\documentclass{article}

\usepackage{amsmath,amsthm,amscd,amsxtra,amsfonts,amssymb,latexsym}

\begin{document}

\newcommand{\TeXButton}[2]{#2}
\newcommand{\stackunder}[2]{\underset{#1}{#2}}
\newcommand{\NEG}{\not}
\newcommand{\QATOP}[2]{#1 \atop #2}

\setcounter{secnumdepth}{3}

\title{Semigroup extensions of isometry groups of flat spacetimes compactified over
lightlike lattices}

\author{Hanno Hammer \\ Departement of Applied Mathematics and Theoretical Physics \\ Silver Street, Cambridge, UK}

\maketitle

\begin{abstract}
We examine some peculiarities of the subset of lattice preserving elements
in a pseudo-Euclidean group, when the lattice under consideration contains a
lightlike vector, or more generally, when the restriction of a
pseudo-Euclidean metric to the real linear enveloppe of the lattice is not
definite. For the case of a Lorentzian metric, it is shown in detail that
the isometry group of the spacetime compactified over such a lattice admits
a natural extension to a semigroup. We explain why such an extension is not
available for spacelike lattices. Furthermore, we argue that for any
Lagrangian defined on such a lightlike compactified spacetime, the elements
of the semigroup relate sectors of the theory belonging to different
discrete compactification radii, and hence connect different superselection
sectors of the theory. This mapping occurs as a one-way process owing to the
non-invertibility of the semigroup elements on the lattice. These structures
might therefore be of relevance to matrix theory.
\end{abstract}

\section{Introduction}

In this work we examine the structure of subsets of maps, in particular of
elements of the isometry group $E_t^n$ of a flat pseudo-Euclidean space $%
\TeXButton{R}{\mathbb{R}}_t^n$, that preserve the points of a lattice $lat$,
whose associated real vector space $\left[ lat\right] $, called the $%
\TeXButton{R}{\mathbb{R}}$-linear enveloppe of the lattice, is lightlike. In
this case, the restriction of the metric $\eta $ with signature $\left(
-t,+s\right) $ to $\left[ lat\right] $ is no longer definite. This gives
rise to the possibility of having lattice-preserving transformations in the
overall pseudo-Euclidean group $E_t^n$ that are injective, but no longer
surjective on the lattice; in other words, their inverses do not preserve $%
lat$. The set of all these transformations therefore will no longer be a
group, but only a semigroup. Since it is precisely the lattice preserving
transformations that descend to the quotient of $\TeXButton{R}{\mathbb{R}}%
_t^n$ over the discrete group of primitive lattice translations $\Gamma $,
i.e. to the ''compactified'' spacetime $\TeXButton{R}{\mathbb{R}}_t^n/\Gamma
=\TeXButton{R}{\mathbb{R}}^{n-m}\times T^m$, these semigroup elements
constitute an extension of the isometry group $I\left( \TeXButton{R}
{\mathbb{R}}_t^n/\Gamma \right) $, which act non-invertibly on the
compactified space. We present in detail the case of the compactification of
a  Lorentzian spacetime over a lightlike lattice $\Gamma $, where it is
shown that the non-invertible elements wind the lightlike circle $k$ times
around itself. We argue that this should map the different sectors of a
Lagrange theory on $\TeXButton{R}{\mathbb{R}}_t^n/\Gamma $, as labelled by
the lightlike compactification radius, in a one-way process into each other.
In the case under consideration, this map will be accomplished by finite
discrete transformations generated by the ''mass'' generator of the
centrally extended Galilei group; since it is known that the eigenvalues of
this generator label different superselection sectors of a theory, we argue
that our semigroup transformations connect different superselection sectors
of any Lagrange theory on a lightlike compactified spacetime.

The plan of the paper is as follows: In section \ref{Sc.1} we provide some
background on orbit spaces and the associated fibre preserving sets. In
section \ref{Sc.2} we introduce our notation conventions. In section \ref
{Sc.3} and \ref{Sc.4} we introduce our concepts of lattice preserving
transformations, and examine some of their structure. In particular, we show
in a theorem why no semigroup extensions are available in a Euclidean
background space, or more generally, for a lattice whose $\TeXButton{R}
{\mathbb{R}}$-linear enveloppe $\left[ lat\right] $ is spacelike. In section 
\ref{Sc.5} we construct the sets normalizing a lightlike lattice in a
Minkowski spacetime. In section \ref{Sc.6} we show how the semigroup
transformations act on the lightlike circle of the spacetime, and how they
relate theories belonging to different compactification radii.

\section{Orbit spaces and normalizing sets \label{Sc.1}}

Assume that a group $G$ has a left action on a topological space $X$ such
that the map $G\times X\ni \left( g,x\right) \mapsto gx$ is a homeomorphism.
When a discrete subgroup $\Gamma \subset G$ acts properly discontinuously
and freely on $X$, then the natural projection $p:X\rightarrow X/\Gamma $ of 
$X$ onto the space of orbits, $X/\Gamma $, can be made into a covering map,
and $X$ becomes a covering space of $X/\Gamma $ (e.g. \cite
{Fulton,Jhnich,Massey}) . More specially, if $X=M$ is a connected
pseudo-Riemannian manifold with a metric $\eta $, and $G=I\left( M\right) $
is the group of isometries of $M$, so that $\Gamma $ is a discrete subgroup
of isometries acting on $M$, then there is a unique way to make the quotient 
$M/\Gamma $ a pseudo-Riemannian manifold (e.g. \cite{Wolf,ONeill}); in this
construction one stipulates that the projection $p$ be a local isometry,
which determines the metric on $M/\Gamma $. In such a case, we speak of $%
p:M\rightarrow M/\Gamma $ as a pseudo-Riemannian covering.

In both cases, the quotient $p:X\rightarrow $ $X/\Gamma $ can be regarded as
a principal fibre bundle with bundle space $X$, base $X/\Gamma $, and $%
\Gamma $ as structure group, the fibre over $m\in X/\Gamma $ being the orbit
of any element $x\in p^{-1}\left( m\right) $ under $\Gamma $, i.e. $%
p^{-1}\left( m\right) =\Gamma x=\left\{ \gamma x\mid \gamma \in \Gamma
\right\} $. If $g\in G$ induces the homeomorphism $x\mapsto gx$ of $X$ (or
an isometry of $M$), then $g$ gives rise to a well-defined map $%
g_{\#}:X/\Gamma \rightarrow X/\Gamma $ {\bf only} when $g$ preserves all
fibres, i.e. when $g\left( \Gamma x\right) \subset \Gamma \left( gx\right) $
for all $x\in X$. This is equivalent to saying that $g\Gamma g^{-1}\subset
\Gamma $. If this relation is replaced by the stronger condition $g\Gamma
g^{-1}=\Gamma $, then $g$ is an element of the {\it normalizer }$N\left(
\Gamma \right) $ of $\Gamma $ in $G$, where 
\begin{equation}
\label{pp3fo1}N\left( \Gamma \right) =\left\{ g\in G\mid g\Gamma
g^{-1}=\Gamma \right\} \quad . 
\end{equation}
The normalizer is a group by construction. It contains all fibre preserving
elements $g$ of $G$ such that $g^{-1}$ is fibre preserving as well. In
particular, it contains the group $\Gamma $, which acts trivially on the
quotient space; this means, that for any $\gamma \in \Gamma $, the induced
map $\gamma _{\#}:X/\Gamma \rightarrow X/\Gamma $ is the identity on $%
X/\Gamma $, $\gamma _{\#}=\left. id\right| _{X/\Gamma }$. This follows,
since the action of $\gamma _{\#}$ on the orbit $\Gamma x$, say, is defined
to be $\gamma _{\#}\left( \Gamma x\right) =\Gamma \left( \gamma x\right)
=\Gamma x$, where the last equality holds, since $\Gamma $ is a group.

In this work we are interested in relaxing the equality in the condition
defining $N\left( \Gamma \right) $; to this end we introduce what we call
the {\it extended normalizer}, denoted by $eN\left( \Gamma \right) $,
through 
\begin{equation}
\label{pp3fo2}eN\left( \Gamma \right) :=\left\{ g\in G\mid g\Gamma
g^{-1}\subset \Gamma \right\} \quad . 
\end{equation}
The elements $g\in G$ which give rise to well-defined maps $g_{\#}$ on $%
X/\Gamma $ are therefore precisely the elements of the extended normalizer $%
eN\left( \Gamma \right) $, as we have seen in the discussion above. Such
elements $g$ are said to {\it descend} to the quotient space $X/G$. Hence $%
eN\left( \Gamma \right) $ contains all homeomorphisms of $X$ (isometries of $%
M$) that descend to the quotient space $X/\Gamma $ ($M/\Gamma $); the
normalizer $N\left( \Gamma \right) $, on the other hand, contains all those $%
g$ for which $g^{-1}$ descends to the quotient as well. Thus, $N\left(
\Gamma \right) $ is the group of all $g$ which descend to {\bf invertible}
maps $g_{\#}$ (homeomorphisms; isometries) on the quotient space. In the
case of a semi-Riemannian manifold $M$, for which the group $G$ is the
isometry group $I\left( M\right) $, the normalizer $N\left( \Gamma \right) $
therefore contains all isometries of the quotient space, the only point
being that the action of $N\left( \Gamma \right) $ is not effective, since $%
\Gamma \subset N\left( \Gamma \right) $ acts trivially on $M/\Gamma $.
However, $\Gamma $ is a normal subgroup of $N\left( \Gamma \right) $, so
that the quotient $N\left( \Gamma \right) /\Gamma $ is a group again, which
is now seen to act effectively on $M/\Gamma $, and the isometries of $%
M/\Gamma $ which descend from isometries of $M$ are in a 1--1 relation to
elements of this group. Thus, denoting the isometry group of the quotient
space $M/\Gamma $ as $I\left( M/\Gamma \right) $, we have the well-known
result that 
\begin{equation}
\label{pp3fo3}I\left( M/\Gamma \right) =N\left( \Gamma \right) /\Gamma \quad
. 
\end{equation}

Now we turn to the extended normalizer. For a general element $g\in eN\left(
\Gamma \right) $ the induced map $g_{\#}$ is no longer injective on $%
X/\Gamma $. To see this assume that for a fixed element $g\in G$, the
inclusion in definition (\ref{pp3fo2}) is proper, i.e. $g\Gamma g^{-1}%
\stackunder{\neq }{\subset }\Gamma $. Take an arbitrary $x\in M$, then $%
g\left( \Gamma x\right) \stackunder{\neq }{\subset }\Gamma \left( gx\right) $%
; this means that there exists an element $z\in \Gamma \left( gx\right) $
that is not the image under $g$ of any $\gamma x$ in the orbit $\Gamma x$.
Hence, since $g$ is invertible on $X$, there exists an $x^{\prime }\in X$,
whose orbit $\Gamma x^{\prime }$ is different from the orbit $\Gamma x$,
such that $z=gx^{\prime }$. Since $g$ preserves fibres we have $g\left(
\Gamma x^{\prime }\right) \stackunder{\neq }{\subset }\Gamma \left(
gx^{\prime }\right) =\Gamma z=\Gamma \left( gx\right) $, the last equality
following, since $z$ lies in the orbit of $gx$. This implies that the
induced map $g_{\#}$ maps the distinct orbits $\Gamma x\neq \Gamma x^{\prime
}$ into the same orbit $\Gamma \left( gx\right) $, which expresses that $%
g_{\#}$ is not injective. In particular, if $g$ was an isometry of $M$, then 
$g_{\#}$ can no longer be an isometry on the quotient space, since it is not
invertible. It also follows that $eN\left( \Gamma \right) $ will not be a
group, in general. For this reason, the elements of the extended normalizer
seem to have attracted limited attention in the literature so far.

In this work, however, we will show that the extended normalizer naturally
emerges when we study identification spaces $M/\Gamma $, where $M=\left( 
\TeXButton{R}{\mathbb{R}}^n,\eta \right) $ is flat $\TeXButton{R}{\mathbb{R}}%
^n$ endowed with a symmetric bilinear form $\eta $ with signature $\left(
-t,+s\right) $ or index $t$; to such a space $M$ we will also refer to as $M=%
\TeXButton{R}{\mathbb{R}}_t^n$. The group $\Gamma $ will be realized as a
discrete group of translations in $M$, the elements being in 1--1
correspondence with the points of a {\it lattice }$lat\subset \TeXButton{R}
{\mathbb{R}}_t^n$, which is regarded as a subset of $\TeXButton{R}
{\mathbb{R}}_t^n$. We will find that in the Lorentzian case, the fact that
the identity component $SO_{1,n-1}^{+}\subset O_{1,n-1}$ is no longer
compact will give rise to a natural extension of the isometry group $N\left(
\Gamma \right) /\Gamma $ of the quotient $M/\Gamma $ to the set $eN\left(
\Gamma \right) /\Gamma $, provided that the $\TeXButton{R}{\mathbb{R}}$%
-linear enveloppe of the lattice is a {\bf lightlike} subvector space (we do
not consider lattices whose associated $\TeXButton{R}{\mathbb{R}}$-linear
vector space is timelike; this would give rise to ''compactifications along
a time direction''). We will show that $eN\left( \Gamma \right) /\Gamma $ in
general has the structure of a {\it semigroup}, naturally containing the
isometry group $N\left( \Gamma \right) /\Gamma $ as a subgroup. This will be
compared with the orthogonal case, and it will be shown that the compactness
of $SO_n$ obstructs such an extension. That is probably why such extensions
have not been studied in crystallography in the past.

\section{Notations and conventions \label{Sc.2}}

--- If a subgroup $H$ of a group $G$ is normal in $G$, we denote this fact
by $H\lhd G$.

--- The isometry group $I\left( \TeXButton{R}{\mathbb{R}}_t^n\right) $ of $%
\TeXButton{R}{\mathbb{R}}_t^n$ is the semi-direct product 
\begin{equation}
\label{pp3not1}I\left( \TeXButton{R}{\mathbb{R}}_t^n\right) =E_t^n=%
\TeXButton{R}{\mathbb{R}}^n\odot O_{t,n-t}\quad , 
\end{equation}
called {\it pseudo-Euclidean group}, where the translational factor $%
\TeXButton{R}{\mathbb{R}}^n$ is normal in $E_t^n$, $\TeXButton{R}{\mathbb{R}}%
^n\lhd E_t^n$. Elements of $E_t^n$ will be denoted by $\left( t,R\right) $
with group law $\left( t,R\right) \left( t^{\prime },R^{\prime }\right)
=\left( Rt^{\prime }+t,RR^{\prime }\right) $. Projections onto the first and
second factor of $E_t^n$ are defined as $p_1:E_t^n\rightarrow \TeXButton{R}
{\mathbb{R}}^n$, $p_1\left( t,R\right) =\left( t,1\right) $; $%
p_2:E_t^n\rightarrow O_{t,n-t}$, $p_2\left( t,R\right) =\left( 0,R\right) $.
Elements of the form $\left( 0,R\right) $ will be referred to as
''rotations'', although in general they are pseudo-orthogonal
transformations. The Lie algebra of the pseudo-Euclidean group $E_t^n$ will
be denoted by 
\begin{equation}
\label{pp3not2}euc_t^n\equiv Lie\left( E_t^n\right) 
\end{equation}
henceforth. -- For $t=1$, $E_1^n$ is the Poincare group.

--- Given a subset $S\subset \TeXButton{R}{\mathbb{R}}_t^n$, the $%
\TeXButton{R}{\mathbb{R}}${\it -linear span} of $S$ is the vector subspace
of $\TeXButton{R}{\mathbb{R}}_t^n$ generated by elements of $S$, i.e. the
set of all finite linear combinations of elements in $S$ with coefficients
in $\TeXButton{R}{\mathbb{R}}$; we denote the $\TeXButton{R}{\mathbb{R}}$%
-linear span of $S$ by $\left[ S\right] _{\TeXButton{R}{\mathbb{R}}}$ or
simply $\left[ S\right] $, if no confusion is likely. If $S=\left\{
u_1,\ldots ,u_m\right\} $ is finite, one also writes $\left[ S\right]
=\sum_{i=1}^m\TeXButton{R}{\mathbb{R}}\cdot u_i$. This contains the $%
\TeXButton{Z}{\mathbb{Z}}${\it -linear span }$\left[ S\right] _{\TeXButton{Z}
{\mathbb{Z}}}=\sum_{i=1}^m\TeXButton{Z}{\mathbb{Z}}\cdot u_i$ as a proper
subset.

--- The {\it index} $ind\left( W\right) $ of a vector subspace $W\subset 
\TeXButton{R}{\mathbb{R}}_t^n$ is the maximum in the set of all integers
that are the dimensions of $\TeXButton{R}{\mathbb{R}}$-vector subspaces $%
W^{\prime }\subset W$ on which the restriction of the metric $\left. \eta
\right| W^{\prime }$ is negative definite, see e.g. \cite{ONeill}. Hence $%
0\le ind\left( W\right) \le m$, and $ind\left( W\right) =0$ if and only if $%
\left. \eta \right| W$ is positive definite. In the Lorentzian case, i.e. $M=%
\TeXButton{R}{\mathbb{R}}_1^n$, we call $W$ timelike $\Leftrightarrow $ $%
\left. \eta \right| W$ nondegenerate, and $ind\left( W\right) =1$; $W$
lightlike $\Leftrightarrow $ $\left. \eta \right| W$ degenerate, and $W$
contains a $1$-dimensional lightlike vector subspace, but no timelike
vector; and $W$ spacelike $\Leftrightarrow $ $\left. \eta \right| W$ is
positive definite and hence $ind\left( W\right) =0$.

\section{Lattices and their symmetries \label{Sc.3}}

In this section we introduce our conventions of lattices in $\TeXButton{R}
{\mathbb{R}}_t^n$ and their associated sets of symmetries. These notions
will be appropriate to examine the peculiarities that arise when the vector
space $\TeXButton{R}{\mathbb{R}}_t^n$ is non-Euclidean. On the one hand,
they extend the usual terminology encountered in crystallography. On the
other hand, our definition of a lattice is adapted to the purposes of this
paper, and therefore somewhat simplified compared with the most general
definitions possible in crystallography. This means that a full adaption of
our terminology introduced here with well-established crystallographic
notions would have been a tedious task with no contribution to deeper
understanding; we therefore have made no attempt to do so.

In this work we restrict attention to lattices that contain the origin $0\in 
\TeXButton{R}{\mathbb{R}}_t^n$ as a lattice point, which suffices for our
purposes. Let $1\le m\le n$, let $\underline{u}\equiv \left( u_1,\ldots
,u_m\right) $ be a set of $m$ linearly independent vectors in $\TeXButton{R}
{\mathbb{R}}_t^n$; then the $\TeXButton{Z}{\mathbb{Z}}$-linear span of $%
\underline{u}$, 
\begin{equation}
\label{pp3fo4}lat\equiv \sum_{i=1}^m\TeXButton{Z}{\mathbb{Z}}\cdot
u_i=\left\{ \sum_{i=1}^mz_i\cdot u_i\mid z_i\in \TeXButton{Z}{\mathbb{Z}}%
\right\} \quad , 
\end{equation}
is called the set of {\it lattice points} with respect to $\underline{u}$.
Elements of $lat$ are regarded as points in $\TeXButton{R}{\mathbb{R}}_t^n$
as well as vectors on $T\TeXButton{R}{\mathbb{R}}_t^n$. Let $\left[
lat\right] \equiv \left[ \underline{u}\right] _{\TeXButton{R}{\mathbb{R}}}$
denote the $\TeXButton{R}{\mathbb{R}}$-linear span of $lat$. We define the 
{\it index of the lattice }as the index of its $\TeXButton{R}{\mathbb{R}}$%
-linear span $\left[ lat\right] $, $ind\left( lat\right) \equiv ind\left(
\left[ lat\right] \right) $. In the Lorentzian case, $M=\TeXButton{R}
{\mathbb{R}}_1^n$, the lattice $lat$ is called {\it timelike / lightlike /
spacelike} if the enveloppe $\left[ lat\right] $ is.

The subset $T_{lat}\subset E_t^n$ is the subgroup of all translations in $%
E_t^n$ through elements of $lat$, 
\begin{equation}
\label{pp3fo5}T_{lat}=\left\{ \left( t_z,0\right) \in E_t^n\mid t_z\in
lat\right\} \quad . 
\end{equation}
Elements of $T_{lat}$ are called {\it primitive translations}.

Now we introduce the set $pres\left( lat\right) $, which is defined to be
the set of all diffeomorphisms on $\TeXButton{R}{\mathbb{R}}_t^n$ preserving 
$lat$, i.e. 
\begin{equation}
\label{pp3fo6}pres\left( lat\right) \equiv \left\{ \phi \in diff\left( 
\TeXButton{R}{\mathbb{R}}_t^n\right) \mid \phi \,lat\subset lat\right\}
\quad . 
\end{equation}
Every such $\phi $ is invertible, hence surjective, on $\TeXButton{R}
{\mathbb{R}}_t^n$; however, it need not be surjective on $lat$, which means
that inverses in this set do not necessarily exist. Neither is it required
that $\phi $ be linear. $pres\left( lat\right) $ is therefore a semigroup
with composition of maps as multiplication, and $\left. id\right| _{%
\TeXButton{R}{\mathbb{R}}_t^n}$ as unit element.

We also define the associated set 
\begin{equation}
\label{pp3fo6a}pres^{\times }\left( lat\right) \equiv \left\{ \phi \in
diff\left( \TeXButton{R}{\mathbb{R}}_t^n\right) \mid \phi \,lat=lat\right\}
\subset pres\left( lat\right) \quad . 
\end{equation}
$pres^{\times }\left( lat\right) $ contains all diffeomorphisms of $%
pres\left( lat\right) $ whose restriction to $lat$ is invertible, and hence
is a group.

The intersection 
\begin{equation}
\label{pp3fo7}sym\equiv pres\left( lat\right) \cap E_t^n 
\end{equation}
we term the {\it set of symmetries} of the lattice $lat$, and 
\begin{equation}
\label{pp3fo7a}sym^{\times }\equiv pres^{\times }\left( lat\right) \cap
E_t^n 
\end{equation}
we call the {\it set of invertible symmetries} of $lat$. $sym^{\times }$ is
a group by construction. On the other hand, $sym$ is only a semigroup, since
inverses \underline{in $sym$} do not necessarily exist. Cleary, $sym^{\times
}\subset sym$. This inclusion is not always proper. Below we prove a theorem
that explains the details. First, however, we examine the structure of $%
sym^{\times }$ and $sym$ more closely.

\section{The structure of $sym^{\times }$ and $sym$ \label{Sc.4}}

An immediate statement is

\subsection{Proposition \label{sym}}

\begin{enumerate}
\item  All translations in $sym$ belong to $sym^{\times }$.

\item  All translations in $T_{lat}$ belong to $sym^{\times }$.

\item  There are no pure translations in $sym^{\times }$ other than
primitive translations from $T_{lat}$.

\item  $T_{lat}$ is normal in $sym^{\times }$.
\end{enumerate}

\TeXButton{Beweis}{\raisebox{-1ex}{\it Proof :}
\vspace{1ex}}

The fist two statements are obvious. As for the third, assume $\left(
t,1\right) \in sym^{\times }$ were no element of $T_{lat}$; then it would
map the lattice point $0$ into the lattice point $t$, which is a
contradition. Now prove $\left( 4\right) $: Since, by $\left( 2\right) $,
translations $\left( t_z,0\right) $ belong to $sym^{\times }$, we have that
for a given $\left( t,R\right) \in sym^{\times }$, the product $\left(
t,R\right) \left( t_z,0\right) \left( t,R\right) ^{-1}\in sym^{\times }$ as
well. But this product equals $\left( Rt_z,1\right) $, hence is a pure
translation, hence, by $\left( 3\right) $, must belong to $T_{lat}$, which
proves $T_{lat}\lhd sym^{\times }$. \TeXButton{BWE}
{\hfill
\vspace{2ex}
$\blacksquare$}

These results do not a priori imply, however, that $p_1\left( sym\right)
=T_{lat}$. We show below that this is true for $sym^{\times }$.

We now examine the projection of $sym^{\times }$ onto the second factor $%
p_2\left( sym^{\times }\right) $. A priori it is not clear whether this
projection is a subset of $sym$, $sym^{\times }$, or not. We will see
shortly that indeed $p_2\left( sym^{\times }\right) \subset sym^{\times }$.
We start with observing that the elements $\left( 0,R\right) $ of $p_2\left(
sym^{\times }\right) $ are in 1--1 correspondence with the left cosets $%
T_{lat}\cdot \left( t,R\right) $, where $\left( t,R\right) $ is in the
inverse image $p_2^{-1}\left( 0,R\right) $. This follows, since $T_{lat}$ is
a subgroup of $sym^{\times }$, so that $T_{lat}\cdot \left( t,R\right) $ is
certainly in the inverse image; and furthermore, any two elements in this
coset must differ by a primitive translation, since for $\left( t,R\right) $%
, $\left( t^{\prime },R\right) $ we have $\left( t,R\right) ^{-1}\in
sym^{\times }$, hence $\left( t^{\prime },R\right) \left( t,R\right)
^{-1}=\left( t^{\prime }-t,1\right) \in sym^{\times }$. It is at this point
that we need the condition that $\left( t,R\right) $ be in $sym^{\times }$
rather in $sym$. The last equation says that $\left( t^{\prime }-t,1\right) $
must be a primitive translation, since $sym^{\times }$ contains no other
translations than these. From these considerations we conclude that there
must exist a coset representative, denoted by $\left( \tau _R,R\right) $ of
the coset $T_{lat}\cdot \left( t,R\right) $ such that 
\begin{equation}
\label{pp3fo7b}\tau _R=\sum_{i=1}^mq_i\cdot u_i\quad ,\quad 0\le q_i<1\quad
. 
\end{equation}
This defines a map 
\begin{equation}
\label{pp3fo7c}\tau :\left\{ 
\begin{array}{c}
sym^{\times }\rightarrow W \\ 
\left( t,R\right) =\left( t_z+\tau _R,R\right) \mapsto \tau _R 
\end{array}
\right. , 
\end{equation}
which is constant on the cosets. We remark that if the map $\tau \equiv 0$
is identically zero, then the associated group $sym^{\times }$ is called 
{\it symmorphic} in crystallography (see, e.g., \cite{Corn1}).

We now define the following subgroups of $sym$, $sym^{\times }:$%
\begin{equation}
\label{pp3fo8}rot=\left\{ \Lambda \in sym\mid \Lambda =\left( 0,R\right)
\right\} \quad , 
\end{equation}
\begin{equation}
\label{pp3fo9}rot^{\times }=rot\cap sym^{\times }\quad , 
\end{equation}
i.e. these are the subsets of $sym$, $sym^{\times }$, respectively, that are
pure (pseudo-Euclidean) ''rotations''. We can now prove the result announced
above:

\subsection{Proposition}

The projection of $sym^{\times }$ onto the ''rotational'' factor coincides
with the set of all pure ''rotations'' in $sym^{\times }$, i.e. 
\begin{equation}
\label{pp3fo10}p_2\left( sym^{\times }\right) =rot^{\times }\quad . 
\end{equation}

\TeXButton{Beweis}{\raisebox{-1ex}{\it Proof :}
\vspace{1ex}}

The inclusion $"\supset "$ is trivial. We prove $"\subset ":$\ If $\left(
0,R\right) \in p_2\left( sym^{\times }\right) $, then there exists a vector $%
t\in W$ (not necessarily in $lat$) so that $\left( 0,R\right) =p_2\left(
t,R\right) $, \underline{and} $\left( t,R\right) $ as well as $\left(
t,R\right) ^{-1}$ are elements of $sym^{\times }$. Now let $t_z\in lat$
arbitrary, then $\left( t_z,0\right) \in sym^{\times }$, and so is the
product $\left( t,R\right) \left( t_z,0\right) \left( t,R\right)
^{-1}=\left( Rt_z,1\right) $. The RHS must be a primitive translation, hence 
$Rt_z\in lat$ for all $t_z$, or $Rlat\subset lat$. The same argument holds
for $R^{-1}$, which says that $Rlat=lat$, or $\left( 0,R\right) \in
rot^{\times }$. \TeXButton{BWE}{\hfill
\vspace{2ex}
$\blacksquare$}

We next show

\subsection{Proposition \label{Tau}}

The restriction of the $\tau $-map to $sym^{\times }$ vanishes identically.

\TeXButton{Beweis}{\raisebox{-1ex}{\it Proof :}
\vspace{1ex}}

Let $\left( t,R\right) \in sym^{\times }$, then the projection onto the
second factor is $\left( 0,R\right) \in p_2\left( sym^{\times }\right)
=rot^{\times }\subset sym^{\times }$, where we have used (\ref{pp3fo10}).
Therefore $\left( t,R\right) $ and $\left( 0,R\right) $ both lie in the same
coset $T_{lat}\cdot \left( t,R\right) $; but this means that $\left(
0,R\right) $ is the unique coset representative that determines the value of
the $\tau $-map on the argument $\left( t,R\right) $. Hence $\tau _R=0$. 
\TeXButton{BWE}{\hfill
\vspace{2ex}
$\blacksquare$}

From proposition \ref{Tau} we infer that every element of $sym^{\times }$
has the form $\left( t_z,R\right) $, where $t_z\in lat$. For every $\left(
t,R\right) \in sym^{\times }$ lies in the coset $T_{lat}\cdot \left(
t,R\right) $, which contains the coset representative $\left( 0,R\right) $.
Hence $\left( t,R\right) $ and $\left( 0,R\right) $ must differ by a
primitive translation, which says that $t=t_z\in lat$.

As a corollary we infer the result we have announced after proposition \ref
{sym}, namely 
\begin{equation}
\label{pp3fo11}p_1\left( sym^{\times }\right) =T_{lat}\quad . 
\end{equation}
Thus, all elements of $sym^{\times }$ have a standard decomposition $\left(
t_z,R\right) =\left( t_z,1\right) \left( 0,R\right) \in T_{lat}\times
rot^{\times }$ according to (\ref{pp3fo10}). Furthermore, the groups $%
T_{lat} $ and $rot^{\times }$ have only the unit element $\left( 0,1\right) $
in common, and $T_{lat}$ is normal in $sym^{\times }$. Thus, we have proven
the

\subsection{Proposition}

$sym^{\times }$ is the semidirect product 
\begin{equation}
\label{pp3fo12}sym^{\times }=T_{lat}\odot rot^{\times }\quad . 
\end{equation}

As an immediate conclusion we see that we must have 
\begin{equation}
\label{pp3fo13}T_{lat}\odot rot\subset sym\quad . 
\end{equation}
We have not examined, however, whether this inclusion is proper, or if $sym$
can contain elements $\left( t,R\right) $ for which $t\not \in lat$.

Finally, we present a condition under which $rot$ coincides with $%
rot^{\times }$; this sheds some light on the question under which
circumstances $sym^{\times }$ is actually a proper subset of $sym$.

\subsection{Theorem \label{Bedingung}}

If $ind\left( lat\right) =0$ or $ind\left( lat\right) =m$ (i.e. minimal or
maximal), then $rot=rot^{\times }$.

\TeXButton{Beweis}{\raisebox{-1ex}{\it Proof :}
\vspace{1ex}}

We first assume that $ind\left( W\right) =0$, i.e. $\left. \eta \right| W$
is positive definite. Let $\Lambda \in rot$. Since $\Lambda $
preserves $lat$, it also preserves its $\TeXButton{R}{\mathbb{R}}$-linear
enveloppe $W$, i.e. $\Lambda W\subset W$. Let $x,y\in W$ arbitrary, then $%
\Lambda x,\Lambda y\in W$. This says that%
$$
\left( \left. \eta \right| W\right) \left[ \left( \left. \Lambda \right|
W\right) x,\left( \left. \Lambda \right| W\right) y\right] =\eta \left(
\Lambda x,\Lambda y\right) =\eta \left( x,y\right) =\left( \left. \eta
\right| W\right) \left[ x,y\right] \quad , 
$$
which says that the restriction $\left. \Lambda \right| W$ of $\Lambda $ to
the subvector space $W$ preserves the bilinear form $\left. \eta \right| W$
on this space. But $\left. \eta \right| W$ is positive definite by
assumption, hence $\left. \Lambda \right| W\in O\left( W\right) $, where $%
O\left( W\right) $ denotes the orthogonal group of $W$.

Now we assume that $\Lambda $ has the property 
\begin{equation}
\label{pp3fo14}\Lambda \in sym\quad ,\quad \text{and\quad }\Lambda ^{-1}\NEG
\in sym\quad , 
\end{equation}
in other words, $\Lambda lat\stackrel{\subset }{\neq }lat$. This means that $%
\left. \Lambda \right| lat$ is not surjective. Hence $\exists \;x\in
lat:\Lambda u\neq x$ for all $u\in lat$. $x$ cannot be zero, since $0\in lat$%
, and $\Lambda $ is linear. Hence $r\equiv \left\| x\right\| >0$, where $%
\left\| x\right\| =\eta \left( x,x\right) $ denotes the Euclidean norm on $W$%
. Now let $S_{m-1}$ be the $\left( m-1\right) $-dimensional sphere 
\underline{in $W$}, centered at $0$. Consider the intersection $sct=lat\cap
r\cdot S_{m-1}$, where $r\cdot S_{m-1}$ is the $\left( m-1\right) $%
-dimensional sphere with radius $r$ in $W$. Note that this set coincides
with the orbit $O_{m-1}\cdot x$ of $x$ under the action of the orthogonal
group $O_{m-1}$, which is a compact subset of $W\simeq \TeXButton{R}
{\mathbb{R}}^m$. From the compactness of $r\cdot S_{m-1}$ it follows that
the number of elements $\#sct$ of $sct$ is {\bf finite}, $0\le \#sct<\infty $%
. Then

\begin{enumerate}
\item  $\left. \Lambda \right| W$ orthogonal$\quad \Rightarrow $\quad $%
\Lambda \left( sct\right) \subset r\cdot S_{n-1}$;

\item  $\Lambda $ lattice preserving\quad $\Rightarrow $\quad $\Lambda
\left( sct\right) \subset lat$;

\item  $\Lambda $ injective\quad $\Rightarrow $\quad $\#\Lambda \left(
sct\right) =\#\left( sct\right) $.
\end{enumerate}

The first two statements imply that $\left. \Lambda \right| W$ preserves $%
sct $, $\left( \left. \Lambda \right| W\right) \left( sct\right) \subset sct$%
; from the third we deduce that $\left( \left. \Lambda \right| W\right)
\left( sct\right) =sct$. But this says that all elements of $sct$ are in the
image of $\left( \left. \Lambda \right| W\right) $, hence $x=\left( \left.
\Lambda \right| W\right) \left( x^{\prime }\right) $ for some $x^{\prime
}\in sct$, which is a contradiction to the result above. This says that our
initial assumption (\ref{pp3fo14}) concerning $\Lambda $ was wrong.

Now assume that $ind\left( lat\right) $ is maximal. Then $\left. \eta
\right| W$ is negative definite, but the argument given above clearly still
applies, since $O_{0,m-1}\simeq O_{m-1,0}$, and the only point in the proof
was the compactness of the $O_{m-1}$-orbits. This completes our proof. 
\TeXButton{BWE}{\hfill
\vspace{2ex}
$\blacksquare$}

We see that the structure of the proof relies on the compactness of orbits $%
O\cdot x$ of $x$ under the orthogonal group, which, in turn, comes from the
fact that the orthogonal groups $O$ are compact. If the metric restricted to 
$\left[ lat\right] $ were pseudo-Euclidean instead, we could have
non-compact orbits, related to the non-compactness of the groups $O_{t,s}$.
We have not proved this in full, but we conjecture that the converse of
theorem \ref{Bedingung} should read:

''If $0<ind\left( lat\right) <m=\dim _{\TeXButton{R}{\mathbb{R}}}\left[
lat\right] $, then $rot^{\times }\stackrel{\subset }{\neq }rot$''.

An explicit example of this situation will be constructed now.

\section{Identifications over a lightlike lattice \label{Sc.5}}

Given a lattice $lat$ in a pseudo-Euclidean space $M=\TeXButton{R}
{\mathbb{R}}_t^n$, we have the associated group of primitive translations $%
\Gamma =T_{lat}$. We want to study the quotient space $M/\Gamma $, its
isometry group $I\left( M/\Gamma \right) =N\left( \Gamma \right) /\Gamma $,
and the possible extension $eN\left( \Gamma \right) /\Gamma $ of this
isometry group. We now show how $N\left( \Gamma \right) $ and $eN\left(
\Gamma \right) $ are related to the sets $rot^{\times }$ and $rot$,
respectively.

An element $\left( t,R\right) \in E_t^n$ is in the extended normalizer $%
eN\left( \Gamma \right) $ iff $\left( t,R\right) \Gamma \left( t,R\right)
^{-1}\subset \Gamma $, where $\left( t,R\right) ^{-1}$ is the inverse of $%
\left( t,R\right) $ in $E_t^n$. This is true iff $\left( Rt_z,1\right) \in
\Gamma $ for all $t_z\in lat$, hence iff $Rlat\subset lat$, hence iff $%
\left( 0,R\right) \in sym$, hence iff $\left( 0,R\right) \in rot$. On the
other hand, $\left( t,R\right) $ is in the normalizer $N\left( \Gamma
\right) $ if the same condition holds for $R^{-1}$ as well, i.e. $R^{-1}\in
rot$. But $R,R^{-1}\in rot$ is true iff $R\in rot^{\times }$. Hence 
\begin{equation}
\label{pp3fo15}eN\left( \Gamma \right) =\left\{ \left( t,R\right) \in
E_t^n\mid R\in rot\right\} =\TeXButton{R}{\mathbb{R}}^n\odot rot\quad , 
\end{equation}
\begin{equation}
\label{pp3fo16}N\left( \Gamma \right) =\left\{ \left( t,R\right) \in
E_t^n\mid R\in rot^{\times }\right\} =\TeXButton{R}{\mathbb{R}}^n\odot
rot^{\times }\quad . 
\end{equation}

From now on we focus on the Lorentzian metric, $t=1$, and work in $M=%
\TeXButton{R}{\mathbb{R}}_1^n$. In the following we examine in detail the
extended normalizer of a group $\Gamma =T_{lat}$, whose associated lattice $%
lat$ is given as follows: The basis vectors $\underline{u}=\left(
u_{+},u_1\ldots ,u_{m-1}\right) $ of the lattice contain {\bf one} lightlike
vector, namely $u_{+}$, and $\left( m-1\right) $ spacelike vectors $%
u_1,\ldots ,u_{m-1}$. It is assumed that $u_{+}\perp \left[ u_1,\ldots
,u_{m-1}\right] _{\TeXButton{R}{\mathbb{R}}}$, which is necessary to
guarantee that the $\TeXButton{R}{\mathbb{R}}$-linear span of all basis
vectors, $W\equiv \left[ \underline{u}\right] _{\TeXButton{R}{\mathbb{R}}}$,
is indeed a lightlike vector subspace of $\TeXButton{R}{\mathbb{R}}_1^n$. As
is well known, this means that the restriction $\left. \eta \right| W$ of
the metric to $W$ is degenerate; and furthermore, that $W$ contains a $1$%
-dimensional lightlike vector subspace, in this case given by $\left[
u_{+}\right] _{\TeXButton{R}{\mathbb{R}}}$, but no lightlike vector
otherwise. Applying a suitable Lorentz transformation it can be assumed
without loss of generality that $\left( 1\right) $ the vector subspace $%
U\equiv \left[ u_1,\ldots ,u_{m-1}\right] _{\TeXButton{R}{\mathbb{R}}}$
coincides with the span of the last $\left( m-1\right) $ canonical basis
vectors of $\TeXButton{R}{\mathbb{R}}_1^n$, i.e. $U=\left[ e_{n-m+1},\ldots
,e_{n-1}\right] _{\TeXButton{R}{\mathbb{R}}}$, where $\TeXButton{R}
{\mathbb{R}}_1^n=\left[ e_0,\ldots ,e_{n-1}\right] _{\TeXButton{R}
{\mathbb{R}}}$; and $\left( 2\right) $, that $u_{+}=\frac{e_0+e_1}{\sqrt{2}}$%
. We want to compute $N\left( \Gamma \right) $ and $eN\left( \Gamma \right) $
for this lattice, where $\Gamma =T_{lat}$. According to (\ref{pp3fo15}, \ref
{pp3fo16}) our first task is to identify the sets $rot$ and $rot^{\times }$.
We do this in several steps. Firstly, we identify the subset of $E_1^n$ that
preserves the $1$-dimensional lightlike subspace $\left[ u_{+}\right] _{%
\TeXButton{R}{\mathbb{R}}}$.

\subsection{Preservation of a lightlike $1$-dimensional subspace}

Let $\underline{e}\equiv \left( e_0,e_1,\ldots ,e_{n-1}\right) $ denote the
canonical basis of $\TeXButton{R}{\mathbb{R}}_1^n$. Construct two lightlike
vectors $u_{+,-}\equiv \frac 1{\sqrt{2}}\left( e_0\pm e_1\right) $, and
consider the new basis $\underline{b}\equiv \left( u_{+},u_{-},e_2,\ldots
,e_{n-1}\right) $. The transformation between the two bases is accomplished
by $T=diag\left( \frac 1{\sqrt{2}}\left( 
\begin{array}{cc}
1 & 1 \\ 
1 & -1 
\end{array}
\right) ,{\bf 1}\right) $, with $T^2=1$, so that $\underline{b}=\underline{e}%
T$ and $\underline{e}=\underline{b}T$. In the $\underline{b}$-basis, the
matrix $\eta _{\underline{b}}$ of $\eta $ takes the form $\eta _{\underline{b%
}}=diag\left( \left( 
\begin{array}{cc}
0 & -1 \\ 
-1 & 0 
\end{array}
\right) ,{\bf 1}\right) $.

We want to find the set $rot\left( \left[ u_{+}\right] _{\TeXButton{R}
{\mathbb{R}}};O_{1,n-1}\right) $ of elements $R\in O_{1,n-1}$ that preserve
this subspace, i.e. $\Phi \left( R\right) u_{+}=\lambda \cdot u_{+}$ with $%
\TeXButton{R}{\mathbb{R}}\ni \lambda \neq 0$. Here $\Phi \left( R\right) $
denotes the linear operator associated with $R$, acting according to $\Phi
\left( R\right) \underline{b}=\underline{b}R$. Note the notation
conventions: We write $R$ for the abstract group elemenent as well as for
the matrix representing $\Phi \left( R\right) $ in a particular basis.

From $\Phi \left( R\right) \underline{b}=\underline{b}R$ we see that the
matrix representing $\Phi \left( R\right) $ must take the general form
$
R=\left( 
\begin{array}{cc}
\begin{array}{c}
{\lambda } \\ {\bf 0} 
\end{array}
& * 
\end{array}
\right) $
, where $"*"$ denotes ''something''. We can make an Ansatz for $R$
according to this form, and then impose the condition $R^T\eta _{\underline{b%
}}R=\eta _{\underline{b}}$ that expresses that $R$ is a Lorentz
transformation. This yields a set of matrices $R=\left( V,a,C\right) $
parametrized by $V\in \TeXButton{R}{\mathbb{R}}^{n-2}\subset \TeXButton{R}
{\mathbb{R}}_1^n$ canonically embedded in $\TeXButton{R}{\mathbb{R}}_1^n$
according to $V\leftrightarrow \left( 
\begin{array}{c}
0 \\ 
0 \\ 
V 
\end{array}
\right) $; $a\in \TeXButton{R}{\mathbb{R}}^{\times }$, where $\TeXButton{R}
{\mathbb{R}}^{\times }=\TeXButton{R}{\mathbb{R}}-\left\{ 0\right\} $ denotes
the multiplicative group of units of $\TeXButton{R}{\mathbb{R}}$; and $C\in
O_{n-1}\subset O_{1,n-1}$, canonically embedded according to $%
C\leftrightarrow \left( 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & C 
\end{array}
\right) \in O_{1,n-1}$. The matrices $\left( V,a,C\right) $ take the form 
\begin{equation}
\label{pp3fo17}\left( V,a,C\right) =\left( 
\begin{array}{ccc}
a & \frac{V^2}a & - 
\sqrt{2}V^TC \\ 0 & \frac 1a & 0 \\ 
0 & -\frac{\sqrt{2}V}a & C 
\end{array}
\right) \quad , 
\end{equation}
where $V^2=\sum_{i=2}^{n-1}V_i^2$ denotes the Euclidean quadratic form on $%
\TeXButton{R}{\mathbb{R}}^{n-2}$. These matrices satisfy the group law 
\begin{equation}
\label{pp3fo18}\left( V,a,C\right) \left( V^{\prime },a^{\prime },C^{\prime
}\right) =\left( aCV^{\prime }+V,aa^{\prime },CC^{\prime }\right) \quad , 
\end{equation}
with unit $\left( 0,1,1\right) $, and inverses 
\begin{equation}
\label{pp3fo19}\left( V,a,C\right) ^{-1}=\left( -\frac 1aC^{-1}V,\frac
1a,C^{-1}\right) \quad . 
\end{equation}
Using the group law (\ref{pp3fo18}) we find the standard decomposition of
elements 
\begin{equation}
\label{pp3fo20}\left( V,a,C\right) =\left( V,1,1\right) \left( 0,a,1\right)
\left( 0,1,C\right) \quad , 
\end{equation}
this decomposition being chosen so that factors which form normal subgroups
stand to the left, as it will be shown now. We firstly identify three
subgroups of $G=rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};O_{1,n-1}\right) $: The set of all $\left( V,1,1\right) $ forms an Abelian
subgroup of $G$, which is isomorphic to $\TeXButton{R}{\mathbb{R}}^{n-2}$,
as can be seen from the group law 
\begin{equation}
\label{pp3fo21}\left( V,1,1,\right) \left( V^{\prime },1,1\right) =\left(
V+V^{\prime },1,1\right) \quad . 
\end{equation}
The set of all $\left( 0,a,1\right) $ is a subgroup of $G$ with group law $%
\left( 0,a,1\right) \left( 0,a^{\prime },1\right) =\left( 0,aa^{\prime
},1\right) $, which will continue to be denoted by $\TeXButton{R}{\mathbb{R}}%
^{\times }$, and the set of all $\left( 0,1,C\right) $ clearly is a subgroup
isomorphic to $O_{n-2}$. Since $\left( 0,a,1\right) \left( 0,1,C\right)
=\left( 0,1,C\right) \left( 0,a,1\right) $, the last two subgroups form a
direct product subgroup $\TeXButton{R}{\mathbb{R}}^{\times }\otimes O_{n-2}$
of $G$. Furthermore, using the group law (\ref{pp3fo18}) again, we see that
conjugation $I\left( U,a,C\right) $ [where $I\left( g\right) h=ghg^{-1}$] of
an element $\left( V,1,1\right) $ of $\TeXButton{R}{\mathbb{R}}^{n-2}$
yields again a translation, 
\begin{equation}
\label{pp3fo22}I\left( U,a,C\right) \left( V,1,1\right) =\left(
aCV,1,1\right) \quad , 
\end{equation}
from which it follows that $\TeXButton{R}{\mathbb{R}}^{n-2}$ is a normal
subgroup of $G$. This implies that $G$ has the structure of a semidirect
product 
\begin{equation}
\label{pp3fo23}G=rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};O_{1,n-1}\right) \;\simeq \;\TeXButton{R}{\mathbb{R}}^{n-2}\odot \left[ 
\TeXButton{R}{\mathbb{R}}^{\times }\otimes O_{n-2}\right] \quad , 
\end{equation}
where $"\odot "$ denotes a semidirect product, and the normal factor $%
\TeXButton{R}{\mathbb{R}}^{n-2}$ stands to the left.

We see that this group has four connected components: They are obtained by
pairing the two connected components $\left( \TeXButton{R}{\mathbb{R}}_{+},%
\TeXButton{R}{\mathbb{R}}_{-}\right) $ of $\TeXButton{R}{\mathbb{R}}^{\times
}$ with the two connected components $\left( SO_{n-2},O_{n-2}^{-}\right) $
of $O_{n-2}$. $\TeXButton{R}{\mathbb{R}}_{-}$ reverses the time direction,
whereas $O_{n-2}^{-}$ reverses spatial orientation. The identity component $%
G_0$ of $G$ is obviously 
\begin{equation}
\label{pp3fo24}G_0=\TeXButton{R}{\mathbb{R}}^{n-2}\odot \left[ \TeXButton{R}
{\mathbb{R}}_{+}\otimes SO_{n-2}\right] \quad . 
\end{equation}
In what follows we shall restrict attention to $G_0$. If we had started this
section with $SO_{1,n-1}^{+}$, then our analysis would naturally render the
identity component $G_0$ for $rot\left( \left[ u_{+}\right] _{\TeXButton{R}
{\mathbb{R}}};O_{1,n-1}\right) $. We adapt our notation to this fact by
denoting as $rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};H\right) $ the set of all elements in the group $H\subseteq O_{1,n-1}$
that preserve $\left[ u_{+}\right] $. Then we can conclude this section with
the results 
\begin{equation}
\label{pp3fo25}rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};O_{1,n-1}\right) \;\simeq \;\TeXButton{R}{\mathbb{R}}^{n-2}\odot \left[ 
\TeXButton{R}{\mathbb{R}}^{\times }\otimes O_{n-2}\right] \quad , 
\end{equation}
\begin{equation}
\label{pp3fo26}rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};SO_{1,n-1}^{+}\right) \;\simeq \;\TeXButton{R}{\mathbb{R}}^{n-2}\odot
\left[ \TeXButton{R}{\mathbb{R}}_{+}\otimes SO_{n-2}\right] \quad . 
\end{equation}

We now give the explicit form of the matrices $\left( V,a,C\right) $ etc. in
the basis $\underline{e}$. Performing a similarity transformation with $T$
then yields, using (\ref{pp3fo17}), 
\begin{equation}
\label{pp3fo27}\left( V,a,C\right) =\left( 
\begin{array}{ccc}
\frac{a+\frac 1a}2+\frac{V^2}{2a} & \frac{a-\frac 1a}2-\frac{V^2}{2a} & 
-V^TC \\ 
\frac{a-\frac 1a}2+\frac{V^2}{2a} & \frac{a+\frac 1a}2-\frac{V^2}{2a} & 
-V^TC \\ 
-\frac Va & \frac Va & {\bf 1}_{n-2} 
\end{array}
\right) \quad . 
\end{equation}
This gives, in particular, 
\begin{equation}
\label{pp3fo28}\left( V,1,1\right) =\left( 
\begin{array}{ccc}
1+\frac{V^2}{2a} & -\frac{V^2}{2a} & -V^T \\ 
\frac{V^2}{2a} & 1-\frac{V^2}{2a} & -V^T \\ 
-V & V & {\bf 1}_{n-2} 
\end{array}
\right) \quad , 
\end{equation}
\begin{equation}
\label{pp3fo29}\left( 0,a,1\right) =\left( 
\begin{array}{cc}
{\rm sgn}\left( a\right) \cdot \left( 
\begin{array}{cc}
\cosh \phi & \sinh \phi \\ 
\sinh \phi & \cosh \phi 
\end{array}
\right) & 0 \\ 
0 & {\bf 1}_{n-2} 
\end{array}
\right) \quad , 
\end{equation}
where $\cosh \phi =\left| \frac{a+\frac 1a}2\right| $ and 
\begin{equation}
\label{pp3fo30}\left( 0,1,C\right) =\left( 
\begin{array}{cc}
{\bf 1}_2 & 0 \\ 
0 & C 
\end{array}
\right) \quad . 
\end{equation}

\subsection{Conformal algebra $cf_{n-2}$}

Before we investigate the Lie algebra of the Lie group $rot\left( \left[
u_{+}\right] _{\TeXButton{R}{\mathbb{R}}};SO_{1,n-1}^{+}\right) $, we
briefly explain the relation between the conformal algebra $cf_{n-2}$ in $%
\left( n-2\right) $ Euclidean dimensions and the Lorentz algebra $so_{1,n-1}$%
. $cf_{n-2}$ is spanned by generators $\left[ \left( L_{ij}\right) _{2\le
i<j\le n-1};\left( K_i,S_j\right) _{i,j=2,\ldots ,n-1};\Delta \right] $,
where $L_{ij}$ and $K_i$ span the Euclidean algebra $so_{n-2}$, $s_j$ are
the generators of special conformal transformations, and $\Delta $ generates
dilations. This basis obeys the relations 
\begin{equation}
\label{pp3fo31}
\begin{array}{c}
\left[ L_{ij},L_{km}\right] =\delta _{ik}\cdot L_{jm}+\delta _{jm}\cdot
L_{ik}-\delta _{im}\cdot L_{jk}-\delta _{jk}\cdot L_{im}\quad , \\ 
\left[ L_{ij},K_k\right] =\delta _{ik}\cdot K_j-\delta _{jk}\cdot K_i\quad ,
\\ 
\left[ L_{ij},S_k\right] =\delta _{ik}\cdot S_j-\delta _{jk}\cdot S_i\quad ,
\\ 
\left[ S_i,K_j\right] =2\left( L_{ij}-\delta _{ij}\cdot \Delta \right) \quad
, \\ 
\left[ K_i,K_j\right] =\left[ S_i,S_j\right] =0\quad , \\ 
\left[ L_{ij},\Delta \right] =0\quad , \\ 
\left[ \Delta ,K_i\right] =K_i\quad , \\ 
\left[ \Delta ,S_j\right] =-S_j\quad . 
\end{array}
\end{equation}
The first two lines contain the $so_{n-2}$ subalgebra.

The generators of $so_{1,n-1}$, on the other hand, are real $\left(
n,n\right) $ matrices $L_{\mu \nu }$ defined by 
\begin{equation}
\label{pp3fo32}\left( L_{\mu \nu }\right) _{\;b}^a=-\delta _\mu ^a\cdot \eta
_{\nu b}+\delta _\nu ^a\cdot \eta _{\mu a}\quad , 
\end{equation}
satisfying 
\begin{equation}
\label{pp3fo33}\left[ L_{\mu \nu },L_{\rho \sigma }\right] =\eta _{\mu \rho
}\cdot L_{\nu \sigma }+\eta _{\nu \sigma }\cdot L_{\mu \rho }-\eta _{\mu
\sigma }\cdot L_{\nu \rho }-\eta _{\nu \rho }\cdot L_{\mu \sigma }\quad . 
\end{equation}
Now we transform the basis $\left( L_{\mu \nu }\right) $ to a new basis%
$$
\left( -L_{01}\,;\,L_{0k}+L_{1k}\,;\,L_{0k}-L_{1i}\,;\,L_{ij}\right) \;= 
$$
\begin{equation}
\label{pp3fo34}=\;\left( \Delta \,;\,K_k\,;\,S_k\,;\,L_{ij}\right) \quad
,\quad k=2,\ldots ,n-1\quad ;\quad 2\le i<j\le n-1\quad . 
\end{equation}
Using (\ref{pp3fo17}) it is easy to verify that this new basis satisfies the
algebra (\ref{pp3fo31}), so that we have the well-known isomorphism of Lie
algebras 
\begin{equation}
\label{pp3fo35}cf_{n-2}\simeq so_{1,n-1}\quad . 
\end{equation}

We now can turn to evaluate the Lie algebra $\hat g_0$ of $rot\left( \left[
u_{+}\right] _{\TeXButton{R}{\mathbb{R}}};SO_{1,n-1}^{+}\right) $. The
generators of the subgroup $\TeXButton{R}{\mathbb{R}}^{n-2}$ with elements $%
\left( V,1,1\right) $ are obtained from (\ref{pp3fo28}) by 
\begin{equation}
\label{pp3fo36}\frac d{dt}\left( t\cdot V,1,1\right)
_{t=0}=\sum_{i=2}^{n-1}V^i\cdot \left( L_{0i}+L_{1i}\right)
=\sum_{i=2}^{n-1}V^i\cdot K_i\quad , 
\end{equation}
with $L_{\mu \nu }$ from (\ref{pp3fo32}), and using the new basis (\ref
{pp3fo34}). Similarly, 
\begin{equation}
\label{pp3fo37}\frac d{dt}\left( 0,t\cdot a,1\right) _{t=0}=-L_{01}=\Delta
\quad . 
\end{equation}
The generators of the $SO_{n-2}$-factor clearly are the elements $\left(
L_{ij}\right) _{2\le i<j\le n-1}$. Thus we see that the Lie algebra $\hat
g_0 $ is a Lie subalgebra of the conformal algebra $cf_{n-2}$ in $\left(
n-2\right) $ Euclidean dimensions; $\hat g_0$ is spanned precisely by those
generators of $cf_{n-2}$, that either annihilate the lightlike vector $u_{+}$%
, or leave it invariant, i.e. 
\begin{equation}
\label{pp3fo38}L_{ij}u_{+}=K_iu_{+}=0\quad ,\quad \Delta u_{+}=u_{+}\quad . 
\end{equation}

\subsection{Preservation of $W=\left[ u_{+}\right] _{\TeXButton{R}
{\mathbb{R}}}\oplus U$}

Consider the subvector spaces $U=\left[ e_{n-m+1},\ldots ,e_{n-1}\right] _{%
\TeXButton{R}{\mathbb{R}}}$ and $W=\left[ u_{+}\right] _{\TeXButton{R}
{\mathbb{R}}}\oplus U$ introduced at the beginning of section \ref{Sc.5}.
Our next question is: Which elements $R$ of $SO_{1,n-1}^{+}$ preserve $W$ in
the sense that $\Phi \left( R\right) W\subset W$? For the sake of
simplicity, we restrict attention to the identity component $SO_{1,n-1}^{+}$
here. This set of elements is again a subgroup and will be denoted by $%
rot\left( W;SO_{1,n-1}^{+}\right) \ $or $rot\left( W\right) $, if no
confusion is likely. Clearly, $rot\left( W\right) \stackunder{\neq }{\subset 
}rot\left( \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}%
};SO_{1,n-1}^{+}\right) $, hence we only need to examine which of the
generators $L_{ij},K_k,\Delta $ discussed in the previous section map $%
e_{n-m+1},\ldots ,e_{n-1}$ into $W$. An easy computation shows that $\Delta $
annihilates $U$, i.e. 
\begin{equation}
\label{pp3fo38a}\Delta U=0\quad , 
\end{equation}
that 
\begin{equation}
\label{pp3fo38b}
\begin{array}{ccc}
K_iU=0 & \text{for} & 2\le i\le n-m \\ 
K_ie_j=-\sqrt{2}\delta _{ij}\cdot u_{+} & \text{for} & n-m+1\le i,j\le n-1 
\end{array}
\quad , 
\end{equation}
and that all generators $L_{ij}$ with $2\le i\le n-m$ but $n-m+1\le j\le n-1$
are broken, so that we the remaining $L$-generators satisfy 
\begin{equation}
\label{pp3fo38c}
\begin{array}{ccc}
L_{ij}U=0 & \text{for} & 2\le i<j\le n-m \\ 
L_{ij}U\stackrel{\text{orth.}}{\subset }U & \text{for} & n-m+1\le i<j\le n-1 
\end{array}
\quad . 
\end{equation}
For the sake of simplicity we now denote indices ranging in $\left\{
2,\ldots ,n-m\right\} $ as $a,b$, etc.; those ranging in $\left\{
n-m+1,\ldots ,n-1\right\} $ as greek $\mu ,\nu $, etc.; and those ranging in 
$\left\{ 2,\ldots ,n-1\right\} $ as $i,j$, etc.\ Then the remaining
generators that preserve $W$ can be written as $\left( \Delta
;K_i;L_{ab};L_{\mu \nu }\right) $; their algebra is 
\begin{equation}
\label{pp3fo39}
\begin{array}{c}
\left[ \Delta ,K_i\right] =K_i\quad . \\ 
\left[ \Delta ,L_{ab}\right] =\left[ \Delta ,L_{\mu \nu }\right] =0\quad .
\\ 
\left[ K_i,K_j\right] =0\quad . \\ 
\left[ L_{ab},K_c\right] =\delta _{ac}\cdot K_b-\delta _{bc}\cdot K_a\quad .
\\ 
\left[ L_{ab},K_\rho \right] =0\quad . \\ 
\left[ L_{\mu \nu },K_a\right] =0\quad . \\ 
\left[ L_{\mu \nu },K_\rho \right] =\delta _{\mu \rho }\cdot K_\nu -\delta
_{\nu \rho }\cdot K_\mu \quad . \\ 
\left[ L_{ab},L_{cd}\right] =\delta _{ac}\cdot L_{bd}+\delta _{bd}\cdot
L_{ac}-\delta _{ad}\cdot L_{bc}-\delta _{bc}\cdot L_{ad}\quad . \\ 
\left[ L_{\mu \nu },L_{\gamma \delta }\right] =\delta _{\mu \gamma }\cdot
L_{\nu \delta }+\delta _{\nu \delta }\cdot L_{\mu \gamma }-\delta _{\mu
\delta }\cdot L_{\nu \gamma }-\delta _{\nu \gamma }\cdot L_{\mu \delta
}\quad . \\ 
\left[ L_{ab},L_{\mu \nu }\right] =0\quad . 
\end{array}
\end{equation}
This is the Lie algebra $Lie\left( rot\left( W\right) \right) $ of $%
rot\left( W\right) $. We see immediately that we have two subalgebras
isomorphic to the Euclidean algebras $Lie\left( E^{n-m-1}\right) $ and $%
Lie\left( E^{m-1}\right) $, which are spanned by $\left( L_{ab};K_c\right) $
and $\left( L_{\mu \nu };K_\rho \right) $, respectively. These subalgebras
commute. Their direct sum $Lie\left( E^{n-m-1}\right) \oplus Lie\left(
E^{m-1}\right) $ is an ideal in the full algebra, in which the dilation
generator $\Delta $ acts non-trivially only on the generators $K_i$. On the
other hand, we can combine the $K$-generators with $\Delta $ to define a
subalgebra $A=\left[ \Delta ,K_i\right] _{\TeXButton{R}{\mathbb{R}}}$, which
is also an ideal in $Lie\left( rot\left( W\right) \right) $.

On exponentiation of this algebra we obtain a covering group of $rot\left(
W\right) $; hence we must have 
\begin{equation}
\label{pp3fo40}rot\left( W\right) \simeq \left[ E^{n-m-1}\otimes
E^{m-1}\right] \odot \TeXButton{R}{\mathbb{R}}_{+}\quad , 
\end{equation}
where the normal factor $E^{n-m-1}\otimes E^{m-1}$ is written to the left of
the multiplicative subgroup $\TeXButton{R}{\mathbb{R}}_{+}$. The group law
can be derived from (\ref{pp3fo18}), if we make a split 
\begin{equation}
\label{pp3fo41}V=\sum_{a=2}^{n-m}V^ae_a\;+\sum_{\mu =n-m+1}^{n-1}V^\mu e_\mu
\;=V_1+V_2\quad , 
\end{equation}
and 
\begin{equation}
\label{pp3fo42}
\begin{array}{c}
C=C_1C_2=C_2C_1\quad ; \\ 
C_1\in SO_{n-m-1}\subset SO_{1,n-1}\quad ;\quad C_2\in SO_{m-1}\subset
SO_{1,n-1}\quad . 
\end{array}
\end{equation}
We observe that, according to the algebra (\ref{pp3fo39}), $C_1$ acts
trivially on $V_2$ and $C_2$ acts trivially on $V_1$, and that $C_1$
commutes with $C_2$. Thus, elements of $rot\left( W\right) $ will be denoted
by 
\begin{equation}
\label{pp3fo43}\left( V,a,C\right) \equiv \left( V_1,C_1,V_2,C_2,a\right)
\quad , 
\end{equation}
With these remarks, the group law for elements (\ref{pp3fo43}) can be
derived from (\ref{pp3fo18}) to be 
$$
\left( V_1,C_1,V_2,C_2,a\right) \left( V_1^{\prime },C_1^{\prime
},V_2^{\prime },C_2^{\prime },a^{\prime }\right) \;= 
$$
\begin{equation}
\label{pp3fo44}=\;\left( aC_1V_1^{\prime }+V_1,C_1C_1^{\prime
},aC_2V_2^{\prime }+V_2,C_2C_2^{\prime },aa^{\prime }\right) \quad . 
\end{equation}
Thus, elements $\left( V_1,C_1,V_2,C_2,a\right) $ decompose according to 
\begin{equation}
\label{pp3fo45}
\begin{array}{c}
\left( V_1,C_1,V_2,C_2,a\right) =\left( V_1,C_1,0,1,1\right) \left(
0,1,V_2,C_2,1\right) \left( 0,1,0,1,a\right) \quad , \\ 
\left( V_1,C_1,0,1,1\right) =\left( V_1,1,0,1,1\right) \left(
0,C_1,0,1,1\right) \quad , \\ 
\left( 0,1,V_2,C_2,1\right) =\left( 0,1,V_2,1,1\right) \left(
0,1,0,C_2,1\right) \quad . 
\end{array}
\end{equation}

We now describe the relationship between exponentiated elements of the Lie
algebra (\ref{pp3fo39}) and the group elements $\left(
V_1,C_1,V_2,C_2,a\right) $. Using straightforward matrix algebra, the
commutation relations (\ref{pp3fo39}) and the decomposition (\ref{pp3fo45})
we find that 
$$
\exp \left[ \sum_{a=2}^{n-m}V_1^a\cdot K_a+\sum_{\mu =n-m+1}^{n-1}V_2^\mu
\cdot K_\mu \right] =\left( V_1,1,V_2,1,1\right) = 
$$
\begin{equation}
\label{pp3fo46}=\left( V_1^2,\ldots ,V_1^{n-m};1;V_2^{n-m+1},\ldots
,V_2^{n-1};1;1\right) \quad , 
\end{equation}
\begin{equation}
\label{pp3fo47}\exp \left( \phi \cdot \Delta \right) =\left( 0,1,0,1,\exp
\phi \right) \quad , 
\end{equation}
and 
$$
\exp \left( \sum_{2\le a<b\le n-m}\omega _1^{ab}\cdot L_{ab}\right) =\left(
0,C_1\left( \omega _1\right) ,0,1,1\right) \quad ; 
$$
\begin{equation}
\label{pp3fo48}\exp \left( \sum_{n-m+1\le \mu <\nu \le n-1}\omega _2^{\mu
\nu }\cdot L_{\mu \nu }\right) =\left( 0,1,0,C_2\left( \omega _2\right)
,1\right) \quad . 
\end{equation}

We finish this subsection with computing the action of group elements $%
\left( V_1,C_1,V_2,C_2,a\right) $ on the transformed basis $\underline{b}$.
We use the relations (\ref{pp3fo38a}-\ref{pp3fo38c}), which we supplement by
the action of the $\left( \Delta ,K_a,K_\mu ,L_{ab},L_{\mu \nu }\right) $%
-basis on the basis vectors of the $\TeXButton{R}{\mathbb{R}}$-linear span 
\begin{equation}
\label{pp3fo49}M^{\prime }\equiv \left[ u_{-},e_2,\ldots ,e_{n-m}\right]
\quad . 
\end{equation}
From the basis transformation introduced at the beginning of this subsection
we see that we have 
\begin{equation}
\label{pp3fo50}M=\left[ u_{+}\right] \oplus M^{\prime }\oplus U\quad , 
\end{equation}
where $M=\TeXButton{R}{\mathbb{R}}_1^n$. The action of $\left( \Delta
,K_a,K_\mu ,L_{ab},L_{\mu \nu }\right) $ on the basis $\underline{b}$ of $M$
is now given by 
\begin{equation}
\label{pp3fo51}
\begin{array}{rclcrclc}
\Delta u_{+} & = & u_{+} & . & \Delta u_{-} & = & -u_{-} & . \\ 
K_au_{+} & = & 0 & . & K_au_{-} & = & -\sqrt{2}e_a & . \\ 
K_\mu u_{+} & = & 0 & . & K_\mu u_{-} & = & -\sqrt{2}e_\mu & . \\ 
L_{ab}u_{+} & = & 0 & . & L_{ab}u_{-} & = & 0 & . \\ 
L_{\mu \nu }u_{+} & = & 0 & . & L_{\mu \nu }u_{-} & = & 0 & . 
\end{array}
\end{equation}
\begin{equation}
\label{pp3fo52}
\begin{array}{rclcrclc}
\Delta e_a & = & 0 & . & \Delta e_\mu & = & 0 & . \\ 
K_ae_b & = & -\sqrt{2}\delta _{ab}\cdot u_{+} & . & K_ae_\mu & = & 0 & . \\ 
K_\mu e_a & = & 0 & . & K_\mu e_\nu & = & -\sqrt{2}\delta _{\mu \nu }\cdot
u_{+} & . \\ 
L_{ab}e_c & = & \delta _{ac}\cdot e_b-\delta _{bc}\cdot e_a & . & 
L_{ab}e_\mu & = & 0 & . \\ 
L_{\mu \nu }e_a & = & 0 & . & L_{\mu \nu }e_\rho & = & \delta _{\mu \rho
}\cdot e_\nu -\delta _{\nu \rho }\cdot e_\mu & . 
\end{array}
\end{equation}

With the help of (\ref{pp3fo46}-\ref{pp3fo48}), the commutation relations (%
\ref{pp3fo39}) and formulas (\ref{pp3fo51}-\ref{pp3fo52}) we can derive the
action of elements $\left( V_1,C_1,V_2,C_2,a\right) $ on $\left[
u_{+}\right] \oplus M^{\prime }\oplus U$. A calculation gives 
\begin{equation}
\label{pp3fo53}
\begin{array}{rclc}
\left( 0,1,0,1,e^\phi \right) u_{+} & = & e^\phi \cdot u_{+} & . \\ 
\left( V_1,1,0,1,1\right) u_{+} & = & u_{+} & . \\ 
\left( 0,1,V_2,1,1\right) u_{+} & = & u_{+} & . \\ 
\left( 0,C_1,0,1,1\right) u_{+} & = & u_{+} & . \\ 
\left( 0,1,0,C_2,1\right) u_{+} & = & u_{+} & . 
\end{array}
\end{equation}
\begin{equation}
\label{pp3fo54}
\begin{array}{rclc}
\left( 0,1,0,1,e^\phi \right) u_{-} & = & e^{-\phi }\cdot u_{-} & . \\ 
\left( V_1,1,0,1,1\right) u_{-} & = & \left( V_1\right) ^2\cdot u_{+}+u_{-}-%
\sqrt{2}V_1 & . \\ 
\left( 0,1,V_2,1,1\right) u_{-} & = & \left( V_2\right) ^2\cdot u_{+}+u_{-}-%
\sqrt{2}V_2 & . \\ 
\left( 0,C_1,0,1,1\right) u_{-} & = & u_{-} & . \\ 
\left( 0,1,0,C_2,1\right) u_{-} & = & u_{-} & . 
\end{array}
\end{equation}
Here $\left( V_1\right) ^2=\sum_{a=1}^{n-m}\left( V_1^a\right) ^2$, $\left(
V_2\right) ^2=\sum_{\mu =n-m+1}^{n-1}\left( V_2^\mu \right) ^2$.
Furthermore, 
\begin{equation}
\label{pp3fo55}
\begin{array}{rclc}
\left( 0,1,0,1,e^\phi \right) e_a & = & e_a & . \\ 
\left( V_1,1,0,1,1\right) e_a & = & -\sqrt{2}V_1^a\cdot u_{+}+e_a & . \\ 
\left( 0,1,V_2,1,1\right) e_a & = & e_a & . \\ 
\left( 0,C_1,0,1,1\right) e_a & = & \sum_{b=2}^{n-m}\left( C_1\right)
_{ab}\cdot e_b & . \\ 
\left( 0,1,0,C_2,1\right) e_a & = & e_a & . 
\end{array}
\end{equation}
\begin{equation}
\label{pp3fo56}
\begin{array}{rclc}
\left( 0,1,0,1,e^\phi \right) e_\mu & = & e_\mu & . \\ 
\left( V_1,1,0,1,1\right) e_\mu & = & e_\mu & . \\ 
\left( 0,1,V_2,1,1\right) e_\mu & = & -\sqrt{2}V_2^\mu \cdot u_{+}+e_\mu & .
\\ 
\left( 0,C_1,0,1,1\right) e_\mu & = & e_\mu & . \\ 
\left( 0,1,0,C_2,1\right) e_\mu & = & \sum_{b=n-m+1}^{n-1}\left( C_2\right)
_{\mu \nu }\cdot e_\nu & . 
\end{array}
\end{equation}

We can decompose $\TeXButton{R}{\mathbb{R}}_1^n$ in the $\underline{b}$%
-basis as 
\begin{equation}
\label{pp3fo57}\TeXButton{R}{\mathbb{R}}_1^n=\left[ u_{+}\right] \oplus
\left[ u_{-}\right] \oplus \,\left[ e_2,\ldots ,e_{n-m}\right] \,\,\oplus
U\quad . 
\end{equation}
Accordingly, we write a general element of $\TeXButton{R}{\mathbb{R}}_1^n$
as $X_{+}+X_{-}+X+Y$, where 
\begin{equation}
\label{pp3fo58}X_{+}=x^{+}\cdot u_{+}\quad ;\quad X_{-}=x^{-}\cdot
u_{-}\quad ;\quad X=\sum_{a=2}^{n-m}x^a\cdot e_a\quad ;\quad Y=\sum_{\mu
=n-m+1}^{n-1}y^\mu \cdot e_\mu \quad . 
\end{equation}

\subsection{Preservation of $lat=\left[ u_{+}\right] _{\TeXButton{Z}
{\mathbb{Z}}}\oplus \left[ u_1,\ldots ,u_{m-1}\right] _{\TeXButton{Z}
{\mathbb{Z}}}$ \label{prlat}}

Having identified the group $rot\left( W;SO_{1,n-1}^{+}\right) $ that
preserves the $\TeXButton{R}{\mathbb{R}}$-linear enveloppe 
\begin{equation}
\label{pp3fo64}\left[ lat\right] =\left[ u_{+}\right] _{\TeXButton{R}
{\mathbb{R}}}\oplus \left[ u_1,\ldots ,u_{m-1}\right] _{\TeXButton{R}
{\mathbb{R}}} 
\end{equation}
of $lat$, we eventually can turn to reduce this group down to the set 
\begin{equation}
\label{pp3fo65}rot\left( lat\right) \cap SO_{1,n-1}^{+}\equiv rot_0\quad , 
\end{equation}
where $rot_0$ is that part of the set $rot$ that lies in the identity
component $SO_{1,n-1}^{+}$ of $O_{1,n-1}$. To this end we must restrict the
enveloppe (\ref{pp3fo64}) to the original lattice points 
\begin{equation}
\label{pp3fo66}lat=\left[ u_{+}\right] _{\TeXButton{Z}{\mathbb{Z}}}\oplus
\left[ u_1,\ldots ,u_{m-1}\right] _{\TeXButton{Z}{\mathbb{Z}}}=\left[ 
\underline{u}\right] _{\TeXButton{Z}{\mathbb{Z}}}\quad . 
\end{equation}
Now we ask, which of the elements (\ref{pp3fo43}) in $rot\left( W\right) $
preserve this set; the answer is found in formulas (\ref{pp3fo53}-\ref
{pp3fo56}) :

Elements $\left( V_1,1,0,1,1\right) $ act as identity on $W$ and hence
preserve $lat$ without further restriction. The same is true for elements $%
\left( 0,C_1,0,1,1\right) $. The set of products $\left(
V_1,C_1,0,1,1\right) $ of these forms a semidirect product subgroup of $%
rot\left( W\right) $ isomorphic to the Euclidean group $E_0^{n-m-1}$ in $%
\left( n-m-1\right) $ dimensions.

Elements $\left( 0,1,V_2,1,1\right) $ map $Y\in U$ into $-\sqrt{2}\left(
Y\bullet V_2\right) \cdot u_{+}+Y$. For $Y\in lat$, this is a lattice vector
if and only if $\sqrt{2}\left( Y\bullet V_2\right) $ is an integer. Since $Y$
now has integer components, we find that the components of $V_2$ must be $%
V_2^\mu =\frac{z^\mu }{\sqrt{2}}$, $z^\mu \in \TeXButton{Z}{\mathbb{Z}}$.

Elements $\left( 0,1,0,C_2,1\right) $ act as identity on $u_{+}$; they must
be further restricted to map the sublattice $lat^{\prime }\equiv \left[
u_1,\ldots ,u_{m-1}\right] _{\TeXButton{Z}{\mathbb{Z}}}$ into itself. Since
the basis lattice vectors are $\TeXButton{R}{\mathbb{R}}$-linearly
independent this will be satisfied only for a finite (hence discrete) subset 
$D\subset SO_{m-1}$. Since the sublattice $lat^{\prime }$ is now {\bf %
spacelike}, theorem \ref{Bedingung} implies that $D$ must be a group [indeed 
$D$ now coincides what in crystallography is called the {\it maximal point
group} of the sublattice $lat^{\prime }$].

The set of products $\left( 0,1,V_2,C_2,1\right) $ forms a discrete subgroup 
$G_{discr}$ of the subgroup $E_0^{m-1}\subset rot\left( W\right) $, where $%
E_0^{m-1}$ is isomorphic to the Euclidean group in $\left( m-1\right) $
dimensions.

The main point comes now: Elements $\left( 0,1,0,1,e^\phi \right) $ must be
restricted to $\left( 0,1,0,1,k\right) $, $k\in \TeXButton{N}{\mathbb{N}}$,
in order to satisfy $\left( 0,k,1\right) u_{+}=k\cdot u_{+}\in lat$.
Although the original set of $\left( 0,1,0,1,e^\phi \right) $ with $e^\phi
\in \TeXButton{R}{\mathbb{R}}_{+}$ was a group, the set of all $\left(
0,1,0,1,k\right) $ is a group no longer, but a semigroup isomorphic to the
semigroup $\left( \TeXButton{N}{\mathbb{N}},\cdot \right) $ of all natural
numbers with multiplicative composition $\left( k,k^{\prime }\right) \mapsto
k\cdot k^{\prime }$, and $1$ as unit. Clearly, $\left( 0,1,0,1,k\right) $ is
still an invertible element of $rot_0\subset SO_{1,n-1}^{+}$; however, as
mentioned above, it has no inverse \underline{in $rot_0$}, since $\left(
0,1,0,1,k\right) ^{-1}=\left( 0,1,0,1,\frac 1k\right) $ is {\bf not}
lattice-preserving, as it maps $u_{+}\mapsto \frac 1k\cdot u_{+}\not \in lat$%
, for $k>1$.

The multiplicative structure of $rot_0$ clearly is the same as that of $%
rot\left( W\right) $, and is given by the group law (\ref{pp3fo44}). Hence
we see that $E^{n-m-1}\otimes G_{discr}$ forms a proper subgroup of $rot_0$.

\subsection{The structure and the Lie algebra of $\TeXButton{R}{\mathbb{R}}%
^n\odot rot\left( W\right) $}

According to (\ref{pp3fo15}, \ref{pp3fo16}), the normalizer and extended
normalizer of $\Gamma $ are given as semidirect products of the
translational group $\TeXButton{R}{\mathbb{R}}^n$ and $rot^{\times }$, $rot$%
, respectively. In order to understand better the Lie algebra of these
normalizing sets, we first examine the Lie algebra of the semidirect product
group $\TeXButton{R}{\mathbb{R}}^n\odot rot\left( W\right) $, in which $%
eN\left( \Gamma \right) =\TeXButton{R}{\mathbb{R}}^n\odot rot$ is embedded.
To this end we again restrict attention to elements lying in $SO_{1,n-1}^{+}$%
, which means that we define 
\begin{equation}
\label{pp3fo67}rot\left( W\right) _0\equiv rot\left( W\right) \cap
SO_{1,n-1}^{+}\quad , 
\end{equation}
and now study the group and Lie algebra 
\begin{equation}
\label{pp3fo68}\TeXButton{R}{\mathbb{R}}^n\odot rot\left( W\right) _0\quad
,\quad Lie\left[ \TeXButton{R}{\mathbb{R}}^n\odot rot\left( W\right)
_0\right] \quad . 
\end{equation}
The elements of the full group $\TeXButton{R}{\mathbb{R}}^n\odot rot\left(
W\right) _0$ now must take the form $\left[ T\mid \Lambda \right] $, where $%
T\in \TeXButton{R}{\mathbb{R}}^n$ and $\Lambda \in rot\left( W\right) _0$.
The group law is the same as that in $E_1^n$, $\left[ T\mid \Lambda \right]
\left[ T^{\prime }\mid \Lambda ^{\prime }\right] =\left[ \Lambda T^{\prime
}+T\mid \Lambda \Lambda ^{\prime }\right] $. In order to determine in detail
how the elements of $rot\left( W\right) _0$ act on the translational factor $%
\TeXButton{R}{\mathbb{R}}^n$, i.e. on elements $\left[ T\mid 1\right] $, we
transform the orthogonal basis $\left( P_0,P_1,\ldots ,P_{n-1}\right) $ of
the Lie algebra $\TeXButton{R}{\mathbb{R}}^n$ of the translational group $%
\TeXButton{R}{\mathbb{R}}^n$ into the new basis 
\begin{equation}
\label{pp3fo69}\left( P_{+},P_{-},P_2,\ldots ,P_{n-m},P_{n-m+1},\ldots
,P_{n-1}\right) \quad ;\quad P_{\pm }=\frac{P_0\pm P_1}{\sqrt{2}}\quad , 
\end{equation}
which is defined in analogy with the $\underline{b}$-basis given above, so
that 
\begin{equation}
\label{pp3fo70}\TeXButton{R}{\mathbb{R}}^n=\left[ P_{+}\right] \oplus \left[
P_{-}\right] \oplus \left[ P_2,\ldots ,P_{n-m}\right] \oplus \left[
P_{n-m+1},\ldots ,P_{n-1}\right] \quad . 
\end{equation}
There is a natural (vector space) isomorphism between $M=\TeXButton{R}
{\mathbb{R}}_1^n$ and the translation algebra $\TeXButton{R}{\mathbb{R}}^n$
defined by 
\begin{equation}
\label{pp3fo71}u_{+}\simeq P_{+}\quad ;\quad u_{-}\simeq P_{-}\quad ;\quad
e_a\simeq P_a\quad ;\quad e_\mu \simeq P_\mu \quad . 
\end{equation}
Accordingly, we write a general element $\left[ Tr\mid 1\right] $ of $%
\TeXButton{R}{\mathbb{R}}^n$ as $Tr=T_{+}+T_{-}+T+T_2\equiv \left[
T_{+},T_{-},T,T_2\mid 1\right] $, where 
\begin{equation}
\label{pp3fo72}T_{+}=t^{+}\cdot P_{+}\quad ;\quad T_{-}=t^{-}\cdot
P_{-}\quad ;\quad T=\sum_{a=2}^{n-m}t^a\cdot P_a\quad ;\quad T_2=\sum_{\mu
=n-m+1}^{n-1}t_2^\mu \cdot P_\mu \quad . 
\end{equation}

In the following, conjugation of a group element $g$ by another group
element $h$ will be denoted by $\left( h,g\right) \mapsto cj\left( h\right)
\cdot g\equiv hgh^{-1}$.

We now present the action of elements $\left[ 0\mid \Lambda \right] $ on $%
\left[ Tr\mid 0\right] $ by conjugation, according to 
\begin{equation}
\label{pp3fo73}\left( \left[ 0\mid \Lambda \right] ,\left[ Tr\mid 0\right]
\right) \mapsto cj\left( \left[ 0\mid \Lambda \right] \right) \cdot \left[
Tr\mid 0\right] \equiv \left[ 0\mid \Lambda \right] \left[ Tr\mid 0\right]
\left[ 0\mid \Lambda \right] ^{-1}=\left[ \Lambda Tr\mid 0\right] \quad , 
\end{equation}
which expresses how $rot\left( W\right) _0$ acts on the (normal)
translational factor $\TeXButton{R}{\mathbb{R}}^n$. These relations can be
directly derived from formulas (\ref{pp3fo53}-\ref{pp3fo56}):

The action of $\left[ 0\mid \Lambda \right] $ on translations $\left[
T_{+},0,0,0\mid 1\right] $ along the lightlike direction $u_{+}$ is given by 
\begin{equation}
\label{pp3fo74}
\begin{array}{rclc}
cj\left( \left[ 0\mid V_1,1,0,1,1\right] \right) \,\cdot \left[
T_{+},0,0,0\mid 1\right] & = & \left[ T_{+},0,0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,C_1,0,1,1\right] \right) \cdot \left[
T_{+},0,0,0\mid 1\right] & = & \left[ T_{+},0,0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,V_2,1,1\right] \right) \cdot \left[
T_{+},0,0,0\mid 1\right] & = & \left[ T_{+},0,0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,C_2,1\right] \right) \cdot \left[
T_{+},0,0,0\mid 1\right] & = & \left[ T_{+},0,0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,1,e^\phi \right] \right) \cdot \left[
T_{+},0,0,0\mid 1\right] & = & \left[ e^\phi \cdot T_{+},0,0,0\mid 1\right]
& \quad . 
\end{array}
\end{equation}
The action of $\left[ 0\mid \Lambda \right] $ on time translations $\left[
0,T_{-},0,0\mid 1\right] $ is given by 
\begin{equation}
\label{pp3fo75}
\begin{array}{rclc}
cj\left( \left[ 0\mid V_1,1,0,1,1\right] \right) \cdot \left[
0,T_{-},0,0\mid 1\right] & = & \left[ t^{-}\left( V_1\right) ^2,T_{-},-t^{-}%
\sqrt{2}V_1,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,C_1,0,1,1\right] \right) \cdot \left[
0,T_{-},0,0\mid 1\right] & = & \left[ 0,T_{-},0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,V_2,1,1\right] \right) \cdot \left[
0,T_{-},0,0\mid 1\right] & = & \left[ t^{-}\left( V_2\right)
^2,T_{-},0,-t^{-}\sqrt{2}V_2\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,C_2,1\right] \right) \cdot \left[
0,T_{-},0,0\mid 1\right] & = & \left[ 0,T_{-},0,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,1,e^\phi \right] \right) \cdot \left[
0,T_{-},0,0\mid 1\right] & = & \left[ 0,e^{-\phi }\cdot T_{-},0,0\mid
1\right] & \quad . 
\end{array}
\end{equation}
The action of $\left[ 0\mid \Lambda \right] $ on spacelike translations $%
\left[ 0,0,T,0\mid 1\right] $ on the space part of the subspace $M^{\prime }$
is given by 
\begin{equation}
\label{pp3fo76}
\begin{array}{rclc}
cj\left( \left[ 0\mid V_1,1,0,1,1\right] \right) \cdot \left[ 0,0,T,0\mid
1\right] & = & \left[ -\sqrt{2}\left( T\bullet V_1\right) ,0,T,0\mid
1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,C_1,0,1,1\right] \right) \cdot \left[ 0,0,T,0\mid
1\right] & = & \left[ 0,0,C_1T,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,V_2,1,1\right] \right) \cdot \left[ 0,0,T,0\mid
1\right] & = & \left[ 0,0,T,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,C_2,1\right] \right) \cdot \left[ 0,0,T,0\mid
1\right] & = & \left[ 0,0,T,0\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,1,e^\phi \right] \right) \cdot \left[
0,0,T,0\mid 1\right] & = & \left[ 0,0,T,0\mid 1\right] & \quad . 
\end{array}
\end{equation}
The action of $\left[ 0\mid \Lambda \right] $ on translations $\left[
0,0,0,T_2\mid 1\right] $ on the enveloppe $U$ of the sublattice $lat^{\prime
}$ is given by 
\begin{equation}
\label{pp3fo77}
\begin{array}{rclc}
cj\left( \left[ 0\mid V_1,1,0,1,1\right] \right) \cdot \left[ 0,0,0,T_2\mid
1\right] & = & \left[ 0,0,0,T_2\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,C_1,0,1,1\right] \right) \cdot \left[ 0,0,0,T_2\mid
1\right] & = & \left[ 0,0,0,T_2\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,V_2,1,1\right] \right) \cdot \left[ 0,0,0,T_2\mid
1\right] & = & \left[ -\sqrt{2}\left( T_2\bullet V_2\right) ,0,0,T_2\mid
1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,C_2,1\right] \right) \cdot \left[ 0,0,0,T_2\mid
1\right] & = & \left[ 0,0,0,C_2T_2\mid 1\right] & \quad . \\ 
cj\left( \left[ 0\mid 0,1,0,1,e^\phi \right] \right) \cdot \left[
0,0,0,T_2\mid 1\right] & = & \left[ 0,0,0,T_2\mid 1\right] & \quad . 
\end{array}
\end{equation}

From formulas (\ref{pp3fo74}-\ref{pp3fo77}) we now can derive the Lie
algebra $Lie\left[ \TeXButton{R}{\mathbb{R}}^n\odot rot\left( W\right)
_0\right] $, using the fact that the conjugating elements $\left[ 0\mid
\Lambda \right] $ occuring on the left hand side of equations (\ref{pp3fo74}-%
\ref{pp3fo77}) are exponentials, as follows from (\ref{pp3fo46}-\ref{pp3fo48}%
). We employ standard Lie algebra machinery, 
\begin{equation}
\label{pp3fo78}\frac d{ds}\left. cj\left( g\right) \cdot \exp sX\right|
_{s=0}=cj\left( g\right) _{*}X=Ad\left( g\right) X\quad , 
\end{equation}
and 
\begin{equation}
\label{pp3fo79}\frac d{ds}\left. Ad\left( \exp sY\right) X\right|
_{s=0}=ad\left( Y\right) X=\left[ Y,X\right] \quad , 
\end{equation}
for elements $X,Y$ in the Lie algebra $Lie\left[ \TeXButton{R}{\mathbb{R}}%
^n\odot rot\left( W\right) _0\right] $. The subalgebra of the generators of $%
rot\left( W\right) _0$ has been derived in (\ref{pp3fo39}) already; hence we
consider commutators $\left[ Y,X\right] $, where $Y\in Lie\left[ rot\left(
W\right) _0\right] =Lie\left[ rot\left( W\right) \right] $, and $X\in 
\TeXButton{R}{\mathbb{R}}^n=Lie\left( \TeXButton{R}{\mathbb{R}}^n\right) $,
this space being decomposed according to (\ref{pp3fo70}): 
\begin{equation}
\label{pp3fo80}
\begin{array}{rclcrclc}
\left[ K_a,P_{+}\right] & = & 0 & \quad .\quad & \left[ K_a,P_{-}\right] & =
& -\sqrt{2}P_a & \quad . \\ 
\left[ L_{ab},P_{+}\right] & = & 0 & \quad .\quad & \left[
L_{ab},P_{-}\right] & = & 0 & \quad . \\ 
\left[ K_\mu ,P_{+}\right] & = & 0 & \quad .\quad & \left[ K_\mu
,P_{-}\right] & = & -\sqrt{2}P_\mu & \quad . \\ 
\left[ L_{\mu \nu },P_{+}\right] & = & 0 & \quad .\quad & \left[ L_{\mu \nu
},P_{-}\right] & = & 0 & \quad . \\ 
\left[ \Delta ,P_{+}\right] & = & P_{+} & \quad .\quad & \left[ \Delta
,P_{-}\right] & = & -P_{-} & \quad . 
\end{array}
\end{equation}
\begin{equation}
\label{pp3fo81}
\begin{array}{rclcrclc}
\left[ K_a,P_b\right] & = & -\sqrt{2}\delta _{ab}\cdot P_{+} & \quad .\quad
& \left[ K_a,P_\mu \right] & = & 0 & \quad . \\ 
\left[ L_{ab},P_c\right] & = & \delta _{ac}\cdot P_b-\delta _{bc}\cdot P_a & 
\quad .\quad & \left[ L_{ab},P_\mu \right] & = & 0 & \quad . \\ 
\left[ K_\mu ,P_a\right] & = & 0 & \quad .\quad & \left[ K_\mu ,P_\nu
\right] & = & -\sqrt{2}\delta _{\mu \nu }\cdot P_{+} & \quad . \\ 
\left[ L_{\mu \nu },P_a\right] & = & 0 & \quad .\quad & \left[ L_{\mu \nu
},P_\rho \right] & = & \delta _{\mu \rho }\cdot P_\nu -\delta _{\nu \rho
}\cdot P_\mu & \quad . \\ 
\left[ \Delta ,P_a\right] & = & 0 & \quad .\quad & \left[ \Delta ,P_\mu
\right] & = & 0 & \quad . 
\end{array}
\end{equation}
For better comparison, we present the algebra of the $rot\left( W\right) $%
-factor again: 
\begin{equation}
\label{pp3fo82}
\begin{array}{c}
\left[ \Delta ,K_i\right] =K_i\quad . \\ 
\left[ \Delta ,L_{ab}\right] =\left[ \Delta ,L_{\mu \nu }\right] =0\quad .
\\ 
\left[ K_i,K_j\right] =0\quad . \\ 
\left[ L_{ab},K_c\right] =\delta _{ac}\cdot K_b-\delta _{bc}\cdot K_a\quad .
\\ 
\left[ L_{ab},K_\rho \right] =0\quad . \\ 
\left[ L_{\mu \nu },K_a\right] =0\quad . \\ 
\left[ L_{\mu \nu },K_\rho \right] =\delta _{\mu \rho }\cdot K_\nu -\delta
_{\nu \rho }\cdot K_\mu \quad . \\ 
\left[ L_{ab},L_{cd}\right] =\delta _{ac}\cdot L_{bd}+\delta _{bd}\cdot
L_{ac}-\delta _{ad}\cdot L_{bc}-\delta _{bc}\cdot L_{ad}\quad . \\ 
\left[ L_{\mu \nu },L_{\gamma \delta }\right] =\delta _{\mu \gamma }\cdot
L_{\nu \delta }+\delta _{\nu \delta }\cdot L_{\mu \gamma }-\delta _{\mu
\delta }\cdot L_{\nu \gamma }-\delta _{\nu \gamma }\cdot L_{\mu \delta
}\quad . \\ 
\left[ L_{ab},L_{\mu \nu }\right] =0\quad . 
\end{array}
\end{equation}

\subsection{The structure and the Lie algebra of $rot_0$}

From (\ref{pp3fo40}) we can now read off the form of $rot_0$, 
\begin{equation}
\label{pp3fo83}rot_0\simeq \left[ E^{n-m-1}\otimes G_{discr}\right] \odot
\left( \TeXButton{N}{\mathbb{N}},\cdot \right) \quad , 
\end{equation}
\begin{equation}
\label{pp3fo84}E^{n-m-1}=\TeXButton{R}{\mathbb{R}}^{n-m-1}\odot
O_{n-m-1}\quad . 
\end{equation}
In analogy with (\ref{pp3fo65}) we furthermore introduce the subgroup $%
rot_0^{\times }$ of the $lat$-invertible transformations that lie in $%
SO_{1,n-1}^{+}$ as 
\begin{equation}
\label{pp3fo85}rot_0^{\times }\equiv rot^{\times }\cap SO_{1,n-1}^{+}\quad . 
\end{equation}
This set contains all Lorentz transformations $R$ belonging to the identity
component of $SO_{1,n-1}$ that preserve the lattice $lat$, such that the
same is true for $R^{-1}$. From the analysis above it is now clear that $%
rot_0^{\times }$ is isomorphic to 
\begin{equation}
\label{pp3fo86}rot_0^{\times }\simeq E^{n-m-1}\otimes G_{discr}\quad . 
\end{equation}
Hence it is the dilations in $\left( \TeXButton{N}{\mathbb{N}},\cdot \right) 
$ that constitute the extension from $rot_0^{\times }$ to $rot_0$, and we
have 
\begin{equation}
\label{pp3fo87}rot_0=rot_0^{\times }\odot \left( \TeXButton{N}{\mathbb{N}}%
,\cdot \right) 
\end{equation}
in this case; i.e., a semidirect product of a group and a semigroup.

The connected component $rot_{00}$ of $rot_0$ can be read off from (\ref
{pp3fo83}); it coincides with the connected component $\left( rot_0^{\times
}\right) _0$ of $rot_0^{\times }$, and is given by 
\begin{equation}
\label{pp3fo88}rot_{00}=\left( rot_0^{\times }\right) _0\simeq
E_0^{n-m-1}\quad . 
\end{equation}
Hence, we have the Lie algebras%
$$
Lie\left( rot\right) =Lie\left( rot_0\right) =Lie\left( rot_{00}\right)
=Lie\left( rot^{\times }\right) =Lie\left( rot_0^{\times }\right) =Lie\left(
rot_0^{\times }\right) _0\simeq 
$$
\begin{equation}
\label{pp3fo89}\simeq euc_0^{n-m-1}\quad . 
\end{equation}

We now can turn eventually to the extended normalizer $eN\left( \Gamma
\right) =\TeXButton{R}{\mathbb{R}}^n\odot rot$, and $N\left( \Gamma \right) =%
\TeXButton{R}{\mathbb{R}}^n\odot rot^{\times }$, as given in (\ref{pp3fo15},%
\ref{pp3fo16}). However, as in the previous subsections, we want to focus on
elements that are in the connected component $SO_{1,n-1}^{+}$. Following (%
\ref{pp3fo65},\ref{pp3fo85}), we accordingly define 
\begin{equation}
\label{pp3fo90}eN\left( \Gamma \right) _0\equiv eN\left( \Gamma \right) \cap
SO_{1,n-1}^{+}=\TeXButton{R}{\mathbb{R}}^n\odot rot_0\quad , 
\end{equation}
\begin{equation}
\label{pp3fo91}N\left( \Gamma \right) _0\equiv N\left( \Gamma \right) \cap
SO_{1,n-1}^{+}=\TeXButton{R}{\mathbb{R}}^n\odot rot_0^{\times }\quad , 
\end{equation}
with identity components 
\begin{equation}
\label{pp3fo92}\left[ eN\left( \Gamma \right) _0\right] _0=\left[ N\left(
\Gamma \right) _0\right] _0=\TeXButton{R}{\mathbb{R}}^n\odot rot_{00}\simeq 
\TeXButton{R}{\mathbb{R}}^n\odot E_0^{n-m-1}\quad . 
\end{equation}
The quotient $eN\left( \Gamma \right) _0/\Gamma $ then takes the form 
\begin{equation}
\label{pp3fo94}eN\left( \Gamma \right) _0/\Gamma =\TeXButton{R}{\mathbb{R}}%
^n/\Gamma \odot rot_0=\left[ \TeXButton{R}{\mathbb{R}}^{n-m}\otimes
T^m\left( R_{+},R_{n-m+1},\ldots ,R_{n-1}\right) \right] \odot rot_0\quad , 
\end{equation}
where $\TeXButton{R}{\mathbb{R}}^{n-m}=\left[ P_{-},P_2,\ldots
,P_{n-m}\right] $, and%
$$
T^m\left( R_{+},R_\mu \right) =T^m\left( R_{+},R_{n-m+1},\ldots
,R_{n-1}\right) \equiv 
$$
\begin{equation}
\label{pp3fo96}=\left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}}/\left[
u_{+}\right] _{\TeXButton{Z}{\mathbb{Z}}}\otimes \left[ u_{n-m+1}\right] _{%
\TeXButton{R}{\mathbb{R}}}/\left[ u_{n-m+1}\right] _{\TeXButton{Z}
{\mathbb{Z}}}\cdots \otimes \left[ u_{n-1}\right] _{\TeXButton{R}{\mathbb{R}}%
}/\left[ u_{n-1}\right] _{\TeXButton{Z}{\mathbb{Z}}}\quad , 
\end{equation}
where we have written 
\begin{equation}
\label{pp3fo97}R_A\equiv \frac{\left\| u_A\right\| _E}{2\pi }\quad ,\quad
\left\| u_A\right\| _E\equiv \sqrt{\sum_{i=0}^{n-1}\left| u_A^i\right| }%
\quad ,\quad A\in \left\{ 0,n-m+1,\ldots ,n-1\right\} 
\end{equation}
i.e. $\left\| u_A\right\| _E$ here is the {\bf Euclidean} norm of $u_A$, and 
$R_A$ are the radii of the associated circles $\left[ u_A\right] _{%
\TeXButton{R}{\mathbb{R}}}/\left[ u_A\right] _{\TeXButton{Z}{\mathbb{Z}}}$. $%
T^m$ is the translation group of a torus $\TeXButton{R}{\mathbb{R}}^m/\Gamma 
$ which we will denote by the same symbol $T^m\left( R_{+},R_\mu \right) $;
coordinates $\left( x^{+},x^{-},x^a,y^\mu \right) $ on this latter torus can
be obtained from (\ref{pp3fo58}). The quotient $\TeXButton{R}{\mathbb{R}}%
^m\rightarrow \TeXButton{R}{\mathbb{R}}^m/\Gamma $ clearly preserves Lie
algebras, so that $Lie\left( T^m\right) =\TeXButton{R}{\mathbb{R}}^m$. $T^m$
acts on $T^m$ as%
$$
\exp \left( t^{+}\cdot P_{+}+t^{-}\cdot P_{-}+\sum_{a=2}^{n-m}t^a\cdot
P_a+\sum_{\mu =n-m+1}^{n-1}t_2^\mu \cdot P_\mu \right) \left(
x^{+},x^{-},x^a,y^\mu \right) = 
$$
\begin{equation}
\label{pp3fo98}=\left( t^{+}+x^{+},t^{-}+x^{-},t^a+x^a,y^\mu +t_2^\mu
\right) \;{\rm mod}\;lat\quad . 
\end{equation}
The full quotient $M/\Gamma $ is $M/\Gamma =\TeXButton{R}{\mathbb{R}}%
^{n-m}\times T^m$. As was explained in section \ref{Sc.5}, the metric on $%
T^{m-1}\left( R^\mu \right) $ is positive definite, whereas it is
identically zero on $T\left( R_{+}\right) =\left[ u_{+}\right] _{%
\TeXButton{R}{\mathbb{R}}}/\left[ u_{+}\right] _{\TeXButton{Z}{\mathbb{Z}}}$.

We now turn to the Lie algebra of $\left[ eN\left( \Gamma \right) _0\right]
_0/\Gamma $. Since $\Gamma $ is a discrete normal subgroup of the connected
component $\left[ eN\left( \Gamma \right) _0\right] _0$, the groups $\left[
eN\left( \Gamma \right) _0\right] _0$ and $\left[ eN\left( \Gamma \right)
_0\right] _0/\Gamma $ are locally isomorphic, hence \cite{SagleWald} they
possess isomorphic Lie algebras. This implies%
$$
Lie\left\{ \left[ eN\left( \Gamma \right) _0\right] _0/\Gamma \right\}
\simeq Lie\left\{ eN\left( \Gamma \right) _0/\Gamma \right\} \simeq
Lie\left\{ N\left( \Gamma \right) _0/\Gamma \right\} \simeq Lie\left\{
N\left( \Gamma \right) /\Gamma \right\} \simeq 
$$
\begin{equation}
\label{pp3fo99}\simeq Lie\left\{ \left[ eN\left( \Gamma \right) _0\right]
_0\right\} \simeq Lie\left\{ N\left( \Gamma \right) \right\} \simeq
Lie\left\{ I\left( M/\Gamma \right) \right\} \simeq Lie\left\{ \TeXButton{R}
{\mathbb{R}}^n\odot E_0^{n-m-1}\right\} \quad . 
\end{equation}
This algebra is spanned by 
\begin{equation}
\label{pp3fo100}\left( P_{+},P_{-},P_a,P_\mu \mid K^a,L_{ab}\right) 
\end{equation}
subject to the relations (\ref{pp3fo80}-\ref{pp3fo82}). We rewrite these
relations in a slightly different form; to this end we redefine 
\begin{equation}
\label{pp3fo101}K_a\mapsto \sqrt{2}K_a\quad ;\quad P_{-}\mapsto -P_{-}\quad
. 
\end{equation}
Then 
\begin{equation}
\label{pp3fo102}\left[ K_a,P_\mu \right] =\left[ L_{ab},P_\mu \right]
=0\quad , 
\end{equation}
and 
\begin{equation}
\label{pp3fo103}
\begin{array}{rclcrclc}
\left[ K_a,P_{+}\right] & = & 0 & \quad .\quad & \left[ K_a,P_{-}\right] & =
& P_a & \quad . \\ 
\left[ L_{ab},P_{+}\right] & = & 0 & \quad .\quad & \left[
L_{ab},P_{-}\right] & = & 0 & \quad . \\ 
\left[ K_a,P_b\right] & = & -\delta _{ab}\cdot P_{+} & \quad .\quad & \left[
L_{ab},P_c\right] & = & \delta _{ac}\cdot P_b-\delta _{bc}\cdot P_a & \quad
. 
\end{array}
\end{equation}
\begin{equation}
\label{pp3fo104}
\begin{array}{c}
\left[ K_i,K_j\right] =0\quad . \\ 
\left[ L_{ab},K_c\right] =\delta _{ac}\cdot K_b-\delta _{bc}\cdot K_a\quad .
\\ 
\left[ L_{ab},L_{cd}\right] =\delta _{ac}\cdot L_{bd}+\delta _{bd}\cdot
L_{ac}-\delta _{ad}\cdot L_{bc}-\delta _{bc}\cdot L_{ad}\quad . 
\end{array}
\end{equation}
This defines a direct sum of Lie algebras,
\begin{equation}
\label{pp3fo104a}A\oplus gal_{ce}^{n-m-1}\quad .
\end{equation}
Here $A$ is an Abelian Lie algebra isomorphic to $\TeXButton{R}{\mathbb{R}}%
^{m-1}$ with generators $\left( P_\mu \right) $. This algebra will play no
further role in what we discuss in the remainder of the paper. The second
algebra has generators $\left( P_{+},P_{-},P_a\mid K^a,L_{ab}\right) $ and
is isomorphic to the centrally extended Galilean algebra $gal_{ce}^{n-m-1}$
in $\left( n-m-1\right) $ dimensions, where $P_{-}$
generates ''time'' translations, $P_a$ generate ''space'' translations, $K_a$
generate Galilei boosts, $L_{ab}$ generate rotations, and $P_{+}$ is the
''mass generator'' spanning a $1$-dimensional central extension, in the
following denoted by $gal_{ce}^{n-m-1}$, of the unextended Galilei algebra $%
gal^{n-m-1}$. . If the generators for $\tau $-, $T$-, $V$-, $R$%
-transformations in the Galilei algebra are denoted by $P_{-}$, $P_a$, $K_a$%
, $L_{ab}$, respectively, we find that these generators satisfy (\ref
{pp3fo102}-\ref{pp3fo104}). Furthermore, if $P_{+}$ denotes the generator of
the central extension $gal_{ce}$ of the Noether charge algebra carried by a
point particle of mass $m$, then the relations of $P_{+}$ with the remaining
generators are given by the left block of formulas (\ref{pp3fo103}), where
the last equation is to be replaced by 
\begin{equation}
\label{pp3fo105}\left[ K_a,P_b\right] =-m\delta _{ab}\cdot P_{+}\quad , 
\end{equation}
which is why $P_{+}$ is called a mass generator (see, e.g. \cite{Azcarraga}).

\section{The effect of semigroup transformations \label{Sc.6}}

Finally, we briefly discuss how the semigroup elements $\Phi \equiv \left(
0,1,0,1,k\right) $ act on the compactified spacetime $\TeXButton{R}
{\mathbb{R}}^{n-m}\times T^m$, and on quantum fields defined on such a
spacetime. From subsection \ref{prlat} we see that the map $\Phi $ acts
trivially on $x^a$-coordinates; on the other hand, the pair $\left(
x^{+},x^{-}\right) $ of coordinates in the $u_{\pm }$-direction is mapped
into $\left( kx^{+},\frac 1kx^{-}\right) $; it corresponds to contraction in
the ''time'' coordinate $x^{-}$, and, more important, to a dilation $%
x^{+}\mapsto kx^{+}$. As $x^{+}$ takes values in $\left[ 0,2\pi R_{+}\right] 
$, its image under $\Phi $ therefore winds $k$ times around the lightlike $%
S_1$-factor. The $2$-cylinder $\left[ u_{-}\right] _{\TeXButton{R}
{\mathbb{R}}}\times \left[ u_{+}\right] _{\TeXButton{R}{\mathbb{R}}}/\left[
u_{+}\right] _{\TeXButton{Z}{\mathbb{Z}}}$ therefore gets contracted and $k$
times wound around itself; this means that the mass generator $P_{+}$ should
correspond to a topological mass term in this context.

If $\phi :\left[ 0,2\pi R_{+}\right] \times \TeXButton{R}{\mathbb{R}}%
^{n-m-1}\times T^{m-1}\rightarrow tgt$ is a field on the compactified
spacetime taking values in some target space $tgt$, and ${\cal L}={\cal L}%
\left[ \phi \right] $ is a Lagrangian governing its dynamics, then 
\begin{equation}
\label{pp3fo106}\int\limits_{\left[ 0,2\pi R_{+}\right] }dx^{+}\int
dx^{-}dx^i\cdot {\cal L}\left[ \phi \right] \;=\;\int\limits_{\left[ 0,\frac{%
2\pi R_{+}}k\right] }dx^{+}\int dx^{-}dx^i\cdot \Phi ^{*}{\cal L}\left[ \phi
\right] \quad , 
\end{equation}
where $\Phi ^{*}{\cal L}\left[ \phi \right] $ is the pullback of ${\cal L}$
to the space $\left[ 0,\frac{2\pi R_{+}}k\right] \times \TeXButton{R}
{\mathbb{R}}^{n-m-1}\times T^{m-1}$. (\ref{pp3fo106}) therefore shows the
important result that, although $\Phi $ is originally a map that preserves
the lattice $lat$, and hence the spacetime $\left[ 0,2\pi R_{+}\right]
\times \TeXButton{R}{\mathbb{R}}^{n-m-1}\times T^{m-1}$, $\Phi $
nevertheless induces a map on actions so as to map a theory on a spacetime
with lightlike compactification radius $R_{+}$ to a theory with the smaller
radius $\frac{R_{+}}k$. This operation corresponds to finite discrete
transformations associated with the ''mass'' generator $P_{+}$, which
commutes with all observables in the Galilei algebra and hence is a
superselection operator for the (spacetime degrees of freedom of the)
theory. This means that it labels different, non-coherent, subspaces of
physical states in the overall Hilbert space of the system; amongst these
different superselection sectors, the superposition principle is no longer
valid. It therefore would seem that the $\Phi $-map, when applied to
actions, relates different superselection sectors of the theory. From the
non-invertibility of $\Phi $ on the lattice we deduce that this is a one-way
operation.

\begin{thebibliography}{9}

\bibitem{Fulton}  William Fulton, ''Algebraic Topology''. Springer Verlag,
1995.

\bibitem{Jhnich}  Klaus J\"ahnich, ''Topology''. Springer-Verlag, 1980.

\bibitem{Massey}  William S. Massey, ''A Basic Course in Algebraic
Topology''. Springer-Verlag, 1991.

\bibitem{ONeill}  Barrett O'Neill, ''Semi-Riemannian Geometry with
Applications to Relativity''. Academic Press, 1983.

\bibitem{SagleWald}  Arthur A. Sagle, Ralph E. Walde, ''Introduction to Lie
Groups and Lie Algebras''. Academic Press, 1973.

\bibitem{Corn1}  J.F. Cornwell, ''Group Theory in Physics'', vol.1.
Academic Press, 1984.

\bibitem{Wolf}  J.A. Wolf, ''Spaces of Constant Curvature''. McGraw Hill,
1967.

\bibitem{Azcarraga}  Jose de Azcarraga and Jose M. Izquierdo, ''Lie groups,
Lie algebras, cohomology and some applications in physics''. Cambridge
University Press, 1995.
\end{thebibliography}

\end{document}
