\cdots s_{\ell }{)}^{-1}{s}_0(s_1{s}_2\cdots s_{\ell })}=e_{s_0\ast s_1\ast s_2\ast \cdots \ast s_{\ell }}=(e_{s_1}{e}_{s_2}\cdots e_{s_{\ell }}{)}^{-1}{e}_{s_0}(e_{s_1}{e}_{s_2}\cdots e_{s_{\ell }}),$$

$$g^{-1}{e}_{x}g=(e_{y_1}^{{ϵ}_1}{e}_{y_2}^{{ϵ}_2}\cdots e_{y_{\ell }}^{{ϵ}_{\ell }}{)}^{-1}{e}_x(e_{y_1}^{{ϵ}_1}{e}_{y_2}^{{ϵ}_2}\cdots e_{y_{\ell }}^{{ϵ}_{\ell }})=e_{x\ast y_1\ast y_2\ast \cdots \ast y_{\ell }}.$$

$$H_2(W,\mathbb{Q})\to H_1(C_W,\mathbb{Q}{)}_W\to H_1(\mathrm{Ad}(Q_W),\mathbb{Q})\to H_1(W,\mathbb{Q})$$

$$\mathrm{rank}(C_W)$$

$$={\dim }_{\mathbb{Q}}{H}_1(\mathrm{Ad}(Q_W),\mathbb{Q})={\dim }_{\mathbb{Q}}(\mathrm{Ad}(Q_W{)}_{\mathrm{Ab}}\otimes \mathbb{Q})=c(W)$$

$$A_W:=⟨a_s(s\in S)\mid (a_s{a}_t{)}_{m(s,t)}=(a_t{a}_s{)}_{m(s,t)}(s,t\in S,s\ne t,m(s,t)<\infty )⟩.$$

$$1\to P_W\hookrightarrow A_W\stackrel{\pi }{\to }W\to 1.$$

$$(e_t{e}_s{)}_{m(s,t)}=e_t{e}_s{e}_t{e}_s\cdots =e_s{e}_{t\ast s}{e}_t{e}_s\cdots =e_s{e}_t{e}_{t\ast s\ast t}{e}_s\cdots =\cdots ,$$

$$(e_t{e}_s{)}_{m(s,t)}=(e_s{e}_t{)}_{m(s,t)-1}{e}_x,$$

$$\begin{array}{rl}x=\stackrel{m(s,t)}{\overbrace{t\ast s\ast t\ast s\ast \cdots }}& =(st{)}_{m(s,t)-1}^{-1}t(st{)}_{m(s,t)-1}\\ & =(st{)}_{m(s,t)-1}^{-1}(ts{)}_{m(s,t)}\\ & =(st{)}_{m(s,t)-1}^{-1}(st{)}_{m(s,t)}.\end{array}$$

$$A_W\stackrel{\psi }{\twoheadrightarrow }\mathrm{Ad}(Q_W)\stackrel{\varphi }{\twoheadrightarrow }W,a_s\mapsto e_s\mapsto s(s\in S).$$

$$R:=\{(st{)}_{m(s,t)}(ts{)}_{m(s,t)}^{-1}\mid s,t\in S,s\ne t,m(s,t)<\infty \},Q:=\{s^2\mid s\in S\}.$$

$$1\to \frac{N(R)N(Q)}{N(R)}\to \frac{F(S)}{N(R)}\stackrel{p}{\to }\frac{F(S)}{N(R)N(Q)}\to 1.$$

$$1$$

$$\psi (g^{-1}{a}_s^{2}g)=\psi (g{)}^{-1}{e}_s^2\psi (g)=e_s^2(s\in S,g\in A_W),$$

$$H^{\ast }(\mathrm{Ad}(Q_W),\mathbb{Q})\stackrel{\cong }{\to }{H}^{\ast }(C_W,\mathbb{Q})\cong {\wedge }_{\mathbb{Q}}(u_1,u_2,\dots ,u_{c(W)})$$

$$E_2^{pq}=H^p(W,H^q(C_W,\mathbb{Q}))\Rightarrow H^{p+q}(\mathrm{Ad}(Q_W),\mathbb{Q})$$

$$E_2^{pq}\cong H^p(W,\mathbb{Q}){\otimes }_{\mathbb{Q}}{H}^q(C_W,\mathbb{Q})\cong \{\begin{array}{ll}{H}^q(C_W,\mathbb{Q})& p=0\\ 0& p\ne 0\end{array}$$

$$1\to C_W\to \mathrm{Ad}(Q_W)\stackrel{\varphi }{\to }W\to 1$$

Full text

1. Introduction

A nonempty set $Q$ equipped with a binary operation $Q\times Q\rightarrow Q$, $(x,y)\mapsto x\ast y$ is called a quandle if it satisfies the following three conditions:

1. (1)

$x\ast x=x$ $(x\in Q)$, 2. (2)

$(x\ast y)\ast z=(x\ast z)\ast(y\ast z)$ $(x,y,z\in Q)$, 3. (3)

For all $y\in Q$, the map $Q\rightarrow Q$ defined by $x\mapsto x\ast y$ is bijective.

If $X$ satisfies (2) and (3) but not necessarily (1), then $X$ is called a rack. Quandles and racks have been studied in low dimensional topology as well as in Hopf algebras (see Nosaka [nosaka-book] and Andruskiewitsch-Graña [MR1994219] for instance). To any quandle or rack $Q$ one can associate a group ${\operatorname{Ad}}(Q)$ called the adjoint group of $Q$ (also called the associated group or the enveloping group in the literature). It is defined by the presentation

[TABLE]

The assignment $Q\mapsto{\operatorname{Ad}}(Q)$ is a functor from the category of quandles or racks to the category of groups. Although adjoint groups play important roles in the study of quandles and racks, not much is known about the structure of them, partly because the definition of ${\operatorname{Ad}}(Q)$ by a possibly infinite presentation is difficult to work with in explicit calculations. We refer Eisermann [MR3205568] and Nosaka [arXiv:1505.03077, nosaka-book] for generality of adjoint groups, and [arXiv:1011.1587, MR3205568, arXiv:1505.03077, nosaka-book] 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 [arXiv:1505.03077], we define the Coxeter quandle $Q_{W}$ associated with $(W,S)$ to be the set of all reflections of $W$:

[TABLE]

The quandle operation is given by the conjugation $x\ast y:=y^{-1}xy=yxy$. The symmetric group $\mathfrak{S}_{n}$ of $n$ letters is a Coxeter group (of type $A_{n-1}$), and the associated Coxeter quandle $Q_{\mathfrak{S}_{n}}$ is nothing but the set of all transpositions. In their paper [MR2799090], Andruskiewitsch-Fantino-García-Vendramin obtained remarkable results concerning with ${\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})$. Namely, they proved that ${\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})$ is an intermediate group between $\mathfrak{S}_{n}$ and the braid group $B_{n}$ of $n$ strands, in the sense that the canonical projection $B_{n}\twoheadrightarrow\mathfrak{S}_{n}$ splits into a sequence of surjections $B_{n}\twoheadrightarrow{\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})\twoheadrightarrow\mathfrak{S}_{n}$, and that ${\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})$ fits into a central extension of the form

[TABLE]

See [MR2799090]*Proposition 3.2 and the proofs therein. Furthermore, the central extension (1.1) turns out to be the unique nontrivial central extension of $\mathfrak{S}_{n}$ by $\mathbb{Z}$ (see Corollary 4.6).

The primary purpose of this paper is to generalize those results to arbitrary Coxeter quandles. We will show that ${\operatorname{Ad}}(Q_{W})$ is an intermediate group between $W$ and the Artin group $A_{W}$ associated with $W$ (Proposition 3.3), and that ${\operatorname{Ad}}(Q_{W})$ fits into a central extension of the form

[TABLE]

where $c(W)$ is the number of conjugacy classes of elements in $Q_{W}$ (Theorem 3.1). As a byproduct of Theorem 3.1, we will determine the rational cohomology ring of ${\operatorname{Ad}}(Q_{W})$ (Corollary 3.6).

In case $c(W)=1$, the central extension (1.2) turns out to be the unique nontrivial central extension of $W$ by $\mathbb{Z}$ (Corollary 4.6). As is known, the central extension (1.2) corresponds to the cohomology class $u_{\phi}\in H^{2}(W,\mathbb{Z}^{c(W)})$. We will construct 2-cocycles of $W$ representing $u_{\phi}$ (Proposition 4.3 and Theorem 4.8).

Alternatively, Eisermann [MR3205568] claimed that ${\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})$ is isomorphic to the semidirect product $A_{n}\rtimes\mathbb{Z}$ where $A_{n}$ 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).

In the final section, we deal with root systems. To each Coxeter system $(W,S)$, one can associate the root system $\Phi_{W}\subset V$ by using the geometric representation $W\rightarrow GL(V)$, where $V$ is a real vector space with the basis $\{\alpha_{s}\mid s\in S\}$. The root system $\Phi_{W}$ turns out to be a rack with the rack operation $\alpha*\beta:=t_{\beta}(\alpha)$ where $t_{\beta}\in GL(V)$ is the reflection along $\beta$. We close this paper by proving ${\operatorname{Ad}}(\Phi_{W})\cong{\operatorname{Ad}}(Q_{W})$ (Theorem 6.3).

Notation*.*

Let $G$ be a group, $g,h\in G$ elements of $G$ and $m\geq 2$ a natural number. Define

[TABLE]

For example, $(gh)_{2}=gh,(gh)_{3}=ghg,(gh)_{4}=ghgh$. Let $G_{\mathrm{Ab}}:=G/[G,G]$ be the abelianization of $G$, $\mathrm{Ab}_{G}:G\rightarrow G_{\mathrm{Ab}}$ the natural projection, and write $[g]:=g[G,G]\in G_{\mathrm{Ab}}$ for $g\in G$. For a quandle $Q$ and elements $x_{k}\in Q$ $(k=1,2,3,\dots)$, we denote

[TABLE]

for simplicity.

2. Coxeter groups and Coxeter quandles

2.1. Coxeter groups

Let $S$ be a finite set and $m:S\times S\rightarrow\mathbb{N}\cup\{\infty\}$ a map satisfying the following conditions:

1. (1)

$m(s,s)=1$ for all $s\in S$; 2. (2)

$2\leq m(s,t)=m(t,s)\leq\infty$ for all distinct $s,t\in S$.

The map $m$ is represented by the Coxeter graph $\Gamma$ whose vertex set is $S$ and whose edges are the unordered pairs $\{s,t\}\subset S$ such that $m(s,t)\geq 3$. The edges with $m(s,t)\geq 4$ are labeled by the number $m(s,t)$. The Coxeter system associated with $\Gamma$ is the pair $(W,S)$, where $W$ is the group generated by $s\in S$ and the fundamental relations $(st)^{m(s,t)}=1$ $(s,t\in S,m(s,t)<\infty)$:

[TABLE]

The group $W$ is called the Coxeter group (of type $\Gamma$), 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 $s\in S$ has order $2$. It is easy to check that the defining relations in (2.1) are equivalent to the following relations

[TABLE]

Finally, the odd subgraph $\Gamma_{odd}$ is a subgraph of $\Gamma$ whose vertex set is $S$ and whose edges are the unordered pairs $\{s,t\}\subset S$ such that $m(s,t)$ is an odd integer. We refer Björner-Brenti [MR2133266], Bourbaki [bourbaki], Davis [MR2360474], and Humphreys [humphreys] for further details of Coxeter groups.

2.2. Conjugacy classes of reflections

Let

[TABLE]

be the set of reflections in $W$ as in the introduction (the underlying set of the Coxeter quandle). Let $\mathcal{O}_{W}$ be the set of conjugacy classes of elements of $Q_{W}$ under $W$, and $\mathcal{R}_{W}$ a complete set of representatives of conjugacy classes. We may choose such that $\mathcal{R}_{W}\subseteq S$. Let $c(W)$ be the cardinality of $\mathcal{O}_{W}$. The following two propositions are well-known and easy to prove (see Björner-Brenti [MR2133266]*Chapter 1, Exercises 16–17 for instance).

Proposition 2.1**.**

The elements of $\mathcal{O}_{W}$ are in one-to-one correspondence with the connected components of $\Gamma_{odd}$. Consequently, $c(W)$ equals to the number of connected components of $\Gamma_{odd}$.

To be precise, the conjugacy class of $s\in S$ corresponds to the connected component of $\Gamma_{odd}$ containing $s\in S$.

Proposition 2.2**.**

$W_{\mathrm{Ab}}$* is the elementary abelian $2$-group with a basis $\{[s]\in W_{\mathrm{Ab}}\mid s\in\mathcal{R}_{W}\}$. In particular, $W_{\mathrm{Ab}}\cong(\mathbb{Z}/2)^{c(W)}$.*

2.3. Coxeter quandles

Now we turn our attention to Coxeter quandles. Let $Q_{W}$ be the Coxeter quandle associated with $(W,S)$ and

[TABLE]

the adjoint group of $Q_{W}$. Observe that

[TABLE]

and

[TABLE]

where (2.4) follows from $e_{y}^{-1}e_{x\ast y}e_{y}=e_{x\ast y\ast y}=e_{x}$.

Proposition 2.3**.**

${\operatorname{Ad}}(Q_{W})$* is generated by $e_{s}$ $(s\in S)$.*

Proof.

Given $x\in Q_{W}$, we can express $x$ as $x=(s_{1}s_{2}\cdots s_{\ell})^{-1}s_{0}(s_{1}s_{2}\cdots s_{\ell})$ for some $s_{i}\in S$ $(0\leq i\leq\ell)$ by the definition of $Q_{W}$. Applying (2.3), we have

[TABLE]

proving the proposition. ∎

Proposition 2.4**.**

${\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}$* is the free abelian group with a basis $\{[e_{s}]\mid s\in\mathcal{R}_{W}\}$. In particular, ${\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}\cong\mathbb{Z}^{c(W)}$.*

Proof.

By the definition of ${\operatorname{Ad}}(Q_{W})$, the abelianization ${\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}$ is generated by $[e_{x}]$ $(x\in Q_{W})$ subject to the relations $[e_{x}]=[e_{y^{-1}xy}]$, $[e_{x}][e_{y}]=[e_{y}][e_{x}]$ $(x,y\in Q_{W})$. Consequently, $[e_{x}]=[e_{y}]\in{\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}$ if and only if $x,y\in Q_{W}$ are conjugate in $W$. We conclude that ${\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}$ is the free abelian group with a basis $\{[e_{s}]\mid s\in\mathcal{R}_{W}\}$. ∎

Let $\phi:{\operatorname{Ad}}(Q_{W})\rightarrow W$ be a surjective homomorphism defined by $e_{x}\mapsto x$, which is well-defined by virtue of (2.3), and let $C_{W}:=\ker\phi$ be its kernel.

Lemma 2.5**.**

$C_{W}$* is a central subgroup of ${\operatorname{Ad}}(Q_{W})$.*

Proof.

Given $g\in C_{W}$, it suffices to prove $g^{-1}e_{x}g=e_{x}$ for all $x\in Q_{W}$. To do so, set $g=e_{y_{1}}^{\epsilon_{1}}e_{y_{2}}^{\epsilon_{2}}\cdots e_{y_{\ell}}^{\epsilon_{\ell}}$ $(\epsilon_{i}\in\{\pm 1\})$. Applying (2.3) and (2.4), we have

[TABLE]

Now $x\ast y_{1}\ast y_{2}\ast\cdots\ast y_{\ell}=(y_{1}y_{2}\cdots y_{\ell})^{-1}x(y_{1}y_{2}\cdots y_{\ell})$, and the proposition follows from $\phi(g)=y_{1}^{\epsilon_{1}}y_{2}^{\epsilon_{2}}\cdots y_{\ell}^{\epsilon_{\ell}}=y_{1}y_{2}\cdots y_{\ell}=1$. ∎

Lemma 2.6**.**

If $x,y\in Q_{W}$ are conjugate in $W$, then $e_{x}^{2}=e_{y}^{2}\in C_{W}$.

Proof.

It is obvious that $e_{x}^{2}\in C_{W}$ for all $x\in Q_{W}$. Since $C_{W}$ is a central subgroup, $e_{x}^{2}=e_{y}^{-1}e_{x}^{2}e_{y}=(e_{y}^{-1}e_{x}e_{y})^{2}=e^{2}_{y^{-1}xy}$ holds for all $x,y\in Q_{W}$, which implies the lemma. ∎

3. Artin groups and the proof of the main result

Now we state the main result of this paper:

Theorem 3.1**.**

The central subgroup $C_{W}$ is the free abelian group with a basis $\{e_{s}^{2}\mid s\in\mathcal{R}_{W}\}$. In particular, $C_{W}\cong\mathbb{Z}^{c(W)}$.

As a consequence of Theorem 3.1, ${\operatorname{Ad}}(Q_{W})$ fits into a central extension of the form $0\rightarrow\mathbb{Z}^{c(W)}\rightarrow{\operatorname{Ad}}(Q_{W})\rightarrow W\rightarrow 1$ as stated in the introduction. We begin with the determination of the rank of $C_{W}$.

Proposition 3.2**.**

${\operatorname{rank}}(C_{W})=c(W)$.

Proof.

The central extension $1\rightarrow C_{W}\rightarrow{\operatorname{Ad}}(Q_{W})\overset{\phi}{\rightarrow}W\rightarrow 1$ yields the following exact sequence for the rational homology of groups:

[TABLE]

(see Brown [brown]*Corollary VII.6.4). Here the co-invariants $H_{1}(C_{W},\mathbb{Q})_{W}$ coincides with $H_{1}(C_{W},\mathbb{Q})$ because $C_{W}$ is a central subgroup of ${\operatorname{Ad}}(Q_{W})$. It is known that the rational (co)homology of a Coxeter group is trivial (see Akita [a-euler, Proposition 5.2] or Davis [MR2360474]*Theorem 15.1.1). As a result, we have an isomorphism $H_{1}(C_{W},\mathbb{Q})\cong H_{1}({\operatorname{Ad}}(Q_{W}),\mathbb{Q})$ and hence we have

[TABLE]

as desired. ∎

To proceed further, we need the notions of Artin groups and pure Artin groups. Given a Coxeter system $(W,S)$, the Artin group $A_{W}$ associated with $(W,S)$ is the group defined by the presentation

[TABLE]

In view of (2.2), there is an obvious surjective homomorphism $\pi:A_{W}\rightarrow W$ defined by $a_{s}\mapsto s$ $(s\in S)$. The pure Artin group $P_{W}$ associated with $(W,S)$ is defined to be the kernel of $\pi$ so that there is an extension

[TABLE]

In case $W$ is the symmetric group on $n$ letters, $A_{W}$ is the braid group of $n$ strands, and $P_{W}$ is the pure braid group of $n$ strands. Artin groups were introduced by Brieskorn-Saito [MR0323910]. Little is known about the structure of general Artin groups. Among others, the following questions are still open.

1. (1)

Are Artin groups torsion free? 2. (2)

What is the center of Artin groups? 3. (3)

Do Artin groups have solvable word problem? 4. (4)

Are there finite $K(\pi,1)$-complexes for Artin groups?

See survey articles by Paris [MR2497781, MR3205598, MR3207280] for further details of Artin groups.

Proposition 3.3**.**

The assignment $a_{s}\mapsto e_{s}$ $(s\in S)$ yields a well-defined surjective homomorphism $\psi:A_{W}\rightarrow{\operatorname{Ad}}(Q_{W})$.

Proof.

As for the well-definedness, it suffices to show $(e_{s}e_{t})_{m(s,t)}=(e_{t}e_{s})_{m(s,t)}$ for all distinct $s,t\in S$ with $m(s,t)<\infty$. Applying the relation $e_{x}e_{y}=e_{y}e_{x\ast y}$ $(x,y\in Q_{W})$ repeatedly as in

[TABLE]

we obtain

[TABLE]

where

[TABLE]

In the last equality we used the relation $(st)_{m(s,t)}=(ts)_{m(s,t)}$. It follows 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 $(e_{s}e_{t})_{m(s,t)-1}e_{x}=(e_{s}e_{t})_{m(s,t)}$ as desired. Finally, the surjectivity follows from Proposition 2.3. ∎

As a result, the adjoint group ${\operatorname{Ad}}(Q_{W})$ is an intermediate group between a Coxeter group $W$ and the corresponding Artin group $A_{W}$, in the sense that the canonical surjection $\pi:A_{W}\twoheadrightarrow W$, $a_{s}\mapsto s$ $(s\in S)$ splits into a sequence of surjections

[TABLE]

Proposition 3.4**.**

$P_{W}$* is the normal closure of $\{a_{s}^{2}\mid s\in S\}$ in $A_{W}$. In other words, $P_{W}$ is generated by $g^{-1}a_{s}^{2}g$ $(s\in S,g\in A_{W})$.*

Proof.

Given a Coxeter system $(W,S)$, let $F(S)$ be the free group on $S$ and put

[TABLE]

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

[TABLE]

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 $F(S)/N(R)N(Q)=F(S)/N(R\cup Q)=W$ by the definition of $W$ and $F(S)/N(R)$ is identified with $A_{W}$ via $s\mapsto a_{s}$ $(s\in S)$. Under this identification, the map $p$ is the canonical surjection $\pi:A_{W}\twoheadrightarrow W$ and hence the left term $N(R)N(Q)/N(R)$ coincides with $P_{W}$. The proposition follows. ∎

Remark 3.5*.*

Digne-Gomi [MR1831679]*Corollary 6 obtained a presentation of $P_{W}$ by using Reidemeister-Schreier method. Their presentation is infinite whenever $W$ is infinite. Although Proposition 3.4 may be read off from their presentation, we wrote down the proof because our proof is much simpler than the arguments in [MR1831679].

Proof of Theorem 3.1.

Consider the commutative diagram

[TABLE]

whose rows are exact. Since $\psi:A_{W}\rightarrow{\operatorname{Ad}}(Q_{W})$ is surjective, one can check that its restriction $\psi|_{P_{W}}:P_{W}\rightarrow C_{W}$ is also surjective. As $P_{W}$ is generated by $g^{-1}a_{s}^{2}g$ $(s\in S,g\in A_{W})$ by Proposition 3.4, $C_{W}$ is generated by

[TABLE]

where the last equality follows from the fact that $C_{W}$ is central by Lemma 2.5. Combining with Lemma 2.6, we see that $C_{W}$ is generated by $\{e_{s}^{2}\mid s\in\mathcal{R}_{W}\}$. Now ${\operatorname{rank}}(C_{W})=c(W)$ by Proposition 3.2 and $|\mathcal{R}_{W}|=c(W)$ by the defintion of $\mathcal{R}_{W}$, $C_{W}$ must be a free abelian group of rank $c(W)$ and $\{e_{s}^{2}\mid s\in\mathcal{R}_{W}\}$ must be a basis of $C_{W}$. ∎

As an immediate cosequence of Theorem 3.1, we can determine the rational cohomology ring of ${\operatorname{Ad}}(Q_{W})$:

Corollary 3.6**.**

For any Coxeter system $(W,S)$, the inclusion $C_{W}\hookrightarrow{\operatorname{Ad}}(Q_{W})$ induces an isomorphism

[TABLE]

with $\deg u_{k}=1$ $(1\leq k\leq c(W))$.

Proof.

In the Lyndon-Hochschild-Serre spectral sequence

[TABLE]

associated with the central extension $1\rightarrow C_{W}\rightarrow{\operatorname{Ad}}(Q_{W})\rightarrow W\rightarrow 1$, one has

[TABLE]

because $H^{p}(W,\mathbb{Q})=0$ for $p\not=0$ (see the proof of Proposition 3.2). The first isomorphism follows immediately. The second isomorphism follows from the fact $C_{W}\cong\mathbb{Z}^{c(W)}$. ∎

Remark 3.7*.*

For any right-angled Coxeter group $W$, Kishimoto [arXiv:1706.06209]*Theorem 5.3 determined the mod 2 cohomology ring of ${\operatorname{Ad}}(Q_{W})$.

4. Construction of $2$-cocycles

Throughout this section, we assume that the reader is familiar with group cohomology and Coxeter groups. The central extension

[TABLE]

corresponds to the cohomology class $u_{\phi}\in H^{2}(W,C_{W})$ (see Brown [brown]*§IV.3 for precise). In this section, we will construct 2-cocycles representing $u_{\phi}$. Before doing so, we claim $u_{\phi}\not=0$. The claim is equivalent to the following lemma.

Lemma 4.1**.**

The central extension $1\rightarrow C_{W}\rightarrow{\operatorname{Ad}}(Q_{W})\overset{\phi}{\rightarrow}W\rightarrow 1$ is nontrivial.

Proof.

If the central extension is trivial, then ${\operatorname{Ad}}(Q_{W})\cong C_{W}\times W$. But this is not the case because ${\operatorname{Ad}}(Q_{W})_{\mathrm{Ab}}\cong\mathbb{Z}^{c(W)}$ by Proposition 2.4 while $(C_{W}\times W)_{\mathrm{Ab}}\cong C_{W}\times W_{\mathrm{Ab}}\cong\mathbb{Z}^{c(W)}\times(\mathbb{Z}/2)^{c(W)}$ by Proposition 2.2. ∎

Now we invoke the celebrated Matsumoto’s theorem:

Theorem 4.2** (Matsumoto [MR0183818]).**

Let $(W,S)$ be a Coxeter system, $M$ a monoid and $f:S\rightarrow M$ a map such that $(f(s)f(t))_{m(s,t)}=(f(t)f(s))_{m(s,t)}$ for all $s,t\in S$ with $s\not=t$, $m(s,t)<\infty$. Then there exists a unique map $F:W\rightarrow M$ such that $F(w)=f(s_{1})\cdots f(s_{k})$ whenever $w=s_{1}\cdots s_{k}$ $(s_{i}\in S)$ is a reduced expression.

The proof also can be found in [bourbaki]*Chapitre IV, §1, Proposition 5 and [MR1778802]*Theorem 1.2.2. Define a map $f:S\rightarrow{\operatorname{Ad}}(Q_{W})$ by $s\mapsto e_{s}$, then $f$ satisfies the assumption of Theorem 4.2 as in the proof of Proposition 3.3, and hence there exists a unique map $F:W\rightarrow{\operatorname{Ad}}(Q_{W})$ such that $F(w)=f(s_{1})\cdots f(s_{k})=e_{s_{1}}\cdots e_{s_{k}}$ whenever $w=s_{1}\cdots s_{k}$ $(s_{i}\in S)$ is a reduced expression. It is clear that $F:W\rightarrow{\operatorname{Ad}}(Q_{W})$ is a set-theoretical section of $\phi:{\operatorname{Ad}}(Q_{W})\rightarrow W$. Define $c:W\times W\rightarrow C_{W}$ by

[TABLE]

The standard argument in group cohomology (see Brown [brown]*§IV.3) implies the following result:

Proposition 4.3**.**

$c$* is a normalized $2$-cocycle and $[c]=u_{\phi}\in H^{2}(W,C_{W})$.*

Remark 4.4*.*

In case $W$ is the symmetric group of $n$ letters $\mathfrak{S}_{n}$, Proposition 4.3 was stated in [MR2799090]*Remark 3.3.

Now we deal with the case $c(W)=1$ more precisely. If $c(W)=1$ then the odd subgraph $\Gamma_{odd}$ of $W$ is connected and hence $W$ must be irreducible. All finite irreducible Coxeter groups of type other than $B_{n}$ $(n\geq 2)$, $F_{4}$, $I_{2}(m)$ ($m$ even) satisfy $c(W)=1$. Among affine irreducible Coxeter groups, those of type $\widetilde{A}_{n}$ $(n\geq 2)$, $\widetilde{D}_{n}$ $(n\geq 4)$, $\widetilde{E}_{6}$, $\widetilde{E}_{7}$ and $\widetilde{E}_{8}$ fulfill $c(W)=1$. For simplifying the notation, we will identify $C_{W}$ with $\mathbb{Z}$ by $e_{s}^{2}\mapsto 1$ and denote our central extension by

[TABLE]

Proposition 4.5**.**

If $c(W)=1$ then $H^{2}(W,\mathbb{Z})\cong\mathbb{Z}/2$.

Proof.

A short exact sequence $0\rightarrow\mathbb{Z}\hookrightarrow\mathbb{C}\rightarrow\mathbb{C}^{\times}\rightarrow 1$ of abelian groups, where $\mathbb{C}\rightarrow\mathbb{C}^{\times}$ is defined by $z\mapsto\exp(2\pi\sqrt{-1}z)$, induces the exact sequence

[TABLE]

(see Brown [brown]*Proposition III.6.1). It is known that $H^{k}(W,\mathbb{C})=0$ for $k>0$ (see the proof of Proposition 3.2), which implies that the connecting homomorphism $\delta:H^{1}(W,\mathbb{C}^{\times})\rightarrow H^{2}(W,\mathbb{Z})$ is an isomorphism. We claim that $H^{1}(W,\mathbb{C}^{\times})=\operatorname{Hom}(W,\mathbb{C}^{\times})\cong\mathbb{Z}/2$. 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 $\operatorname{Hom}(W,\mathbb{C}^{\times})$ consists of the trivial homomorphism and the homomorphism $\rho$ defined by $\rho(s)=-1$ $(s\in S)$. ∎

Combining Lemma 4.1 and Proposition 4.5, we obtain the following corollary.

Corollary 4.6**.**

If $c(W)=1$ then $0\rightarrow\mathbb{Z}\rightarrow{\operatorname{Ad}}(Q_{W})\overset{\phi}{\rightarrow}W\rightarrow 1$ is the unique nontrivial central extension of $W$ by $\mathbb{Z}$.

In general, given a homomorphism of groups $f:G\rightarrow\mathbb{C}^{\times}$, the cohomology class $\delta\!f\in H^{2}(G,\mathbb{Z})$, where $\delta:\operatorname{Hom}(G,\mathbb{C}^{\times})\rightarrow H^{2}(G,\mathbb{Z})$ is the connecting homomorphism as above, can be described as follows. For each $g\in G$, choose a branch of $\log(f(g))$. We argue such that $\log(f(1))=0$. Define $\tau_{f}:G\times G\rightarrow\mathbb{Z}$ by

[TABLE]

By a diagram chase, one can prove the following:

Proposition 4.7**.**

$\tau_{f}$* is a normalized $2$-cocycle and $[\tau_{f}]=\delta\!f\in H^{2}(G,\mathbb{Z})$.*

Assuming $c(W)=1$, let $\rho:W\rightarrow\mathbb{C}^{\times}$ be the homomorphism defined by $\rho(s)=-1$ $(s\in S)$ as in the proof of Proposition 4.5. Note that $\rho(w)=(-1)^{\ell(w)}$ $(w\in W)$ where $\ell(w)$ is the length of $w$ (see Humphreys [humphreys]*§5.2). For each $w\in W$, choose a branch of $\log(\rho(w))$ as

[TABLE]

Applying (4.2), the corresponding map $\tau_{\rho}:W\times W\rightarrow\mathbb{Z}$ is given by

[TABLE]

Combining Lemma 4.1, Corollary 4.6, and Proposition 4.7, we obtain the following theorem:

Theorem 4.8**.**

If $c(W)=1$ then $\tau_{\rho}:W\times W\rightarrow\mathbb{Z}$ defined by $(\ref{eq-cocycle-coxeter-conn})$ is a normalized $2$-cocycle and $[\tau_{\rho}]=u_{\phi}\in H^{2}(W,\mathbb{Z})$.

5. Commutator subgroups of adjoint groups

As was stated in the introduction, Eisermann [MR3205568]*Example 1.18 claimed that ${\operatorname{Ad}}(Q_{\mathfrak{S}_{n}})$ is isomorphic to the semidirect product $A_{n}\rtimes\mathbb{Z}$ where $A_{n}$ is the alternating group on $n$ letters. We will generalize his result to Coxeter quandles by showing the following theorem:

Theorem 5.1**.**

$\phi:{\operatorname{Ad}}(Q_{W})\rightarrow W$* induces an isomorphism*

[TABLE]

Proof.

Consider the following commutative diagram with exact rows and columns.

[TABLE]

From Proposition 2.2 and Proposition 2.4, we see that $\ker\phi_{\mathrm{Ab}}$ is the free abelian group with a basis $\{[e_{s}^{2}]\mid s\in\mathcal{R}_{W}\}$, which implies that $\mathrm{Ab}_{{\operatorname{Ad}}(Q_{W})}:C_{W}\rightarrow\ker\phi_{\mathrm{Ab}}$ is an isomorphism because it assigns $[e_{s}^{2}]$ to $e_{s}^{2}$ $(s\in\mathcal{R}_{W})$. Since $\phi$ is surjective, it is obvious that $\phi([{\operatorname{Ad}}(Q_{W}),{\operatorname{Ad}}(Q_{W})])=[W,W]$. We will show that $\phi:[{\operatorname{Ad}}(Q_{W}),{\operatorname{Ad}}(Q_{W})]\rightarrow[W,W]$ is injective. To do so, let $g\in[{\operatorname{Ad}}(Q_{W}),{\operatorname{Ad}}(Q_{W})]$ be an element with $g\in\ker\phi=C_{W}$. Then $[g]:=\mathrm{Ab}_{{\operatorname{Ad}}(Q_{W})}(g)=1$ by the exactness of the middle row. But it implies $g=1$ because $\mathrm{Ab}_{{\operatorname{Ad}}(Q_{W})}:C_{W}\rightarrow\ker\phi_{\mathrm{Ab}}$ is an isomorphism. ∎

Corollary 5.2**.**

There is a group extension of the form

[TABLE]

If $c(W)=1$ then the extension splits and ${\operatorname{Ad}}(Q_{W})\cong[W,W]\rtimes\mathbb{Z}$.

Since $c(\mathfrak{S}_{n})=1$ and $[\mathfrak{S}_{n},\mathfrak{S}_{n}]=A_{n}$, we recover the claim by Eisermann mentioned above.

Remark 5.3*.*

By using Theorem 5.1, Kishimoto [arXiv:1706.06209]*Theorem 2.5 proved that the following commutative square which appeared in the proof of Theorem 5.1 is actually a pull-back.

[TABLE]

6. Root systems

As was pointed out by Andruskiewitsch-Graña [MR1994219]*§1.3.5, Brieskorn [MR975077]*p. 58, Example 9, and Fenn-Rourke [MR1194995]*Example 10, a root system turns out to be a rack. In this final section, we turn our attention to root systems. Let us recall the definition and properties of the root system associated with a Coxeter system $(W,S)$. See Björner-Brenti [MR2133266]*Chapter 4 or Humphreys [humphreys]*Part II, Chapter 5 for precise. Let $V$ be the real vector space with a basis $\{\alpha_{s}\mid s\in S\}$ in one to one correspondence with $S$. Define a symmetric bilinear form $B(-,-)$ on $V$ by

[TABLE]

Here we understand $B(\alpha_{s},\alpha_{t})=-1$ in case $m(s,t)=\infty$. For each $s\in S$, define a reflection $\sigma_{s}\in GL(V)$ by

[TABLE]

The geometric representation of $W$ (or the canonical representation in the literature) is the unique homomorphism $\sigma:W\rightarrow GL(V)$ satisfying $s\mapsto\sigma_{s}$ $(s\in S)$. The geometric representaion is faithful and preserves the form $B(-,-)$. For simplicity, we may write $w(\lambda)$ in place of $\sigma(w)(\lambda)$ $(w\in W,\lambda\in V)$. The root system $\Phi_{W}$ associated with $(W,S)$ is defined by

[TABLE]

For each root $\alpha\in\Phi_{W}$, define a reflection $t_{\alpha}\in GL(V)$ by $t_{\alpha}(\lambda):=\lambda-2B(\alpha,\lambda)\alpha$. Note that if $\alpha=\alpha_{s}$ $(s\in S)$ then $t_{\alpha}=\sigma_{s}$.

Lemma 6.1**.**

The root system $\Phi_{W}$ turns out to be a rack with the rack operation $\alpha\ast\beta=t_{\beta}(\alpha)$ $(\alpha,\beta\in\Phi_{W})$.

Proof.

It suffices to prove $(\alpha\ast\beta)\ast\gamma=(\alpha\ast\gamma)\ast(\beta\ast\gamma)$ for $\alpha,\beta,\gamma\in\Phi_{W}$. One has

[TABLE]

while

[TABLE]

and the assertion follows from $B(t_{\gamma}(\beta),t_{\gamma}(\alpha))=B(\beta,\alpha)$. Note that $\Phi_{W}$ is not a quandle because $\alpha\ast\alpha=t_{\alpha}(\alpha)=-\alpha$. ∎

Now let $Q_{W}$ be the Coxeter quandle associated with a Coxeter system $(W,S)$ as before, and define a map $p:\Phi_{W}\to Q_{W}$ by $\alpha\mapsto\ t_{\alpha}$. Here we identify elements of $\sigma(W)$ with those of $W$, which is possible since $\sigma$ is injective. Note that $t_{\alpha}$ belongs to $Q_{W}$ because $t_{\alpha}=wsw^{-1}$ holds for $\alpha=w(\alpha_{s})$, and that the map $p$ is surjective and two-to-one (see Humphreys [humphreys]*p. 116 or Björner-Brenti [MR2133266]*p. 104). Indeed, $p$ maps $\pm\alpha\in\Phi_{W}$ to the same element $t_{\alpha}=t_{-\alpha}$.

Lemma 6.2**.**

The map $p\colon\Phi_{W}\to Q_{W}$ is a morphism of racks.

Proof.

In general, $wt_{\alpha}w^{-1}=t_{w(\alpha)}$ holds for $\alpha\in\Phi_{W}$ and $w\in W$ (see Björner-Brenti [MR2133266]*p. 104). Now for $\alpha,\beta\in\Phi_{W}$, set $\gamma:=t_{\beta}(\alpha)$ and we have

[TABLE]

as desired. ∎

We close this paper by proving the following result:

Theorem 6.3**.**

The morphism $p\colon\Phi_{W}\to Q_{W}$ of racks induces an isomorphism ${\operatorname{Ad}}(\Phi_{W})\cong{\operatorname{Ad}}(Q_{W})$.

Proof.

Recall that ${\operatorname{Ad}}(\Phi_{W})$ is defined by the presentation

[TABLE]

For each $\alpha\in\Phi_{W}$, we have $e_{\alpha\ast\alpha}=e_{-\alpha}$ while $e_{\alpha}^{-1}e_{\alpha}e_{\alpha}=e_{\alpha}$, which proves that $e_{-\alpha}=e_{\alpha}$. On the other hand, as $p:\Phi_{W}\to Q_{W}$ is surjective, ${\operatorname{Ad}}(Q_{W})$ may be defined as

[TABLE]

Since $p^{-1}(\alpha)=\{\pm\alpha\}$ and $p(\alpha\ast\beta)=p(\alpha)\ast p(\beta)$ by Lemma 6.2, the assignment $e_{\alpha}\mapsto e_{p(\alpha)}$ $(\alpha\in\Phi_{W})$ induces an isomorphism ${\operatorname{Ad}}(\Phi_{W})\rightarrow{\operatorname{Ad}}(Q_{W})$. ∎

Remark 6.4*.*

For a (crystallographic) root system $\Phi$ and its Weyl group $W(\Phi)$, one can prove that ${\operatorname{Ad}}(\Phi)\cong{\operatorname{Ad}}(Q_{W(\Phi)})$ in a similar way.

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 [MR1831679] and careful reading of drafts of this paper. This work was partially supported by JSPS KAKENHI Grant Number 26400077, and by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.