#  ${\rm SL}_{*}$ over local and adèle rings:
$*$-euclideanity and
Bruhat generators

Luis Gutiérrez Frez  ,  Luis Lomelí  and  José Pantoja∗

(Date: February 2021)

###### Abstract.

Let $(R,*)$ be a ring with involution and let $A={\rm M}(n,R)$ be the matrix
ring endowed with the $*$-transpose involution. We study ${\rm SL}_{*}(2,A)$
and the question of Bruhat generation over commutative and non-commutative
local and adèlic rings $R$. An important tool is the property of a ring being
$*$-Euclidean. In this regard, we introduce the notion of a $*$-local ring
$R$, prove that $A$ is $*$-Euclidean and explore reduction modulo the Jacobson
radical for such rings. Globally, we provide an affirmative answer to the
question wether a commutative adèlic ring $R$ leads towards the ring $A$ being
$*$-Euclidean; while the non-commutative adèlic quaternions are such that $A$
is $*$-Euclidean and ${\rm SL}_{*}$ is generated by its Bruhat elements if and
only if the characteristic is $2$.

###### Key words and phrases:

${\rm SL}_{*}$ groups; $*$-Euclidean rings; Bruhat generators

2020 _Mathematics Subject Classification_. Primary 20G35, 20F05, 16L30

††footnotetext: (*) The authors were partially supported by Fondecyt Grant
1171583.

## Introduction

We study ${\rm SL}_{*}$ groups, introduced by Pantoja and Soto-Andrade [5],
over local and adèle rings with involution, where we work over commutative and
non-commutative rings with identity. At the base of the algebraic properties
of ${\rm SL}_{*}$ groups are its Bruhat elements and the question of Bruhat
generation. In [6], the notion of a $*$-Euclidean ring is introduced, which
provides a powerfult tool that allows us to infer that ${\rm SL}_{*}(2,A)$ is
generated by its Bruhat elements when $A$ is such a ring with involution.

The non-commutative $*$-analogue of a special linear group, poses many
interesting questions that are in accordance with the classical theory of Weil
representations, Bruhat presentations and the Langlands program. In this
article, we study for the first time the questions of $*$-Euclideanity and
Bruhat generation for adèle rings, including the quaternions in characteristic
zero and charactersitic $p$. We develop general machinery along the way, where
we are in many places guided by global questions posed in the adèlic setting.
For example, we make a careful study of the local question of reduction modulo
$\mathfrak{p}$, in fact, modulo the Jacobson radical. It is curious that we
quickly arrive at the notion of a _$*$ -local ring_, for which we introduce a
proper set of hypothesis in order to characterize a ring with a unique ideal
that is stable under the involution and is maximal with this property.

Many examples arise by considering the ring of matrices $A={\rm M}(n,R)$, when
$R$ is a $*$-local ring or an adèle ring with involution. The ring $A$ is
itself endowed with the $*$-transpose involution; the involution of $A$
induced from that of $R$. We prove that $A$ is $*$-Euclidean for a list of
examples of involutive rings $(R,*)$ provided in Theorem 4.3.1:

  * (i)

$R$ a $*$-local ring.

  * (ii)

$R$ the ring of adèles $\mathbb{A}_{F}$ of a global field $F$, with trivial
involution.

  * (iii)

$R=\mathbb{A}_{E}$, where $E/F$ is a separable quadratic field extension with
the involution dictated by the non-trivial Galois automorphism.

  * (iv)

$R=\mathbb{A}_{\mathbb{H}}$, where $\mathbb{H}$ is a quaternion algebra over a
global field $F$, ${\rm char}(F)=2$.

In all of these cases, ${\rm SL}_{*}(2,A)$ is generated by its Bruhat
elements. It is noteworthy to observe in case (iv), of a quaternion algebra
over a global field $F$, that the ring ${\rm M}(n,\mathbb{A}_{\mathbb{H}})$ is
$*$-Euclidean and is generated by its Bruhat elements if and only if ${\rm
char}(F)=2$.

We now give a more detailed account of the contents of this article. The ${\rm
SL}_{*}$ functor of Pantoja and Soto-Andrade for not necessarily commutative
rings with involution and basic properties are reviewed in § 1; the main
references being [5, 6]. In particular, in § 1.3 we recall the notion of a
$*$-Euclidean ring.

We arrive at the notion of a $*$-local ring $R$ in § 2, where we quickly
characterize these rings in the basic Lemma 2.1.1. They are $1$\- or $2$-local
rings with involution. We refer to the former simply as a local ring with
involution, where the Jacobson radical is the unique maximal prime ideal
$\mathcal{J}=\operatorname{\mathfrak{p}}$. While the latter case has two
distinct maximal prime ideals $\operatorname{\mathfrak{p}}$ and
$\operatorname{\mathfrak{p}}^{*}$, its Jacobson radical being
$\mathcal{J}=\operatorname{\mathfrak{p}}\cap\operatorname{\mathfrak{p}}^{*}$.
We then proceed to study symmetric and invertible elements for the matrix ring
$A={\rm M}(n,R)$, endowed with the involution induced from the $*$-local ring
$R$, and their behavior under reduction modulo the Jacobson radical
$\mathcal{J}$ of $A$. More precisely, we treat both cases of $1$\- and
$2$-local rings with involution in a single swoop, study the projection map
$\pi:A\xrightarrow{\ \pi\ }\overline{A}=A/\mathcal{J}$ and produce a section
map $\sigma$ that is compatible with symmetry and invertibility in Lemma
2.2.1.

We prove that the matrix ring $A={\rm M}(n,R)$ over a $*$-local ring is
$*$-Euclidean in Theorem 3.1.1; a result that Soto-Andrade proved when $R$ is
a field [8], and, for a division ring $R$, it is part of the “Co-prime Lemma”
of Pantoja and Soto-Andrade [5]. For this, we show in Lemma 3.1.2 that if
$\overline{A}=A/\mathcal{J}$ is $*$-Euclidean, then so is $A$. And, in Lemma
3.1.3 we establish $*$-Euclideanity for the new case of $R=D_{1}\times D_{2}$,
with $D_{1}$ and $D_{2}$ division rings related by an anti-automorphism
$\varphi:D_{1}\rightarrow D_{2}$ and the $\varphi$-flip transpose involution
on $A={\rm M}(n,R)$.

The adèles over a commutative global field are studied in § 3.2, where $A={\rm
M}(n,\mathbb{A}_{F})$ is proved to be $*$-Euclidean in Theorem 3.2.1. A
quadratic global field extension $E/F$, which we address in § 3.3, leads to
$A={\rm M}(n,\mathbb{A}_{E})$ being $*$-Euclidean for the involution induced
from the non-trivial Galois element of ${\rm Gal}(E/F)$, Theorem 3.3.1.
Locally, we require Lemmas 3.1.2 and 3.1.3 for the semi-local ring
$R=\mathcal{O}_{F}\times\mathcal{O}_{F}$ and a finite residue ring
$R/\mathcal{J}=k_{F}\times k_{F}$, respectively, each of them with the flip
involution.

The $*$-Euclidean property for the quaternions $\mathbb{H}$ over a local or a
global field $F$, is studied in § 3.4. The general basic theory is expounded
in [10] for central simple algebras in a manner that is independent of the
characteristic, and is detailed in [9] for $\mathbb{H}$. We here encounter
fundamental differences depending on the characteristic being $2$ or not, see
Theorem 3.4.3. Locally, we may have non-split or split quaternions at a place
$v$ of $F$. The finitely many non-split cases $\mathbb{H}_{v}$ are division
algebras over a local field $F_{v}$, while at the remaining infinitely many
split places we have that $\mathbb{H}_{v}$ are matrix quaternions. The case of
a finite place $v$ of $F$, leads to a non-Archimedean local field $F_{v}$ with
ring of integers $\mathcal{O}_{v}$. And we also have quaternioninc rings
$\mathcal{Q}_{v}$ at the non-Archimedean places, which can also be split or
non-split.

Before continuing to inspect the global quaternions, we need a couple of local
lemmas that we prove in the slightly more general settings of quaternion
matrix rings and quaternion division algebras that exhibit the main difference
of the characteristic being different than $2$ or not, Lemmas 3.4.1 and 3.4.2,
respectively. The main $*$-Euclideanity result for the adèlic quaternions is
Theorem 3.4.3, whose proof also requires two general local lemmas that are of
independent interest for local split matrix rings of quaternions and for
division algebras, namely Lemmas 3.4.4 and 3.4.5. In short, the ring of
matrices $A={\rm M}(n,\mathbb{A}_{\mathbb{H}})$ over the quaternion adèle ring
is $*$-Euclidean if and only if ${\rm char}(\mathbb{H})=2$.

In the final section, we begin with Dieudonné and his non-conmutative
determinant, and we extend a basic criterion for invertibility to general
linear groups over $*$-local rings. The case of a commutative $2$-local ring
with involution naturally arises while studying unitary groups at a split
place of a quadratic extension of global fields, where there is a known
connection to general linear groups. We review the connection between ${\rm
SL}_{*}$ groups and the even unitary groups in § 4.2.

It is in § 4.3 where we arrive at our main application of the $*$-Euclidean
property, that is, ${\rm SL}_{*}$ groups and Bruhat generation for a list of
examples over local and adèle rings, which includes unitary groups and ${\rm
SL}_{*}$ over quaternion rings. We recall the definition of the Bruhat
elements, provide basic properties and state the main application in the form
of Theorem 4.3.1. Having done much of the work for local and adèle rings, a
large part of the theorem follows from the results of § 3, once we incorporate
the result of Pantoja and Soto-Andrade that ${\rm SL}_{*}(2,A)$ has a set of
Bruhat generators when $A$ is $*$-Euclidean [6]. However, special care must be
taken in the case of the quaternions.

The quaternions over local and adèle rings, their connection to ${\rm SL}_{*}$
groups and the question of Bruhat generation are scrutinized in § 4.4. In
addition to completing the proof of Theorem 4.3.1, we record a couple of
interesting quaternionic facts along the way. For example, a quaternion
algebra $\mathbb{H}$ over a non-Archimedean local field $F$ leads towards
special, Dieudonné special and $*$-analogue special linear groups: ${\rm
SL}(2,F)$, ${\rm SL}(2,\mathbb{H})$ and ${\rm SL}_{*}(2,\mathbb{H})$. All
three groups are distinct, and the first and the third are related by

${\rm SL}_{*}(2,\mathbb{H})=D_{\mathbb{H}}\cdot{\rm SL}(2,F),$

where $D_{\mathbb{H}}$ is the subgroup of ${\rm SL}_{*}(2,\mathbb{H})$
consisting of diagonal elements. Furthermore, it is curious to observe that
each of ${\rm SL}(2,F)$ and ${\rm SL}(2,\mathbb{H})$, by Ihara’s theoerem, has
a further decomposition as an amalgamated product involving the corresponding
congruence subgroup over the ring of integers and Iwahori subgroup; see [7],
for example.

We conclude by refining our theorem on the algebra $A={\rm
M}(n,\mathbb{A}_{\mathbb{H}})$, obtaining in this case that ${\rm
SL}_{*}(2,A)$ has Bruhat generation precisely when ${\rm char}(\mathbb{H})=2$.
This is done by incorporating a result of [3] on Bruhat generation for finite
fields that generalizes to the setting of a local non-Archimedean split
quaternionic algebra ring, together with the results of § 3.4.

##  1\. Preliminaries on ${\rm SL}_{*}$

Let $R$ be a ring with $1$, endowed with an anti-automorphism
$\alpha:R\rightarrow R$, $r\mapsto r^{*}$, with $\alpha$ either trivial or of
order $2$. Notice that $\alpha={\rm id}$ is only possible when $R$ is
commutative. In general, $R$ is a not necessarily commutative ring with
involution.

###  1.1. ${\rm SL}_{*}$ groups

Let $\mathcal{A}_{R}$ be the category of involutive rings with identity
$(A,*)$, such that $R\subset A$ and the involution of $A$ is compatible with
that of the involutive ring $R$. Our main examples are obtained by taking the
ring of $n\times n$ matrices $A={\rm M}(n,R)$ with entries in $R$. In this
case, the involution is given by

$a^{*}=(a_{ji}^{*}),\text{ for }a=(a_{ij})\in{\rm M}(n,R).$ (1.1)

Let $(A,*)\in\mathcal{A}_{R}$. We denote the center of $A$ by $Z_{A}$ and its
group of invertible elements by $A^{\times}$. We let

$A^{\text{sym}}=\\{a\in A\mid a^{*}=a\\},$

called the set of symmetric elements, and let

${Z^{\times\,\text{sym}}_{A}}=Z_{A}\cap A^{\times}\cap A^{\text{sym}},$

be the set of central invertible symmetric elements of $A$.

If we let $\mathcal{G}$ denote the category of groups, then we have a functor

$\displaystyle\mathcal{SL}:\mathcal{A}_{R}$
$\displaystyle\rightarrow\mathcal{G}$ $\displaystyle(A,*)$
$\displaystyle\mapsto{\rm SL}_{*}(2,A),$

originally defined by Pantoja and Soto-Andrade. We next recall several basic
properties of ${\rm SL}_{*}$-groups, proved in [5].

###  1.2. General setting

Let $(A,*)\in\mathcal{A}_{R}$. The involution on $A$ induces the involution on
the ring of matrices ${\rm M}(2,A)$ given by (1.1). If we write

$J=\left(\begin{array}[]{rr}\ 0&1\\\ -1&0\end{array}\right),$

then ${\rm GL}_{*}(2,A)$ denotes the set of invertible matrices in $g\in{\rm
M}(2,A)$ such that

$g^{*}Jg=\delta(g)J,$

for some $\delta(g)\in Z^{\times\,\text{sym}}_{A}$.

The set ${\rm GL}_{*}(2,A)$ forms a group under matrix multiplication, in
fact, $g\in{\rm GL}_{*}(2,A)$ implies $\delta(g^{-1})=\delta(g)^{-1}$,
$g^{*}\in{\rm GL}_{*}(2,A)$ and $\delta(g)=\delta(g^{*})$. Furthermore, we
have an epimorphism

${\rm det}_{*}:{\rm GL}_{*}(2,A)\rightarrow Z_{A}^{\times\,{\rm sym}},$

given by

${\rm det}_{*}(g)=ad^{*}-bc^{*},\quad\quad g=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in{\rm GL}_{*}(2,A).$

######  Definition 1.2.1.

The group ${\rm SL}_{*}(2,A)$ is the kernel of the epimorphism $\det_{*}$.

Making explicit the conditions on the matrices, we see that ${\rm
SL}_{*}(2,A)$ is the group of matrices

$g=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right),$ (1.2)

with $a,b,c,d\in A$, satisfying the following:

$ad^{*}-bc^{*}=a^{*}d-c^{*}b=1\quad\text{and}\quad
ab^{*},cd^{*},a^{*}c,b^{*}d\in A^{\rm sym}.$ (1.3)

We observe that ${\rm SL}_{*}(2,A)$ may often be viewed as the isometry group,
(and ${\rm GL}_{*}(2,A)$ as the similitude group with multiplier $\delta$) of
the hermitian form

$h(x,y)=x^{*}Jy,\quad x,y\in A^{2}.$

Here, $x\in A^{2}$ is identified with a column vector, and the involution $*$
of $A$ is extended to a map from column vectors to row vectors, by taking the
involution $*$ of $A$ entry-wise and the transpose.

Another observation is that one retrieves the groups ${\rm GL}(2,A)$ and ${\rm
SL}(2,A)$ when $A$ is commutative with trivial $*$.

###  1.3. $*$-Euclidean rings

Let $(A,*)\in\mathcal{A}_{R}$. We say that $A$ is _$*$ -Euclidean_ if given
$a,c\in A$ such that

$Aa+Ac=A,\quad a^{*}c\in A^{\rm sym},$

and setting $a=r_{-1}$, $c=r_{0}$, then there exist elements
$s_{0},\ldots,s_{n-1}\in A^{\text{sym}}$, $r_{1},\ldots,r_{n-1}\in A$ and
$r_{n}\in A^{\times}$ such that

$r_{i-1}=s_{i}r_{i}+r_{i+1},$

for $i=0,\ldots,n-1$. When this is so, we say $n$ is the decomposition length
of the pair $(a,c)$. The minimum length valid for all possible pairs, if it
exists, is the length of the $*$-Euclidean ring $A$.

As examples of $*$-Euclidean rings $A$ we have: the ring of matrices ${\rm
M}(n,F)$ with entries in a field [8]; and, ${\rm M}(n,D)$ with entries in a
division ring [5]. In § 3 we will prove that ${\rm M}(n,R)$ is $*$-Euclidean,
where $R$ is a $1$-local, $2$-local or a commutative adèle ring.

##  2\. Local rings with involution

We say that a ring $R$ with identity is _l-maximal_ if every left maximal
ideal is an ideal. There is the similar notion of _r-maximal_ , involving
right maximal ideals.

Assume that $(R,*)$ is a ring with involution, then $R$ is $l$-maximal if and
only if it is $r$-maximal. A stable ideal $\mathfrak{a}$ of $R$ is one such
that $\mathfrak{a}^{*}=\mathfrak{a}$.

An $l$-maximal ring with involution $(R,*)$ is Dedekind finite, i.e., a ring
where every element with a right inverse also has a left inverse;
equivalently, a left invertible element is invertible. To see this, suppose
$x\in R$ is right invertible, but not left invertible. Then there exists a
left maximal ideal $\operatorname{\mathfrak{m}}$ of $R$ containing $x$. But
$x$ being right invertible implies $\operatorname{\mathfrak{m}}R=R$,
contradicting the assumption on $R$ that $\operatorname{\mathfrak{m}}$ is also
a right ideal.

We observe that a finite direct product of local rings is $l$-maximal.
However, the ring of matrices $A={\rm M}(n,R)$ over a local ring has left
maximal ideals but no maximal ideals for $n>1$; its Jacobson radical is a
stable ideal, the unique maximal stable ideal. In this case, we know that $A$
is semilocal [4], hence Dedekind finite.

###  2.1. On the notion of a $*$-local ring

Define a _$*$ -local ring_ to be an $l$-maximal ring with involution $(R,*)$
having a unique stable maximal ideal.

######  Lemma 2.1.1.

A $*$-local ring $R$ has maximal spectrum

${\rm
MSpec}(R)=\left\\{\operatorname{\mathfrak{p}},\operatorname{\mathfrak{p}}^{*}\right\\}\subset{\rm
Spec}(R),$

and unique stable maximal ideal given by the Jacobson radical

$\operatorname{\mathcal{J}}=\operatorname{\mathfrak{p}}\cap\operatorname{\mathfrak{p}}^{*}.$

Furthermore, $D=R/\operatorname{\mathfrak{p}}$ is a division ring, and we have

$x\in R^{\times}\ \Longleftrightarrow\
\bar{x}\in\left(R/\mathcal{J}\right)^{\times}\ \Longleftrightarrow\
x\notin\operatorname{\mathfrak{p}},\
x^{*}\\!\notin\operatorname{\mathfrak{p}}.$

###### Proof.

Let $\mathfrak{s}$ be the unique stable maximal ideal of $R$. Observe that
$\mathfrak{s}$ is contained in a maximal ideal $\mathfrak{m}$ of $R$, and we
have that $\mathfrak{m}\cap\mathfrak{m}^{*}$ is a stable ideal. By hypothesis
$\mathfrak{s}\supset\mathfrak{m}\cap\mathfrak{m}^{*}$. If $x$ were an element
of $\mathfrak{s}\setminus\mathfrak{m}\cap\mathfrak{m}^{*}$, then
$x\notin\mathfrak{m}$ or $x^{*}\notin\mathfrak{m}$. If
$x^{*}\notin\mathfrak{m}$, for example, we would then have

$R=Rx^{*}+\mathfrak{m}\subset\mathfrak{s}+\mathfrak{m}\subset\mathfrak{m},$

a contradiction; and, similarly if $x\notin\mathfrak{m}$. Hence, we must have
$\mathfrak{s}=\mathfrak{m}\cap\mathfrak{m}^{*}$.

Now, if $\mathfrak{a}\in{\rm MSpec}(R)$ were distinct from $\mathfrak{m}$,
$\mathfrak{m}^{*}$, then so would $\mathfrak{a}^{*}\in{\rm MSpec}(R)$. But
then

$\mathfrak{s}\supset(\mathfrak{m}\cap\mathfrak{m}^{*})+(\mathfrak{a}\cap\mathfrak{a}^{*})=R,$
(2.1)

where the last equality can be seen by using the Chinese Remainder Theorem.
However, equation (2.1) gives a contradiction. Therefore, we must have

${\rm MSpec}(R)=\left\\{\mathfrak{m},\mathfrak{m}^{*}\right\\}\text{ and
}\mathfrak{s}=\operatorname{\mathcal{J}},$

the Jacobson radical.

An application of the Chinese Remainder Theorem, tells us that

$x+\operatorname{\mathcal{J}}\in\left(R/\mathcal{J}\right)^{\times}\
\Longleftrightarrow\
x+\mathfrak{m}\in\left(R/\mathfrak{m}\right)^{\times}\\!\\!,\
x+\mathfrak{m}^{*}\in\left(R/\mathfrak{m}^{*}\right)^{\times}.$

Now, the Jacobson radical has the property that $1+y$ is invertible for every
$y\in\operatorname{\mathcal{J}}$. From here, we can infer that

$x\in R^{\times}\ \Longleftrightarrow\
x+\operatorname{\mathcal{J}}\in\left(R/\mathcal{J}\right)^{\times}\
\Longleftrightarrow\ x\notin\mathfrak{m},\ x\notin\mathfrak{m}^{*}.$

Finally, $R$ is semilocal, hence Dedekind finite [4]. Then a maximal ideal
$\operatorname{\mathfrak{m}}\in{\rm MSpec}(R)$ is prime, i.e.
$\operatorname{\mathfrak{m}}=\operatorname{\mathfrak{p}}\in{\rm Spec}(R)$, and
the quotient $D=R/\operatorname{\mathfrak{p}}$ is a division ring. ∎

We thus have two possibilities for a $*$-local ring $R$, depending if it has
one or two maximal prime ideals. We refer to the former case as a $1$-_local
ring with involution_ , or simply a _local ring with involution_ , since $R$
has a unique maximal ideal
$\operatorname{\mathfrak{p}}=\operatorname{\mathfrak{p}}^{*}$. And call the
latter a _$2$ -local ring with involution_, where
$\operatorname{\mathfrak{p}}\neq\operatorname{\mathfrak{p}}^{*}$.

To give an example of a $2$-local ring with involution, take a local ring $R$
with maximal ideal $\operatorname{\mathfrak{p}}$, then we form the semilocal
ring $S=R\oplus R$ and provide it with the flip involution
$(r,r^{\prime})^{*}=(r^{\prime},r)$. Then $S$ is a $*$-local ring that is not
a local ring.

###  2.2. Reduction mod $\operatorname{\mathfrak{p}}$ for $*$-local rings

Let $R$ be a $*$-local ring with

${\rm
MSpec}=\\{\operatorname{\mathfrak{p}},\operatorname{\mathfrak{p}}^{*}\\}.$

We consider the ring $S=R\oplus R$ with flip∗ involution

$(x,y)^{*}=(y^{*},x^{*}).$

Then $R$ is isomorphic to the diagonal subring

$R_{\Delta}=\left\\{(z,z)\mid z\in R\right\\}\subset S.$

In this setting, the involution on $R$ is compatible with the flip∗ involution

${R}$${R}$${R_{\Delta}}$${R_{\Delta}}$$\scriptstyle{*}$$\scriptstyle{\wr}$$\scriptstyle{\wr}$$\scriptstyle{{\rm
flip}^{*}}$

We fix a maximal prime ideal $\operatorname{\mathfrak{p}}$ of $R$, and reduce
mod $\operatorname{\mathcal{J}}$. We write

$\overline{R}=R/\mathcal{J},\quad\overline{S}=R/\\!\operatorname{\mathfrak{p}}\,\oplus\,R/\\!\operatorname{\mathfrak{p}}^{*},$

where we have two projection maps

$x\in R\mapsto\bar{x}=x+\operatorname{\mathfrak{p}}\in
R/\operatorname{\mathfrak{p}}\text{ and }y\in
R\mapsto\tilde{y}=y+\operatorname{\mathfrak{p}}^{*}\in
R/\operatorname{\mathfrak{p}}^{*},$

giving rise to a projection from $S$ to $\overline{S}$,
$\pi:(x,y)\mapsto(\bar{x},\tilde{y})$. Let

$\overline{R}_{\operatorname{\mathfrak{p}}}=\left\\{(\bar{z},\tilde{z})\mid
z\in R\right\\}\subset\overline{S},$

so that we obtain a non-canonical projection map $\pi$.

${R_{\Delta}}$${R}$${R/\mathcal{J}}$${\overline{R}_{\operatorname{\mathfrak{p}}}}$$\scriptstyle{\sim}$$\scriptstyle{\pi}$$\scriptstyle{\rm
proj}$$\scriptstyle{\wr}$

Here, $\overline{R}_{\operatorname{\mathfrak{p}}}$ is isomorphic to
$R/\operatorname{\mathfrak{p}}$ when $R$ is a local ring, and is the degree
$2$ separable algebra $R/\operatorname{\mathfrak{p}}\oplus
R/\operatorname{\mathfrak{p}}^{*}\simeq R/\mathcal{J}$ when $R$ is a $2$-local
ring, equipped with the involution

$(\bar{z},\tilde{z})\mapsto(\bar{z}^{*},\tilde{z}^{*}).$

In the latter case, notice that the isomorphism
$R/\mathcal{J}\simeq\overline{R}_{\operatorname{\mathfrak{p}}}$ is obtained
via the Chinese Remainder Theorem, where every $(\bar{x},\tilde{y})$
corresponds to a
$(\bar{z},\tilde{z})\in\overline{R}_{\operatorname{\mathfrak{p}}}$, $z\in R$.

We wish to construct a non-canonical section map $\sigma$ for $\pi$, in such a
way that it is compatible with symmetry and invertibility. From Lemma 2.1.1,
we know that

$z\in R^{\times}\simeq R_{\Delta}^{\times}\ \Longleftrightarrow\
\pi(z)=(\bar{z},\tilde{z})\in\overline{R}_{\operatorname{\mathfrak{p}}}^{\times}.$

We build a set consisting of pairs of representatives

$R_{\sigma}^{\times}=\\{(z_{\sigma},z_{\sigma})\in
R_{\Delta}^{\times}\mid(\bar{z}_{\sigma},\tilde{z}_{\sigma})\in\overline{R}_{\operatorname{\mathfrak{p}}}^{\times}\\},$

which satisfy

$z=z^{*}\ \Longleftrightarrow\ z_{\sigma}=z_{\sigma}^{*}.$

We enlarge this set to obtain a section from $R_{\operatorname{\mathfrak{p}}}$
to $R_{\Delta}$, in such a way that it respects symmetry,

$R_{\sigma}=R_{\sigma}^{\times}\cup\left\\{(z_{\sigma},z_{\sigma})\mid
z_{\sigma}\in\operatorname{\mathfrak{p}}\cup\operatorname{\mathfrak{p}}^{*}\right\\}.$

Hence, by construction, we have

$z\in R^{\times}\simeq R_{\Delta}^{\times}\ \Longleftrightarrow\ z_{\sigma}\in
R^{\times}.$

and

$z\in R^{\,\rm sym}\simeq R_{\Delta}^{\,\rm sym}\ \Longleftrightarrow\
z_{\sigma}\in R^{\,\rm sym}.$

We next extend this further to $A={\rm M}(n,R)$, where
$(A,*)\in\mathcal{A}_{R}$ for the involution given by (1.1), where in this
context $\operatorname{\mathcal{J}}$ denotes the Jacobson radical of $A$. We
let

$\overline{A}_{\operatorname{\mathfrak{p}}}={\rm
M}(n,\overline{R}_{\operatorname{\mathfrak{p}}})=\left\\{(\bar{a},\tilde{a})\in{\rm
M}(n,R/\\!\operatorname{\mathfrak{p}})\oplus{\rm
M}(n,R/\\!\operatorname{\mathfrak{p}}^{*})\mid a\in{\rm M}(n,R)\right\\},$

which has involution

$(\bar{a},\tilde{a})\mapsto(\bar{a}^{*},\tilde{a}^{*}).$

Hence, we also have a non-canonical projection $\pi$ in this setting.

${A\simeq{\rm
M}(n,R_{\Delta})}$${\overline{A}_{\operatorname{\mathfrak{p}}}\simeq
A/\mathcal{J}.}$$\scriptstyle{\rm proj}$$\scriptstyle{\pi}$ (2.2)

We continue by extending the section map on $R$, entry-wise for the elements
of $A$. Setting

$A_{\sigma}=\left\\{a_{\sigma}=(a_{ij})\in{\rm M}(n,R)\mid a_{ij}\in
R_{\sigma}\right\\},$

allows for writing $A$ as a sum of two sets

$A=A_{\sigma}+\mathcal{J},$

with $A_{\sigma}\cap\mathcal{J}=\\{0\\}$. Given $a\in A$, we obtain a unique
decomposition

$a=a_{\sigma}+a_{\j},\quad a_{\sigma}\in A_{\sigma},\ a_{\j}\in\mathcal{J}.$
(2.3)

Furthermore, symmetric elements are such that

$a=a^{*}\ \Longleftrightarrow\ a_{\sigma}=a_{\sigma}^{*}\text{ and
}a_{\j}=a_{\j}^{*}.$ (2.4)

We summarize the basic properties of the above construction in the following.

######  Lemma 2.2.1.

Let $R$ be a $*$-local ring, and form the matrix ring $A={\rm M}(n,R)$. For
the projection map

$A\xrightarrow{\ \pi\ }\overline{A}=A/\mathcal{J},\quad a\mapsto
a+\operatorname{\mathcal{J}},$

there exists a non-canonical section map

$\overline{A}\xrightarrow{\ \sigma\ }A,\quad
a+\operatorname{\mathcal{J}}\mapsto a_{\sigma}.$

The maps preserve the involutions on $A$ and $\overline{A}$, and the following
properties hold:

  * (i)

The projection $\pi(a)$ is symmetric if and only if the section $a_{\sigma}$
is symmetric.

  * (ii)

The units satify

$a\in A^{\times}\ \Longleftrightarrow\ \pi(a)\in\overline{A}^{\times}\
\Longleftrightarrow\ a_{\sigma},a_{\sigma}^{*}\in A^{\times}.$

###### Proof.

By definition, $\pi$ preserves the involutions. The existence of $\sigma$ is
due to the decomposition (2.3), where we have

$a^{*}=a_{\sigma}^{*}+a_{\j}^{*}.$

Hence, the section preserves the involutions, namely,

$(a_{\sigma})^{*}=(a^{*})_{\sigma}.$

Property (i) follows from (2.4).

For invertibility, first suppose $a_{\sigma},a_{\sigma}^{*}\in A^{\times}$.
The elements of the form

$1+x,\ \text{ with }x\in\mathcal{J},$

are known to be invertible by Bass’ Lemma 6.4 of [2]. Hence

$a_{\sigma}^{-1}a=1+a_{\sigma}^{-1}a_{\j}\in A^{\times},$

and we conclude that $a\in A^{\times}$. Clearly $a\in
A^{\times}\,\Longrightarrow\,\pi(a)\in\overline{A}^{\times}$. Now, suppose
that $\pi(a)\in\overline{A}^{\times}$; and, for brevity write
$\bar{a}=\pi(a)$, $\bar{b}=\pi(b)$. Then
$\bar{a}\cdot\bar{b}=\bar{b}\bar{a}=\bar{1}$ for some
$\bar{b}\in\overline{A}^{\times}$. By writing $a=a_{\sigma}+a_{\j}$,
$b=b_{\sigma}+b_{\j}$, we obtain
$a_{\sigma}b_{\sigma}=b_{\sigma}a_{\sigma}=1$. Thus $a_{\sigma}\in
A^{\times}$, and $a_{\sigma}^{*}\in A^{\times}$. ∎

##  3\. $*$-Euclideanity over local and adèle rings

We study involutive matrix rings over adèlic rings and their $*$-Euclidean
property. This global situation, naturally poses questions for rings with
involution in the local setting. Hence, we begin by establishing for $*$-local
rings, a result that Soto-Andrade proved when $R$ is a field in Chapitre III,
§ 1.2, Lemme 3 of [8]. And, for division rings $R$, it is part of the “Co-
prime Lemma” of Pantoja and Soto-Andrade [5]. We then extend to adèlic rings
over a global field $F$, we consider separable quadratic extensions of global
fields $E/F$, in addition to quaternion algebras over $F$.

###  3.1. Local setting

Throughout this subsection, we let $R$ be a $*$-local ring, and form the ring

$A={\rm M}(n,R),\quad(A,*)\in\mathcal{A}_{R},$

with the involution induced from that of $R$. We let
$\operatorname{\mathcal{J}}$ denote the Jacobson radical of $A$, or sometimes
the Jacobson radical of $R$, and it should be clear from context which one is
being used.

######  Theorem 3.1.1.

The ring of $n\times n$ matrices $A$ over a $*$-local ring is $*$-Euclidean.

We need a couple of results in order to prove this theorem, where the key
point is reduction mod $\operatorname{\mathcal{J}}$.

######  Lemma 3.1.2.

If the ring $\overline{A}=A/\mathcal{J}$ is $*$-Euclidean, then so is $A$.

###### Proof.

We have elements $a,c\in A$, which we can reduce mod $\mathcal{J}$, namely, we
look at $\bar{a},\bar{c}\in\overline{A}={\rm M}(n,\overline{R})$. We have the
hypothesis

$\displaystyle a^{*}c\in A^{\rm sym}\ $ $\displaystyle\Longrightarrow\
\bar{a}^{*}\bar{c}\in\overline{A}^{\,\rm sym}$ $\displaystyle Aa+Ac=A\ $
$\displaystyle\Longrightarrow\
\overline{A}\bar{a}+\overline{A}\bar{c}=\overline{A}.$

Identify $A$ with ${\rm M}(n,R_{\Delta})$ and $\overline{A}=A/\mathcal{J}$
with $\overline{A}_{\operatorname{\mathfrak{p}}}$, as in (2.2), where we have
a projection map $\pi$. Setting $\bar{a}=\bar{r}_{-1}$ and
$\bar{c}=\bar{r}_{0}$, then, by assumption, there exist elements
$\bar{s}_{0},\ldots,\bar{s}_{n-1}\in A^{\text{sym}}$,
$\bar{r}_{1},\ldots,\bar{r}_{n-1}\in A$ and $\bar{r}_{n}\in A^{\times}$ such
that

$\bar{r}_{i-1}=\bar{s}_{i}\bar{r}_{i}+\bar{r}_{i+1},$

for $i=0,\ldots,n-1$.

The section map of Lemma 2.2.1 gives

$a_{\sigma},\ c_{\sigma},\ s_{i,\sigma},\ r_{i,\sigma}\in A_{\sigma}\subset
A,$

where $s_{i,\sigma}\in A^{\text{sym}}$, $r_{i,\sigma}\in A^{\text{sym}}$ and
$r_{n,\sigma}\in A^{\times}$. We then have

$\overline{r}_{i-1,\sigma}=\overline{s_{i,\sigma}r_{i,\sigma}+r_{i+1,\sigma}}\text{
in }\overline{A}=A/\mathcal{J}.$

Hence, there exists in every step an $x_{i}\in\mathcal{J}$ such that

$r_{i-1,\sigma}+r_{i-1,\j}=s_{i,\sigma}r_{i,\sigma}+(r_{i+1,\sigma}+x_{i}).$

The last term parenthesis, when $i+1=n$, is the sum of a unit of $A$ plus an
element of the radical of $A$, so it is a unit by Property (ii) of Lemma
2.2.1. Hence, the result follows. ∎

After reducing mod $\operatorname{\mathcal{J}}$, there are two possibilities
for $R$: it is either a $1$-local or a $2$-local ring with involution. The
former gives $\overline{A}={\rm M}(n,D)$, where $D$ is a division ring, a case
proved by Pantoja and Soto-Andrade in Proposition 3.3 of [5]. The latter leads
to a sum of two division rings after reduction mod
$\operatorname{\mathcal{J}}$, and we now prove the $*$-Euclidean property in
this case.

######  Lemma 3.1.3.

Let $D_{1}$ and $D_{2}$ be division rings, together with an anti-automorphism
$\varphi:D_{1}\rightarrow D_{2}$. Let $A_{i}={\rm M}(n,D_{i})$, $i=1$, $2$,
and extend $\varphi:A_{1}\rightarrow A_{2}$, componentwise. Then, consider the
ring $A=A_{1}\oplus A_{2}$, with $\varphi$-flip transpose involution

$(a_{1},a_{2})^{*}=(\varphi^{-1}(a_{2})^{t},\varphi(a_{1})^{t}).$

Let $a,c\in A$ be such that

$Aa+Ac=A,\quad a^{*}c=c^{*}a.$

Then, there exist

$s\in A^{\rm sym}=\\{(x,\varphi(x)^{t})\mid x\in A_{1}\\},\quad r\in
A^{\times},$

satisfying

$a=sc+r.$

###### Proof.

Let us observe that any $s\in A^{\rm sym}$ and $r\in A^{\times}$ satisfying
$a=sc+r$, must also be such that

$Ac+Ar=A.$ (3.1)

And, the symmetry relation $a^{*}c=c^{*}a$ leads to

$c^{*}r=r^{*}c.$ (3.2)

We write

$a=(a_{1},a_{2}),\ c=(c_{1},c_{2}),\ s=(s_{1},s_{2}),\ r=(r_{1},r_{2}).$

Now, we have by hypothesis

$A_{1}a_{1}+A_{1}c_{1}=A_{1}.$

Since $A_{1}$ consists of $n\times n$ matrices with entries in a division ring
$D_{1}$, this equation tells us that $a_{1}$ and $c_{1}$ must satisfy

${\rm rank}(a_{1})+{\rm rank}(c_{1})\geq n.$

Because of this, we can multiply $a_{1}$ and $c_{1}$ by products of elementary
matrices, $e$ and $f$, in such a way that

$ea_{1}+fc_{1}=u,$

where $u$ is a unipotent matrix, in particular, $u\in A_{1}^{\times}$. We can
now go back and choose $s$, where we note that we only need to define the
first component, since the symmetry requirement, $s\in A^{\rm sym}$, fixes the
second component

$s_{1}=-e^{-1}f\ \Longrightarrow\ r=a-sc.$

With such a choice, $r_{1}=u\in A_{1}^{\times}$. Next, we need the second
component of $r$ to be a unit, and for this we look at equation (3.2), which
gives

$\varphi(r_{1})^{t}c_{2}=\varphi(c_{1})^{t}r_{2}.$

And, incorporating (3.1) leads to

$\displaystyle A_{2}c_{2}+A_{2}r_{2}=A_{2}\ $ $\displaystyle\Longrightarrow\
A_{2}\varphi(r_{1})^{t}c_{2}+A_{2}r_{2}=A_{2}$ $\displaystyle\Longrightarrow\
A_{2}\varphi(c_{1})^{t}r_{2}+A_{2}r_{2}=A_{2}$ $\displaystyle\Longrightarrow\
A_{2}r_{2}=A_{2}\ \Longrightarrow\ r_{2}\in A_{2}^{\times}.$

∎

With the two lemmas at hand, the proof of Theorem 3.1.1 is complete. We
observe that, in both cases, the $*$-Euclidean length is $1$.

###  3.2. The adèles

Let $F$ be a global field, i.e., either a number field or a function field,
and let $\mathcal{O}_{F}$ be its ring of integers and $\mathbb{A}_{F}$ its
ring of adèles [10]. Given a place or valuation $v$ of $F$, we let $F_{v}$
denote its completion. If $v$ is non-Archimedean, we let $\mathcal{O}_{v}$ be
the corresponding ring of integers.

We wish to study ${\rm SL}_{*}$ groups over the adèles, in fact, over $A={\rm
M}(n,\mathbb{A}_{F})$. For this, we fix some notation. At every place $v$ of
$F$, we write $A_{v}$ for ${\rm M}(n,F_{v})$. There are finitely many infinite
places of $F$, where we write $v\mid\infty$, and have two possibilities:
$A_{v}={\rm M}(n,\mathbb{R})$ or $A_{v}={\rm M}(n,\mathbb{C})$. The condition
$v\mid\infty$ being empty in the case of function fields. On the other hand,
finite places of $F$ are in correspondence with nonzero prime ideals of the
ring of integers $\mathcal{O}_{F}$:

$\operatorname{\mathfrak{p}}\ \longleftrightarrow\ v.$

At every finite place $v$ of $F$, we write $O_{v}$ for the maximal compact
open subgroup ${\rm M}(n,\mathcal{O}_{v})$. We recall that the matrix ring of
adèles is a restricted direct product

$A={\rm M}(n,\mathbb{A}_{F})=\prod{}^{\prime}(A_{v}:O_{v}).$

Let $S$ be a finite set of places containing all $v\mid\infty$, and let

$A^{S}=\prod_{v\in S}A_{v}\times\prod_{v\notin S}O_{v}\subset A.$

Given an element $a\in A$, there is an $S$ as above such that $a\in A^{S}$,
and we write

$a=(a_{v})=a_{S}\cdot a^{S},$

where $a_{S}$ has coordinates $a_{v}\in A_{v}$ at every place $v\in S$ and is
$1$ for $v\notin S$; and, $a^{S}$ has $1$ for coordinate at every $v\in S$ and
$a_{v}\in O_{v}$ at every $v\notin S$.

######  Theorem 3.2.1.

The ring of matrices $A={\rm M}(n,\mathbb{A}_{F})$ is $*$-Euclidean.

###### Proof.

With notation as above, let $S$ be a finite set of places of $k$ such that

$a,c\in A^{S}.$

For $a_{S},c_{S}\in\prod_{v\in S}A_{v}\hookrightarrow A$, we can go place by
place where the local result, included in Theorem 3.1.1, is known for each of
the fields $F_{v}$ by Soto-Andrade [8], and there are only finitely many
places $v\in S$. We thus have the $*$-Euclidean property with
$s_{S}\in\prod_{v\in S}A_{v}^{\text{sym}}$ and $r_{S}\in\prod_{v\in
S}A_{v}^{\times}$.

Now, at places $v\notin S$, Theorem 3.1.1 gives the $*$-Euclidean property,
where the decomposition length is $1$. Thus we can solve for the equation

$a_{v}=s_{v}c_{v}+r_{v},$

with $s_{v}\in O_{v}^{\text{sym}}$ and $r_{v}\in O_{v}^{\times}$. In this way,
we obtain $s^{S}\in\prod_{v\notin S}O_{v}\hookrightarrow A$ and
$r^{S}\in\prod_{v\notin S}O_{v}^{\times}\hookrightarrow A$.

Finally, by setting

$s=s_{S}\cdot s^{S}\in A\text{ and }r=r_{S}\cdot r^{S}\in A^{\times},$

we obtain the desired $*$-Euclidean property for the ring of adèles with
decomposition length $1$. ∎

###  3.3. Quadratic extensions.

We now let $E/F$ be a separable quadratic field extension of the global field
$F$, where we take the involution given by the non trivial Galois element
$\alpha\in{\rm Gal}(E/F)$. For every finite absolute value $v$ of $F$, there
are two possibilities, either $v$ remains inert with respect to $E$ or $v$ is
split.

$\mathfrak{p}$$\mathfrak{P}$$\mathfrak{P}_{1}$$\mathfrak{P}_{2}$$\mathfrak{p}$$v$$w$$w_{1}$$w_{2}$$v$
(3.3)

In one case, $\mathfrak{P}=\mathfrak{p}\mathcal{O}_{E}$ is a prime ideal of
$\mathcal{O}_{E}$ and we have corresponding prime ideals
$\operatorname{\mathfrak{p}}_{v}$ of $\mathcal{O}_{v}=\mathcal{O}_{F_{v}}$ and
$\mathfrak{P}_{w}$ of $\mathcal{O}_{w}=\mathcal{O}_{E_{w}}$, where we say
there is one place $w$ above $v$, written $w\mid v$. In the other case
$\mathfrak{P}_{1}\mathfrak{P}_{2}=\operatorname{\mathfrak{p}}\mathcal{O}_{E}$,
and we obtain two places $w_{1},w_{2}\mid v$. Now, every infinite place $v$ of
$F$, written $v\mid\infty$, leads towards two possibilities: one place $w\mid
v$, when we must have $E_{w}/F_{v}=\mathbb{C}/\mathbb{R}$; or two places
$w_{1}$, $w_{2}\mid v$, when
$E\otimes_{F}F_{v}\simeq\mathbb{R}\times\mathbb{R}$ or
$E\otimes_{F}F_{v}\simeq\mathbb{C}\times\mathbb{C}$.

For every valuation $v$ of $F$, finite or infinite, we let

$E_{v}=E\otimes_{F}F_{v}\simeq\prod_{w\mid v}E_{w}.$

We thus have that $E_{v}/F_{v}$ is a separable quadratic $F_{v}$-algebra with
involution. When $E_{v}/F_{v}$ is a field extension, the involution for
$E_{v}$ is given by the non-trivial Galois automorphism. When $E_{v}\simeq
E_{w_{1}}\times E_{w_{2}}$, we have $E_{w_{1}}\simeq E_{w_{2}}\simeq F_{v}$,
hence we fix $E_{v}$ to be the $F_{v}$-algebra $E_{v}=F_{v}\times F_{v}$ with
the flip involution. By the $\check{\rm C}$ebotarev density theorem, each case
of one or two places of $E$ above one for $F$ happens with density $1/2$. In
the former case $v$ is inert, while in the latter $v$ is split.

We thus endow $\mathbb{A}_{E}$ with the involution $*$ obtained from the
involution of the $F_{v}$-algebra $E_{v}$ at every place $v$ of $F$. The
involution extends to the adèlic ring of matrices $A={\rm
M}(n,\mathbb{A}_{E})$, as in (1.1), giving
$(A,*)\in\mathcal{A}_{\mathbb{A}_{F}}$.

Using $w$ to denote places of $E$ and $v$ for places of $F$, and writing
$O_{w}={\rm M}(n,\mathcal{O}_{w})$ for finite $w$, the ring of matrices over
the adèles of $E$ is the restricted direct product as before

$A={\rm M}(n,\mathbb{A}_{E})=\prod_{w}{}^{\prime}(A_{w}:O_{w}).$

However, in order to incorporate the involution, we group the places of $E$
according to the places of $F$, by setting for finite places

$R_{v}=\prod_{w\mid v}\mathcal{O}_{w},\quad K_{v}={\rm M}(n,R_{v})\subset
G_{v}={\rm M}(n,E_{v}).$

Then, we can rearrange the restricted direct product to obtain

$A=\prod_{v}{}^{\prime}(G_{v}:K_{v}).$

######  Theorem 3.3.1.

Let $E/F$ be a quadratic extension of global fields. The ring of matrices
$A={\rm M}(n,\mathbb{A}_{E})$ is $*$-Euclidean of decomposition length 1.

###### Proof.

Let $a$, $c\in A$ be such that $Aa+Ac=A$ and $a^{*}c=c^{*}a$. There exists a
finite set of places $S$ of $F$, which includes all $v\mid\infty$, such that

$a,c\in G^{S}=\prod_{v\in S}G_{v}\times\prod_{v\notin S}K_{v}\subset A.$

We write

$a=a_{S}\cdot a^{S},\quad c=c_{S}\cdot c^{S},$

similar to what we did before, however, we are now grouping the places $w$ of
$E$ that lie above each place $v$ of $F$. With these observations, we then
follow the same argument used to prove Theorem 3.2.1 in order to prove the
result. ∎

###  3.4. Quaternions.

We work over a local or global field, where the basic theory is expounded in
[10] for central simple algebras in a manner that is independent of the
characteristic, and is detailed in [9] for quaternion algebras. We do,
however, encounter differences with regards to the $*$-Euclidean property
depending on the characteristic being $2$ or not, see Theorem 3.4.3.

Given a local or a global field $F$, we let $(\mathbb{H},*)\in\mathcal{A}_{F}$
be a quaternion algebra over $F$. Up to isomorphism, there are two options for
$\mathbb{H}$: either it is a division ring over $F$ or it is the ring of
matrices ${\rm M}(2,F)$. Both options are possible except when
$F=\operatorname{\mathbb{C}}$, where the only quaternion algebra is
$\mathbb{H}={\rm M}(2,\operatorname{\mathbb{C}})$. The case of a matrix
algebra over $F$ in general is called the split quaternion algebra, where we
take the involution to be

$h^{*}=Jh^{t}J^{-1},\ h\in\mathbb{H}={\rm M}(2,F).$ (3.4)

The division ring case is called the non-split quaternion algebra over $F$.

If $F$ is a non-Archimedean local field, we denote its ring of integers by
$\mathcal{O}$, and we denote by $(\mathcal{Q},*)\in\mathcal{A}_{\mathcal{O}}$
the quaternionic ring of $\mathbb{H}$. The ring $\mathcal{Q}$ is ${\rm
M}(2,\mathcal{O})$ with the involution given by (3.4) when $\mathbb{H}$ is
split, and it is a non-commutative local ring with involution when
$\mathbb{H}$ is a division ring. In fact, in the non-split case the ring
$\mathcal{Q}$ is locally profinite much like the $\mathfrak{p}$-adic integers
$\mathcal{O}$.

If $F$ is a global field, at every place $v$ of $F$ we let

$\mathbb{H}_{v}=\mathbb{H}\otimes_{F}F_{v},$

where indeed, each $\mathbb{H}_{v}$ is an $F_{v}$-quaternion algebra. If $v$
is a finite place of the global field $F$, we then write $F_{v}$ for the
resulting non-Archimedean local field with ring of integers $\mathcal{O}_{v}$;
furthermore, we denote by $\mathcal{Q}_{v}$ the quaternionic ring of
$\mathbb{H}_{v}$. The ring $\mathbb{H}_{v}$ is split at almost every place,
Chapter XI of [10].

Writing $\mathbb{A}_{F}$ for the ring of adèles of a global field $F$, we have
the adèlic quaternions

$\mathbb{A}_{\mathbb{H}}=\mathbb{H}\otimes_{F}\mathbb{A}_{F}.$

They can equivalently be seen as a restricted direct product

$\mathbb{A}_{\mathbb{H}}=\prod_{v}{}^{\prime}(\mathbb{H}_{v}:\mathcal{Q}_{v}),$

We have that $(\mathbb{A}_{\mathbb{H}},*)\in\mathcal{A}_{\mathbb{A}_{F}}$ with
the involution obtained from that of $\mathbb{H}_{v}$ at every place, i.e.,

$a^{*}=(a_{v}^{*}),\quad a=(a_{v})\in\mathbb{A}_{\mathbb{H}}.$

Before continuing to inspect $*$-Euclideanity for these rings, we record two
lemmas that arise in the local setting and already mark a difference between
working in characteristic $2$ or not. For this, we extend the involution given
by (3.4) to ${\rm M}(2,R)$ over any ring with identity $R$. However, $a^{*}a$
does not define a quaternion norm for general $R$ as in the commutative case.

######  Lemma 3.4.1.

Let $R$ be such that $2$ is a regular element and let $Q={\rm M}(2,R)$, so
that $(Q,*)\in\mathcal{A}_{R}$ with the involution obtained from (3.4). Then
$Q$ is not $*$-Euclidean.

###### Proof.

Because of the hypothesis on $R$, we have

$Q^{\rm sym}=\left\\{\alpha I_{2}\mid\alpha\in R\right\\}.$

The elements

$a=\left(\begin{array}[]{rr}1&0\\\ 1&0\end{array}\right),\
c=\left(\begin{array}[]{rr}0&1\\\ 0&1\end{array}\right)\in Q,$

are such that

$Qa+Qc=Q\text{ and }a^{*}c=c^{*}a=0.$

However, $a$ and $c$ cannot satisfy the $*$-Euclidean property. ∎

When the underlying ring is a division algebra $D$, we recall that the
Dieudonné determinant on ${\rm M}(2,D)$ is given by

$\det\left(\begin{array}[]{rr}\alpha&\beta\\\
\gamma&\delta\end{array}\right)=\left\\{\begin{array}[]{cl}\alpha\delta&\text{if
}\gamma=0\\\ \alpha\gamma\delta\gamma^{-1}-\gamma\beta&\text{if }\gamma\neq
0\end{array}\right..$

######  Lemma 3.4.2.

Let $D$ be a division ring of characteristic $2$ and let $H={\rm M}(2,D)$, so
that $(H,*)\in\mathcal{A}_{D}$ with the involution obtained from (3.4). If
$a$, $c\in H$ are such that

$Ha+Hc=H,$

then there exists an $s\in H^{\rm sym}$ such that

$r=a+sc\in H^{\times}.$

###### Proof.

If $a$ is invertible or zero, the Lemma is immediate. Hence, we assume that
${\rm rank}(a)=1$. For any unit $u_{0}\in H^{\times}$ and $s\in H^{\rm sym}$,
we observe that

$r=a+sc\in H^{\times}\ \Longleftrightarrow\ ru_{0}=au_{0}+scu_{0}\in
Q^{\times}.$

Thus, after taking $u_{0}$ to be a suitable product of elementary matrices, we
can assume $a$ is of the form

$a=\left(\begin{array}[]{rr}x&0\\\ y&0\end{array}\right).$ (3.5)

Also, we must have ${\rm rank}(c)=1$ or $2$.

Assume $c$ is invertible. When $D=F$ is a field, this case is easy because
$c^{*}c\in F$ is the quaternion norm. For $D$ in general, when $c$ is
invertible, one can take $s\in H^{\rm sym}$ to be of the form

$\left(\begin{array}[]{rr}\alpha&\beta\\\ 0&\alpha\end{array}\right),\
\left(\begin{array}[]{rr}\alpha&0\\\ \gamma&\alpha\end{array}\right)\text{ or
}\left(\begin{array}[]{rr}0&\alpha\\\ \alpha&0\end{array}\right),$ (3.6)

to get $sc$ in triangular or anti-triangular form with non-zero diagonal or
anti-diagonal entries, respectively. Depending on the form of $a$, one can
choose $\alpha$, $\beta$ and $\gamma$ so that

$r=a+sc\in H^{\times}.$

Note that a careful consideration of the three cases for $a$ of the form
$\eqref{eq:a:r1}$ is needed when $D=\operatorname{\mathbb{F}}_{2}$.

Now, suppose ${\rm rank}(c)=1$, then one can choose an appropriate $s\in
A^{\times}$ of one of the forms in (3.6), so that

$sc=\left(\begin{array}[]{rr}x&y\\\ 0&0\end{array}\right)\text{ or
}\left(\begin{array}[]{rr}0&0\\\ x&y\end{array}\right).$

We can take one or the other, depending on the form of $a$, to obtain

$a=sc+r,\ s\in H^{\rm sym},\ r\in H^{\times}.$

∎

We now study the ring $A$ of $n\times n$ matrices with entries in
$\mathbb{A}_{\mathbb{H}}$. The proof of the next theorem gives another example
of how a global question requires us to inspect what is happening locally in
detail, where we prove a pair of lemmas along the way. Note that the
involution on $A$, globally or locally, is obtained by combining the
involution on the quaternion ring with the involution given by (1.1).

######  Theorem 3.4.3.

Let $\mathbb{H}$ be a quaternion algebra over a global field $F$, and let
$A={\rm M}(n,\mathbb{A}_{\mathbb{H}})$ so that
$(A,*)\in\mathcal{A}_{\mathbb{A}_{\mathbb{H}}}$ with the involution induced
from that of $\mathbb{A}_{\mathbb{H}}$. Then

  * (i)

$A$ is $*$-Euclidean of decomposition length $1$ when ${\rm char}(F)=2$.

  * (ii)

$A$ is not $*$-Euclidean when ${\rm char}(F)\neq 2$.

Let $a=(a_{v})$, $c=(c_{v})\in A$ be such that

$Aa+Ac=A\text{ and }a^{*}c=c^{*}a.$

At finite non-split places, $\mathcal{Q}_{v}$ is a local ring with involution,
where we know the result locally holds. Now, the proof of Theorem 3.4.3
follows the general outline that is present in the proofs of Theorems 3.2.1
and 3.3.1. However, care needs to be taken when inspecting ${\rm
M}(n,\mathcal{Q}_{v})$ at finite split places.

Before we continue, we have two lemmas that we write in a slightly more
general setting. Let $R$ be a ring with identity where $2=0$. In this setting,
we have that

$J=\left(\begin{array}[]{rr}\ 0&1\\\ 1&0\end{array}\right)=J^{-1},$

and we take the block diagonal matrix

$B=\left(\begin{array}[]{ccc}\ J&\cdots&0\\\ \vdots&&\vdots\\\
0&\cdots&J\end{array}\right)=B^{-1}.$

Consider the matrix ring ${\rm M}(2n,R)$, with involution given by

$a^{*}=Ba^{t}B.$ (3.7)

We first address the case when $R=F$ is a field.

######  Lemma 3.4.4.

Let $F$ be a field of characteristic $2$ and let $\mathbb{H}={\rm M}(2,F)$ be
the split quaternions. Form the matrix ring $A={\rm M}(n,\mathbb{H})$ with the
involution induced from that of $\mathbb{H}$. Then $A$ is $*$-Euclidean of
length $1$.

###### Proof.

Consider $A$ as the ring of $2n\times 2n$ matrices with entries in $F$, whose
involution is given by (3.7). We then have a non-degenerate bilinear pairing

$\left\langle\cdot\,,\cdot\right\rangle:F^{2n}\times F^{2n}\rightarrow
F,\quad\left\langle v,w\right\rangle=v^{t}Bw,$

where $v$, $w$ are seen as column vectors of $F^{2n}$. The involution
$a\mapsto a^{*}$, is precisely the adjoint for the pairing
$\left\langle\cdot\,,\cdot\right\rangle$, to which we can apply Transversality
and the coprime Lemma of Pantoja and Soto-Andrade, in particular, Proposition
3.3 of [5]. In this way, given $a$, $c\in A$, such that

$Aa+Ac=A,\quad a^{*}c=c^{*}a,$

we obtain an $s\in A^{\rm sym}$ such that

$r=a+sc\in A^{\times}.$

∎

We now require to extend the above to the case of a local ring.

######  Lemma 3.4.5.

Let $R$ be a local ring of characteristic $2$ with maximal prime ideal
$\operatorname{\mathfrak{p}}$. Let $Q={\rm M}(2,R)$, so that
$(Q,*)\in\mathcal{A}_{R}$ with the involution obtained from (3.4). Form the
matrix ring $A={\rm M}(n,Q)$ with the involution induced from that of $Q$.
Then $A$ is $*$-Euclidean of length $1$.

###### Proof.

We can identify $A$ with ${\rm M}(2n,R)$, where the involution is given by
(3.7). We next follow our general construction of § 2.2 for reduction modulo
the Jacobson radical $\operatorname{\mathcal{J}}$, $A\xrightarrow{\ \pi\
}\overline{A}=A/\mathcal{J}$, which produces a section map
$\overline{A}\xrightarrow{\ \sigma\ }A$. More specifically, equations (2.3)
and (2.4) lead towards Lemma 2.2.1 where $\pi$ and $\sigma$ preserve the
involutions on $A$ and $\overline{A}$, and the units satisfy

$a\in A^{\times}\ \Longleftrightarrow\ \pi(a)\in\overline{A}^{\times}\
\Longleftrightarrow\ a_{\sigma}\in A^{\times}.$

Lemma 3.1.2, extended to this setting, shows that if $\overline{A}$ is
$*$-Euclidean the so is $A$. After reducing mod $\operatorname{\mathfrak{p}}$,
we apply Lemma 3.4.4. ∎

We now continue the proof of Theorem 3.4.3. At every split place of $F$, we
have a non-Archimedean local field $F_{v}$ and split quaternions
$\mathbb{H}_{v}={\rm M}(2,F_{v})$. We also have a ring of integers
$\mathcal{O}_{v}$ and ring of quaternions $\mathcal{Q}_{v}={\rm
M}(2,\mathcal{O}_{v})$.

When ${\rm char}(F)=2$, thanks to Lemma 3.4.5, we now have $*$-Euclideanity
for the ring ${\rm M}(n,\mathcal{Q}_{v})$. However, when ${\rm char}(F)\neq
2$, we use Lemma 3.4.1, which gives an immediate negative answer to the case
$n=1$ at every split non-Archimedean place; a result that directly extends to
the case of $n>1$. This concludes the proof of Theorem 3.4.3.

##  4\. Examples and Bruhat generators

We begin with the Dieudonné determinant and $*$-local rings, observing a
connection between ${\rm SL}_{*}$ groups and the general linear group. We also
provide two non-trivial examples for the theory, namely, unitary groups
arising as ${\rm SL}_{*}$ groups and ${\rm SL}_{*}$ over local and adèlic
quaternions.

The $*$-Euclidean property is a strong one for a ring $A$, and leads to the
important result of Pantoja and Soto-Andrade that ${\rm SL}_{*}(2,A)$ has a
set of Bruhat generators [6]. We thus establish Bruhat generation over local
and adèlic rings, to conclude with Theorem 4.3.1.

###  4.1. ${\rm GL}(n)$ and $*$-local rings

Given a division ring $D$, the Dieudonné determinant [1], gives a criterion
for the invertibility of $n\times n$ matrices with entries in $D$:

$x\in{\rm GL}(n,D)\ \Longleftrightarrow\ x\in{\rm M}(n,D)\text{ and
}\det(x)\neq 0.$

Now, let $R$ be a $*$-local ring, with ${\rm
MSpec}(R)=\\{\operatorname{\mathfrak{p}},\operatorname{\mathfrak{p}}^{*}\\}$.
From § 2.2, we have a projection map from $R$ to
$R/\\!\operatorname{\mathfrak{p}}$ and, depending on
$\operatorname{\mathfrak{p}}=\operatorname{\mathfrak{p}}^{*}$ or
$\operatorname{\mathfrak{p}}\neq\operatorname{\mathfrak{p}}^{*}$, also a
projection from $R$ to $\,R/\\!\operatorname{\mathfrak{p}}^{*}$. This leads to
the following map of matrix rings

${\rm M}(n,R)\longrightarrow{\rm
M}(n,R/\\!\operatorname{\mathfrak{p}})\oplus{\rm
M}(n,R/\\!\operatorname{\mathfrak{p}}^{*}),$

where we write

$a\mapsto(\bar{a},\tilde{a}).$

Over the $*$-local ring $R$, the general linear group consists of invertible
$n\times n$ matrices ${\rm GL}(n,R)$, with its principal congruence subgroup

$K=\\{a\in M(n,R)\mid\pi(a)=I_{n}+\operatorname{\mathcal{J}}\\},$

where $\pi:R\rightarrow R/\mathcal{J}$ is the canonical projection map. We now
observe that invertibility is compatible with the Dieudonné determinant and
reduction mod $\operatorname{\mathcal{J}}$.

######  Lemma 4.1.1.

Let $R$ be a $*$-local ring. The following are equivalent for $a\in{\rm
M}(n,R)$:

  * (i)

$a\in{\rm GL}(n,R)$.

  * (ii)

There exists $b\in{\rm M}(n,R)$ such that $ba\in K$.

  * (iii)

$\det(\bar{a})\neq 0$ and $\det(\tilde{a})\neq 0$.

###### Proof.

This follows from Lemma 2.2.1; notice that ${\rm M}(n,R)$ is Dedekind finite.
∎

A basic and important example of a $2$-local ring with involution is
$S=R_{1}\times R_{2}$, obtained from local rings $R_{i}$ that are linked by an
anti-automorphism $\varphi:R_{1}\rightarrow R_{2}$, equipped with the
$\varphi$-flip involution

$r^{*}=(\varphi^{-1}(r_{2}),\varphi(r_{1}))\text{ for }r=(r_{1},r_{2})\in S.$

Form the matrix ring

$A={\rm M}(n,S)=A_{1}\oplus A_{2},\quad A_{i}={\rm M}(n,R_{i}),$

with the $\varphi$-flip transpose involution

$a^{*}=(\varphi^{-1}(a_{2})^{t},\varphi(a_{1})^{t})\text{ for
}a=(a_{1},a_{2})\in A,$

so that $(A,*)\in\mathcal{A}_{R}$. We recall that Proposition 5.1 of [5],
gives an isomorphism

${\rm SL}_{*}(2,A)=\\{a=(a_{1},J(\varphi(a_{1})^{-1})^{t}J^{-1})\in A\mid
a_{1}\in{\rm GL}(2,A_{1})\\}\simeq{\rm GL}(2,A_{1}).$ (4.1)

###  4.2. Unitary groups

Classically, one works over a base field $F$, where we have the unitary group
${\rm U}_{2n}(F)$ associated to a separable quadratic algebra $E$ over $F$
with involution $\alpha:E\rightarrow E$, $x\mapsto\bar{x}$, given by the non-
trivial Galois element $\alpha\in{\rm Gal}(E/F)$ if $E/F$ is a field
extension, and is obtained from the flip involution $(x,y)\mapsto(y,x)$ if
$E\simeq F\times F$. Let

$\Phi_{n}=\begin{pmatrix}\ 0&J_{n}\\\ -J_{n}&0\end{pmatrix},$

where $J_{n}$ is the $n\times n$ matrix $J_{n}=\left(\delta_{i,n-j+1}\right)$,
where $\delta_{i,j}$ denotes Kronecker’s delta function, and define the
hermitian form $h_{n}$ on the $2n$-dimensional vector space $V$ of column
vectors with entries in $E$, defined by

$h_{n}(x,y)=\bar{x}^{t}\,\Phi_{n}y,\quad x,y\in V.$

Then ${\rm U}_{2n}(F)$ is the group of isometries of $h_{n}$.

There is a more general setting for defining unitary groups by considering
$(A,*)\in\mathcal{A}_{R}$ and let $\varepsilon=\pm 1$. Then, a
$\varepsilon$-hermitian form over a free left $A$-module $V$ of finite rank is
a bi-additive map $h:V\times V\to A$ , linear in the second variable and
satisfies

$h(v,u)^{*}=\varepsilon h(u,v),\quad u,v\in V.$

The unitary group ${\rm U}(h)$ associated to $h$ consists then of all
$g\in{\rm GL}(V)$ such that

$h(gu,gv)=h(u,v),\quad u,v\in V.$

One of our examples arises by taking $A={\rm M}(n,\mathbb{A}_{E})$, where we
have the connection between unitary groups and ${\rm SL}_{*}$ groups,

${\rm U}_{2n}(\mathbb{A}_{F})={\rm SL}_{*}(2,\mathbb{A}_{E}).$

We thus may observe that, at every split place $v$ of $F$, we further have a
connection between ${\rm SL}_{*}$ groups and general linear groups. Indeed, in
this case we can fix $E_{v}=F_{v}\times F_{v}$ and take the map $\varphi$ of
(4.1) to be the identity. More precisely, ${\rm SL}_{*}(2,E_{v})$ is
isomorphic to ${\rm GL}(n,F_{v})$ via the projection map

$(g_{0},J(g_{0}^{-1})^{t}J^{-1})\mapsto g_{0}.$

###  4.3. Bruhat Generators

We have the Bruhat elements of ${\rm SL}_{*}(2,A)$, which are the natural
extension of those for ${\rm SL}_{2}(F)$ when $F$ is a field. Namely, the
matrices

$h_{a}=\left(\begin{array}[]{l}a^{*}\,\ 0\\\ 0\ \,\
a^{-1}\end{array}\\!\\!\\!\right),(a\in A^{\times}),\quad
u_{b}=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix},(b\in A^{\rm sym}),\quad\text{
and }\quad\omega=\left(\begin{array}[]{rr}0&1\\\ -1&0\end{array}\right)$

are the Bruhat elements for ${\rm SL}_{*}(2,A)$.

We observe a formal Bruhat relation, valid when one of the entries is a unit.
In particular, if

$g=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\in{\rm SL}_{*}(2,A),$

and $a\in A^{\times}$, then

$g=w^{-1}h_{a}^{-1}\,u_{-a^{*}c}\,w\,u_{a^{-1}b}.$ (4.2)

Note that if $b$, $c$ or $d$ is in $A^{\times}$, then multipliying $g$ on the
left or right by $w$, leads to a matrix in the previous situation. Notice that
each one of the elements appearing in (4.2) is indeed inside ${\rm
SL}_{*}(2,A)$, follows from the defining relations given by (1.3). For
example,

$ab^{*}=ba^{*}\ \Longrightarrow\ a^{-1}b=b^{*}(a^{*})^{-1}\in A^{\rm sym}.$

Hence $u_{a^{-1}b}\in{\rm SL}_{*}(2,A)$.

However, an element of ${\rm SL}_{*}(2,A)$ does not in general satisfy the
property that one of its entries is a unit. When $A={\rm M}(n,R)$, with $R$ a
$*$-local ring, this only happens when $n=1$, because one can reduce the
$*$-determinant relation mod $\mathcal{J}$:

$\bar{a}\bar{d}^{*}-\bar{b}\bar{c}^{*}=\bar{1},$

forcing $\bar{a}\bar{d}^{*}$ or $\bar{b}\bar{c}^{*}$ to be a unit. Hence
$ad^{*}$ or $bc^{*}$ is a unit, and two of the entries are thus units in this
case. For $n>1$, we can arrange for a product

$u_{b_{1}}w\,u_{b_{2}}w^{-1}u_{b_{3}}$

to have all of its entries non-invertible, after suitable choices for
$b_{1},b_{2},b_{3}\in A^{\rm sym}$.

Now, a very interesting problem is to determine when the matrices $h_{t}$,
$u_{s}$, $w$ generate the group ${\rm SL}_{*}(2,A)$. In this sense, there
exists an important relation between $*$-Euclidianity and Bruhat elements.
More precisely, Pantoja and Soto-Andrade proved that if $A$ is $*$-Euclidean,
then $h_{a}$, $u_{b}$ and $w$, with $a\in A^{\times}$ and $b\in A^{\rm sym}$,
generate ${\rm SL}_{*}(2,A)$ [6], § 5 Proposition 4.

Thus, we can apply these observations to the four different settings of
Theorems 3.1.1, 3.2.1 and 3.3.1, from which the next result is a corollary; we
elaborate on ${\rm SL}_{*}$ groups over the quaternions in the following
subsection.

######  Theorem 4.3.1.

Consider involutive rings $(R,*)$ in the following cases:

  * (i)

$R$ is a $*$-local ring; or

  * (ii)

$R$ is the ring of adèles $\mathbb{A}_{F}$ of a global field $F$, with trivial
involution; or

  * (iii)

$R=\mathbb{A}_{E}$, where $E/F$ is a separable quadratic field extension with
the involution dictated by the non-trivial Galois automorphism; or

  * (iv)

$R=\mathbb{A}_{\mathbb{H}}$, where $\mathbb{H}$ is a quaternion algebra over a
global field $F$, ${\rm char}(F)=2$.

Let $A={\rm M}(n,R)$, so that $(A,*)\in\mathcal{A}_{R}$ with the involution
induced from that of $R$. Then ${\rm SL}_{*}(2,A)$ is generated by the Bruhat
elements $h_{a}$, $u_{b}$ and $w$.

When the underlying ring $R$ is local, it was proved in [3], where many
interesting properties are explored like the Weil representations of these
groups. Part (i) of the previous theorem is an extension to the case of
$*$-local rings and provides and alternate proof when $R$ is local.

The unitary groups, as in § 4.2, give an example of case (iii), where the
proof of the theorem follows from the results of § 3. And finally, the
quaternions provide an interesting non-conmutative example of Theorem 4.3.1
for case (i) in general, and case (iv) when ${\rm char}(F)=2$. We now inspect
${\rm SL}_{*}$ groups over the quaternions more closely, explore some
interesting properties and complete the proof of the theorem.

###  4.4. Quaternions

Let $\mathbb{H}$ be a quaternion division algebra over a local or a global
field $F$. We have three related, yet distinct groups in this setting, namely

${\rm SL}(2,F),\ {\rm SL}(2,\mathbb{H})\text{ and }{\rm
SL}_{*}(2,\mathbb{H}).$

The first two are the special linear groups of $F$ and $\mathbb{H}$,
respectively, where ${\rm SL}(2,\mathbb{H})$ is given by the kernel of the
Dieudonné determinant. The third is the ${\rm SL}_{*}$ group of Pantoja and
Soto-Andrade over the division algebra $\mathbb{H}$ endowed with the
quaternionic involution.

Case of a $\operatorname{\mathfrak{p}}$-adic field. Let $F$ be a non-
Archimedean local field $F$, with ring of integers $\mathcal{O}$ and maximal
ideal $\operatorname{\mathfrak{p}}$. We have a finite residue field
$k_{F}=\mathcal{O}/\operatorname{\mathfrak{p}}$.

In this setting, $\mathbb{H}$ is the unique (up to isomorphism) $4$
dimensional central division algebra over $F$. As in § 3.4, we let
$\mathcal{Q}$ denote the ring of integers of $\mathbb{H}$, and $\mathfrak{q}$
its maximal ideal. We obtain a finite quotient field
$k_{\mathbb{H}}=\mathcal{Q}/\mathfrak{q}$.

Consider the ring of matrices $A={\rm M}(n,\mathcal{Q})$ with the involution
induced from $(\mathcal{Q},*)\in\mathcal{A}_{\mathcal{O}}$. Then $A$ is
$*$-Euclidean by Theorem 3.1.1. Hence, by [6] we know that ${\rm SL}_{*}(2,A)$
is generated by the Bruhat elements of § 4.3 and Theorem 4.3.1 is thus valid
for the quaternionic ring $\mathcal{Q}$. In particular, ${\rm
SL}_{*}(2,\mathbb{H})$ is generated by the Bruhat elements of § 4.3.

The special linear groups ${\rm SL}(2,F)$ and ${\rm SL}(2,\mathbb{H})$ are
generated by its Bruhat elements, see for example [1]. We further observe that
we have group homomorphisms obtained from reduction modulo the corresponding
maximal ideal, namely

$\displaystyle\varphi_{F}:{\rm SL}(2,\mathcal{O})\rightarrow{\rm SL}(2,k_{F})$

and

$\displaystyle\varphi_{\mathbb{H}}:{\rm SL}(2,\mathcal{Q})\rightarrow{\rm
SL}(2,k_{\mathbb{H}}).$

The Borel subgroups, $B(k_{F})$ of ${\rm SL}(2,k_{F})$ and $B(k_{\mathbb{H}})$
of ${\rm SL}(2,k_{\mathbb{H}})$, consist of the respective upper triangular
matrices. These lead towards the Iwahori subgroups, defined as the pre-images
of the above maps:

$\mathcal{I}_{F}=\varphi_{F}^{-1}(B(k_{F}))\text{ and
}\mathcal{I}_{\mathbb{H}}=\varphi_{\mathbb{H}}^{-1}(B(k_{\mathbb{H}})).$

Now, consider the subgroups of diagonal elements,

$D_{F}=\left\\{h_{a}\mid a\in F^{\times}\right\\}\text{ and
}D_{\mathbb{H}}=\left\\{h_{a}\mid a\in\mathbb{H}^{\times}\right\\}.$

It then follows from the definitions that we have the following relation

${\rm SL}_{*}(2,\mathbb{H})=D_{\mathbb{H}}\cdot{\rm SL}(2,F).$ (4.3)

It is interesting to note a result of Ihara, which gives an expression of the
special linear group as an amalgamated product, valid over $F$,

${\rm SL}(2,F)={\rm SL}(2,\mathcal{O})*_{\mathcal{I}_{F}}{\rm
SL}(2,\mathcal{O}),$ (4.4)

and over $\mathbb{H}$,

${\rm SL}(2,\mathbb{H})={\rm
SL}(2,\mathcal{Q})*_{\mathcal{I}_{\mathbb{H}}}{\rm SL}(2,\mathcal{Q}).$ (4.5)

Where a proof of equations (4.4) and (4.5) can be found in [7]. Also note that
equation (4.3) can also be written as an amalgamaded product,

${\rm SL}_{*}(2,\mathbb{H})=D_{\mathbb{H}}*_{D_{F}}{\rm SL}(2,F).$ (4.6)

Case of a global field. Let $F$ be either a number field or a global function
field of characteristic $p$. At each place $v$ of $F$, we obtain a quaternion
algebra $\mathbb{H}_{v}$ over the local field $F_{v}$; $\mathbb{H}_{v}$ may be
split or non-split. At a non-Archimedean place $v$ of $F$, we have the ring of
integers $\mathcal{O}_{v}$ of $F_{v}$, in addition to the quaternionic ring
$\mathcal{Q}_{v}$ of $\mathbb{H}_{v}$. At these places, we let $Q_{v}={\rm
M}(n,\mathcal{Q}_{v})\in\mathcal{A}_{\mathcal{Q}_{v}}$.

Let $A={\rm M}(n,\mathbb{A}_{\mathbb{H}})$, and at every place $v$ of $F$ let
$A_{v}={\rm M}(n,\mathbb{H}_{v})$. The matrix ring $A$ (resp. $A_{v}$ at each
$v$) is equipped with the involution induced from that of
$\mathbb{A}_{\mathbb{H}}$ (resp. $\mathbb{H}_{v}$). A straightforward
inspection tells us that we have Bruhat generation for the group ${\rm
SL}_{*}(2,A)$ if and only if we have Bruhat generation for ${\rm
SL}_{*}(2,A_{v})$ at every Archimedean place and for ${\rm SL}_{*}(2,Q_{v})$
at every non-Archimedean place.

The quaternionic ring $\mathbb{H}$ over the global field $F$ forms a central
division algebra over $F$. There is at least one place $v$ of $F$ where
$\mathbb{H}_{v}$ is a division algebra, hence non-split; and, $\mathbb{H}_{v}$
can be non-split at only finitely many $v$ [10].

Fix a split place $v$ of $F$, where we have that $\mathbb{H}_{v}$ and
$\mathcal{Q}_{v}$ are matrix algebras over $F_{v}$ and $\mathcal{O}_{v}$,
respectively. The involutions on $\mathbb{H}_{v}$ and $\mathcal{Q}_{v}$ come
from (3.4). These, in turn, induce the involutions on $A_{v}$ and $Q_{v}$,
respectively. When ${\rm char}(F)=2$, in Lemma 3.4.5 we showed that $Q_{v}$ is
$*$-Euclidean of length $1$. And, Lemma 3.4.4 says that $A_{v}$ is also
$*$-Euclidean of length $1$. By [6], § 5 Proposition 4, ${\rm
SL}_{*}(2,Q_{v})$ is in this case generated by its Bruhat elements at every
non-Archimedean place and so is ${\rm SL}_{*}(2,A_{v})$.

However, continuing with the case of a split place $v$, if ${\rm char}(F)\neq
2$, then it follows directly from Lemma 3.4.1 that $Q_{v}$ is not
$*$-Euclidean. Furthermore, from [3] Example 13.3 for a finite field, we
deduce that ${\rm SL}_{*}(2,Q_{v})$ is not generated by its Bruhat elements.

At a non-split place, we have either a division algebra $\mathbb{H}_{v}$ or a
local ring with involution $\mathcal{Q}_{v}$. Hence, we know that ${\rm
SL}_{*}(2,A_{v})$ and ${\rm SL}_{*}(2,Q_{v})$ are generated by its Bruhat
elements at every non-split place.

Finally, we observe that only in the case of a global function field $F$ with
${\rm char}(F)=2$, do we have that $A_{v}$ is generated by its Bruhat elements
at every place and so is $Q_{v}$ at all non-Archimedean places. This concludes
the proof of Theorem 4.3.1 in the remaining case. In particular, we have the
following:

> _Let $\mathbb{H}$ be a quaternion algebra over a global field $F$ and let
> $\mathbb{A}_{\mathbb{H}}$ be its ring of adèles. Let $A={\rm
> M}(n,\mathbb{A}_{\mathbb{H}})$ be endowed with the involution induced from
> that of $\mathbb{A}_{\mathbb{H}}$. Then ${\rm SL}_{*}(2,A)$ is generated by
> its Bruhat elements if and only if ${\rm char}(F)=2$._

## References

  * [1] E. Artin, _Geometric Algebra_ , Interscience Tracts in Pure and Applied Mathematics 3, New York, 1957.
  * [2] H. Bass, _$K$ -theory and stable algebra_, Pub. Math. I.H.É.S. 22 (1964), 5-60.
  * [3] J. Cruickshank, L. Gutiérrez Frez and F. Szechtman, _Weil representations via abstract data and Heisenberg groups: A comparison_ , J. Algebra 547 (2020), 129-161.
  * [4] T. Y. Lam, _A first course in noncommutative rings_ , $2^{\rm nd}$ ed., Graduate Texts in Mathematics 131, Springer-Verlag, New York, 2001.
  * [5] J. Pantoja and J. Soto-Andrade, _A Bruhat decomposition of the group ${\rm SL}_{*}(2,A)$_, J. Algebra 262 (2003), 401-412.
  * [6] J. Pantoja and J. Soto-Andrade, _Bruhat presentations for $*$-classical groups_, Comm. Algebra 37 (2009), 4170-4191.
  * [7] J.-P. Serre, _Trees_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1980.
  * [8] J. Soto-Andrade, _Représentations de certains groupes symplectiques finis_ , Mémoires S.M.F. 55-56 (1978), 334 p.
  * [9] M.-F. Vignéras, _Arithmétique des algèbres de quaternions_ , Lecture Notes in Mathematics 800, Springer-Verlag, Berlin, 1980.
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Luis Gutiérrez Frez, Instituto de Ciencias Físicas y Matemáticas, Universidad
Austral de Chile, Campus Isla Teja SN, Valdivia, Chile.

_E-mail address:_<EMAIL_ADDRESS>
Luis Lomelí, Instituto de Matemáticas, Pontificia Universidad Católica de
Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile.

_E-mail address:_<EMAIL_ADDRESS>
José Pantoja, Instituto de Matemáticas, Pontificia Universidad Católica de
Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile.

_E-mail address:_<EMAIL_ADDRESS>