%Date: Tue, 26 May 1998 14:47:31 +0200 (METDST)
%From: Arnold Vladimir <arnold@pi.ceremade.dauphine.fr>   

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\begin{document}

\title{ Relatives of the quotient of the complex projective plane by the
complex cojugation}
\author {V.I.Arnold \thanks {Partially supported by RBRF 96-01-01104
and  by the Institut universitaire de France}}
\date{}
\maketitle

\def\CC{{\bf C}}   
\def\RR{{\bf R}} 
\def\HH{{\bf H}} 
\def\P{\mbox{\rm P}} 
\def\conj{\mbox{\rm conj}} 
\def\Conj{\mbox{\rm Conj}}
\def\Aut{\mbox{\rm Aut}}
 
\begin{abstract}

It is proved, that the quotient space of the four-dimensional quaternionic
projective space by the automorphism group of the
quaternionic algebra becomes the 13-dimensional sphere
while quotioned  the the quaternionic conjugation.

This fact and its various generalisations are proved
using the results of the theory of the hyperbolic
partial differential equations, providing
also the proof  of the theorem (which was, it seems,
known to L.S.Pontriagin already in the thirties)
claiming that the quotient of the complex
projective plane by the complex conjugation is
the 4-sphere.

\end{abstract}

\section{Introduction}

In the paper {\cite{1}} on the topology of the real algebraic curves,
written in $1971,$ I have used the fact, that the quotient
space of the complex projective plane by complex
conjugation is the four-dimensional sphere. The
attempts to find a reference for this fact
in the literature were not successful at this time 
{\footnote {among others I had asked N.Kuiper whether
he knew such a reference}}.
However V.A.Rokhlin told me that this result had been
known to L.S.Pontriagin already in the thirties.

I do not know how had Pontriagin proved this theorem.
My proof (published later in{\cite 2} ) was based on the
theory of the hyperbolic partial differential equations.
Thinking on this proof recently once more from the view point
of the systematical complexification and quaternionisation of the results
of {\cite{3}} ( related to the attempts to 
find the quaternionic version of the Berry phase and on the quantum Hall effect,
see {\cite{4,5}}) I have realised that this proof, based on
the complex numbers, has quaternionic and real versions,
leading to the following strange trinity of theorems with a common proof
(of which the first theorem is trivial while the third one
seems to be new):

\bigskip

{\bf Theorem {\RR.}} {\it The real one-dimensional projective space is
homeomorphic to the circle:}
$$ {\RR}\P^1 \approx S^1 .$$

\bigskip

{\bf Theorem {\CC.}} {\it The quotient space of the complex 
projective plane by the complex conjugation is homeomorphic
to the four-dimensional sphere:}
$$ {\CC}\P^2 /{\conj} \approx S^4 .$$

\bigskip

{\bf Theorem {\HH.}} {\it The quotient space of the  four-dimensional
quaternionic space by the automorphisms of the algebra of
the quaternions becomes, after a factorisation by
the quaternionic involution, a $13$-dimensional sphere:}
$$ 
({\HH}\P^4 /{\Aut}\HH) / {\Conj} \approx S^{13} .
$$

\bigskip

{\bf Remark.} I have no doubt that the homeomorphisms constructed below become
diffeomorphisms, if the quotients are equiped with their natural smooth 
structures. However the
description
of these structures is too long and so the diffeomorphisms 
are not proved below (see, however, {\cite {6}}-{\cite {9}}).

\section{Geometry of hyperbolic cones}

{\bf Definition.} A real algebraic hypersurface in the real
projective space is called {\it hyperbolic} with respect to some
point $P$ of its complement, if every real straight line
containing $P$ intersects the hypersurface only in real points
(whose number is equal to the degree of the hypersurface
provided that they are counted taking multiplicities into account).

The point $P$ is then called a {\it time-like point}. The
hyperbolic hypersurface of the projective space $ {\RR}\P^{N -1}$
are represented in the vector-space $ {\RR}^{N}$(whose vectors
define the homogeneous coordinates in the projective 
space) by the algebraic hyperbolic cones (with vertices at the origin).
The time-like points are represented by the {\it time-like 
vectors} of this space. The hyperbolicity condition takes
in the vector-space the following form: every real straight
line in $ {\RR}^{N}$, parallel to the time-like vector, intersects the 
hyperbolic cone
at the real point, whose number is equal to the degree of the cone.

\bigskip

{\bf Example {1}.} Consider the space of the quadratic forms in $ {\RR}^{n}$.
This real vector-space of dimension $N = n(n+1)/2$ contains the cone of
the degenerate forms. This cone is hyperbolic with respect to
the time-like vector, representing any positive-definite quadratic
form (i.e. any Euclidean structure in $ {\RR}^{n}$). This property of the 
cone is essentially the theory of the eigenvalues of the quadratic
forms in the Euclidean space. The singular points of the cone form an algebraic 
variety of codimension
$3$ in the ambient space of forms $ {\RR}^{N}$.

\bigskip

{\bf Example {2}.} Consider the space of the {\it Hermitian quadratic 
forms} in $ {\CC}^{n}$ (of the real quadratic forms in the corresponding
space $ {\RR}^{2n}$, which are invariant under the action of the group $S^1$ of
complex numbers of modulus one, acting as the multiplication of vectors of $ 
{\CC}^{n}$).

This real vector-space of dimension $N= n^2$ contains the cone of
the degenerate forms. This cone is hyperbolic with respect to
the time-like vector, representing any positive-definite Hermitian form 
(i.e. any Hermitian structure in $ {\CC}^{n}$). This property of the cone
 is essentially the theory of the eigenvalues of the Hermitian forms 
in the Hermitian space. The singular points of the cone form an algebraic 
variety
of codimension $4$ in the ambient space of forms,  $ {\RR}^{N}$.

\bigskip

{\bf Example {3}.} Consider the space of the hyperhermitian quadratic forms 
in the quaternionic $n$-dimensional vector-space ${\HH}^{n}$. These are real 
quadratic forms in the
corresponding space $ {\RR}^{4n}$, which are invariant under the 
action of the group $S^3$ of unitary quaternions (acting on the 
vectors multiplying each component by the quaternion
from some side, chosen once forever, say from the left).

This real vector-space of dimension $N= n(2n-1)$ contains the 
cone of the degenerate forms. This cone is hyperbolic with 
respect to the time-like vector, representing any positive-definite 
hyperhermitian form (i.e. any hyperhermitean structure in $ {\HH}^{n}$). 
This property of the cone is essentially the theory of the eigenvalues of the 
hyperhermitian forms. The singular points of the cone form an algebraic variety
of codimension $6$ in the ambient space $ {\RR}^{N}$ of forms.

The main fact of the geometry of the hyperbolic cones that we
shall use is the following theorem (see {\cite {2}}).

Consider the intersection of an hyperbolic cone of degree
$k$ in $ {\RR}^{N}$ with a sphere $S^{N-1}$ centered at the origin.
Fix a time-like point $P$ (a pole) on the sphere and
connect it with $-P$ by the meridians. Consider on each
meridian the first intersection point with the cone (counting
from $P$), the second one, ..., the $k$-th one.

\bigskip

{\bf Theorem.} {\it The $i$-th points situated on all
the meridians, form a semialgebraic variety $V_i$, homeomorphic
to the sphere $S^{N-2}$.}

\bigskip

Indeed, the homeomorphism of $V_i$ onto the equator is defined by the
projection along the meridians.

The proofs of the results formulated in the introduction
are deduced below from the general theorem above, applied to
the variety $V_1$ of the nonnegative degenerate quadratic
forms of a suitable number of variables.

\section{The Gram matrix of a system of vectors.}

Consider $n$ vectors $v_1,\dots ,v_n$ in an Euclidean
space. The {\it Gram matrix} of this system of vectors is
the matrix $a_{i,j} = (v_i, v_j)$ formed by their scalar
products.

\bigskip

{\bf Lemma.} {\it The Gram matrix of a system of vectors is the
matrix of a nonnegative quadratic form, whose rank is equal
to the dimension of the linear hull of the vectors.

Each matrix of a nonnegative quadratic form of rank
$r$ is realizable as the Gram matrix of a system of $n$ vectors
in an $r$-dimensional Euclidean space.

The ordered system of $n$ vectors is defined by its
Gram matrix up to the orthogonal transformation of
the ambient space $ {\RR}^{r}$.}

\bigskip

The proof of this well-known fact is based on the
representation of the orthogonalization of a given basis
of an Euclidean space in terms of its Gram matrix.
The quadratic form, corresponding to the given system
of vectors, is provided by the formula $f(c)= (\sum c_i v_i, \sum c_i v_i)$.
The Lemma implies the

\bigskip

{\bf Corollary.} {\it The variety $V_1$ of the nonnegative degenerate
quadratic forms in $ {\RR}^{n}$ (normed by the condition ``the sum of the
squares of the components of the matrix of the form in
a fixed basis is equal to $1$'') is homeomorphic to the variety
of the ordered sets of $n$ vectors in the Euclidean space $ {\RR}^{n-1}$
(normed by the condition ``the sum of the squares of the scalar
products of the vectors is equal to $1$''), considered up to the
orthogonal transformations of the space $ {\RR}^{n-1}$.}

\bigskip

Applying to $V_1$ the Theorem of Section $2$, we get the following

\bigskip
 
{\bf Theorem.} {\it The variety of the ordered normed sets of $n$ 
vectors in $ {\RR}^{n-1}$, considered up to the orthogonal
transformations of this Euclidean space, is naturally identified
with the sphere $S^M$, $M=n(n+1)/2 - 2$.}

\bigskip

The Theorems of Section $1$ are deduced from this fact,
$n$ being $2,3$ or $5$ ($ {\RR}^{n-1}$ beeing,
correspondingly, $\RR, \CC, \HH$).

\section{The proof of the theorem on the quotient
of the quaternionic projective space.}

Starting from five quaternions $q_0,\dots , q_4 \in \HH^5$, we
construct a nonnegative degenerate quadratic form of five real variables,
$f(c)= (\sum c_i q_i, \sum c_i q_i)$ --- the scalar square of the
linear combination of the five quaternions.

By the Theorem of Section $3$, the variety of the normed
(and hence nonzero) quintuples, quotioned by the action of
the orthogonal group $O(4)$ on $\HH$, is the sphere $S^{13}$.
The quotient space may be constructed in three steps.

\bigskip

{\bf Lemma.} {\it The quotient space $S^{4m+3}/O(4)$ of the
sphere in $\HH^{m+1}$ by the action of the group $O(4)$ on
$\HH$ can be obtained from the quaternionic projective 
space ${\HH}\P^m$ taking first the quotient by the action
of the automorphisms of the quaternions algebra and
than the quotient of the resulting space by the
action of the quaternionic conjugation involution on it:}
$$
S^{4m+3}/O(4) \approx ({\HH}\P^m /{\Aut}\HH) / {\Conj} .
$$

\bigskip

{\bf Proof.} Denote by $S^3 (=SU(2))$ the group of the unitary quaternions.
The direct product $S^3\times S^3$ acts orthogonally on the space $\HH$ of
the quaternions by the left and right multiplications, $q \mapsto uqv^{-1}$.
We have constructed a homomorphism $S^3\times S^3 \rightarrow SO(4) $ which
is the standard universal two-fold covering $Spin(4) \rightarrow SO(4)$,
whose kernel consists of two elements  $(1,1)$ and $(-1,-1)$.

The quaternionic projective space ${\HH}\P^m$ consists of the
equivalence classes of the (nonzero) $m+1$-tuples of the
quaternons $(uq_0, \dots,uq_m) \in \HH^{m+1} \setminus 0$. The automorphisms
of the algebra of quaternions are interior automorphisms (of
the form $q \mapsto uqu^{-1}, u \in S^3$). Such an automorphism
sends the equivalence class of a vector $q \in \HH^{m+1} \setminus 0$ in an
equivalence class (of the vector $qu^{-1}$), since the left
and the right multiplications commute. Thus, $S^{4m+3}/SO(4) \approx {\HH}\P^m 
/{\Aut}\HH$.

The quaternionic conjugation $\Conj$ does {\it not} act on ${\HH}P^m,$ but does 
act 
on the space of the two-sided
equivalence classes that we have constructed:
the element
$ {\Conj}(uqv^{-1})={\Conj}(v^{-1}){\Conj}(q){\Conj}(u)$
belongs to the class of ${\Conj}(q).$


The orthogonal transformation ${\Conj}: {\HH}\rightarrow {\HH} $
changes the orientation of this real 4-space. Hence we find that
$$
 ( {\HH}\P^m/{\Aut}{\HH})/{\Conj}\approx S^{4m+3} / O(4) , 
$$
which proves the lemma.

Applying the Lemma to the case $ m=4,$ we find from the Theorem  of  Section 3,
that
$$
 S^{13}\approx S^{19}/O(4) \approx ({\HH}\P^4/{\Aut}{\HH})/{\Conj}, 
$$
which is the Theorem {\HH} of the Introduction.

Theorems {\RR} and ${\CC}$ are proved by the same reasoning,
one should only replace the quaternions by the real or by the complex
numbers and replace the projective four-space by the one-space
or by the two-space accordingly.

The quotient space by the action of $O(4)$ will be replaced
by the quotient space by the action of $O(1)$ and of $O(2)$
respectively. The real (complex) projective space that one gets at
the first step should be quotient by the action of $O(1)/S^0 = 1$ 
in the real case and of $O(2)/S^1 = S^0 $ in the complex case
(the nontrivial element of the last group of two elements acts as the
complex conjugation on ${\CC}\P^2).$

\section{ The hyperhermitian geometry}

Replacing the quadratic forms by the Hermitian or by the
hyperhermitian ones in the Theorem of Section 3, one gets new
generalisation of the theorems proved above.

\bigskip

{\bf Definition.} An {\it Hermitian form} in a complex vector-space is
a real quadratic form on it, which is invariant under
the multiplication
of the vector of this complex space by the complex
numbers whose modulus is equal to one. The dimension
of the real vector-space of the Hermitian forms in 
${\CC}^m$ is equal to $m^2.$

A positive-definite Hermitian form is called an
{\it Hermitian structure.}
The complex linear transformations, preserving the Hermitian
structure, form the {\it unitary group} $U(m)$ of  dimension
$m^2.$ A complex vector-space equiped with an Hermitian
structure is called the{\it Hermitian space}.

\bigskip

{\bf Theorem.} {\it The variety of the ordered normed $n$-tuples
of vectors in the Hermitian space $\CC^{n-1},$ considered up to
the unitary transformations of this space, is naturally identified with
the sphere $S^M,  M = n^2 - 2.$}

\bigskip

{\bf Example.} For $n=2$ we get the Hopf fibration
$S^2 =  S^3/S^1.$
For $n=3$ we obtain the  quotient space ( which is no longer the
base of a fibration)
$$
 S^7 \approx S^{11}/U(2) = {\CC}\P^5 /SU(2) = {\HH}\P^2 / S^1. 
$$

\bigskip

{\bf Definition.} A {\it hyperhermitian form} in the quaternionic vector-space 
is a real
quadratic form, which is invariant under
the multiplication of all the quaternionic components of
the vector from the left by any unitary quaternion.

\bigskip

{\bf Remark.} It does {\it not} imply in general the invariance under the
multiplication from the right.

\bigskip

The positive-definite hyperhermitian form  are called {\it hyperhermitian 
structures,}
and the quaternionic space equiped with a
 hyperhermitian structure is called the
 {\it hyperhermitian space.}

\bigskip
 
 {\bf Example.}  The function $ F(q) = q_1 \bar q_1 + \dots + q_m \bar q_m $ 
defines
 in ${\HH}^m$ {\it the standard hyperhermitian structure.} Each hyperhermitian 
structure can
 be written in such a form in terms of the components with
 respect to a suitable basis. Fixing the basis, we get for the hyperhermitian 
forms the
 following description.
 
\bigskip

 {\bf Theorem.} {\it Every hyperhermitian form in ${\HH}^m$ can be represented
 using a ( uniquely defined) quaternionic matrix $ f_{i,j} ,$ verifying the 
hyperhermiticy condition
 
 $f_{j,i} = \bar f_{i,j}, $ by the formula}
 
 $$
 \begin{array}{rr}
 \;\;\;\;f(q) = \sum \limits_{i,j=1}^m q_i F_{i,j}\bar q_j. & 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)   
 \end{array}
 $$
 
 \bigskip
 
{\bf Proof.}  $1^o.$  {\it This form is real.} Indeed, $\bar f(q) = \sum q_j 
\bar f_{i,j} \bar q_i = f(q). $


$2^o$. {\it This form is invariants under the multiplication of $q$ from the 
left by a
unitary quaternion $u.$}  Indeed, 

$$
f(uq) =\sum u q_i f_{i,j} \bar q_j \bar u = u f(q) \bar u = f(q),
$$
since the real number $f(q)$ commutes with $u.$

$3^o.$ Thus formula $(1)$ defines a hyperhermitian form.
We shall prove now, that {\it every hyperhermitian form
can be represented by formula $(1).$}

To prove it equip with the usual Euclidean scalar products 
$(p,q)= {\rm {\mbox Re}}(p \bar q)$  the spaces ${\HH}$ and
${\HH}^m = \oplus {\HH}^1_i .$
Our form can be written as
$$
f(q) =\sum (A_{i,j}(q_i), q_j),\;\;\;\;\;\;\;\;\; A^*_{i,j} = A_{j,i}\;,
$$
where $A_{i,j} : \HH \rightarrow \HH $ is a real-linear operator,
the sign $*$ being Euclidean conjugation.
The invariancy condition for $f$ takes the form
$$
(A_{i,j}(ux), uy) = (A_{i,j}(x), y)
$$
for any quaternions $x$ and $y$. Hence $A_{i,j}$ should verify the relation 
$u^{-1}A_{i,j}u =A_{i,j}$
(since $u^*=u^{-1}$). Thus the operator $A_{i,j}$ should
commute with the multiplication of quaternions by $u$ from the left.
 
 \bigskip
 
{\bf Lemma.} {\it Any real-linear operator $A$ in $\HH$, which
commutes with the multiplication of the quaternions
(from the left) by the unitary quaternion $u$, is
the operator of the multiplication by some quaternion 
from the right.}

\bigskip

{\bf Proof.} The commutativity implies that
$A(u) = A(u \cdot 1) = u A(1).$ Hence $A$ acts
on any quaternion $q$ as the multiplication from the
right, $A(q) = qa,$ where $a = A(1).$

\bigskip

It follows from the Lemma that we have thus proved that
$A_{i,j}(q_i)=q_if_{i,j},$ where $f_{i,j}$ is a quaternion.
Since $A$ is symmetric, the resulting quaternionic
matrix $f_{i,j}$ is hyperhermitian, $f_{j,i} =\bar {f_{i,j}}$. In 
particular the diagonal elements are real. We thus get
formula $(1)$: 
$$
f(q) = {\rm {\mbox Re}}\sum q_i f_{i,j} \bar q_j = \sum q_i f_{i,j} \bar q_j.
$$

$4^o.$ {\it The huperhermitian matrix is unambiguosly
defined by the form} $f$. Indeed, if $f=0$, its restriction
to $\HH^1_i \oplus \HH^1_j$ also vanishes. Hence the
operators $A_{i,j}$ (and thus also the quaternions $f_{i,j}$) vanish.
The theorem is thus proved.

\bigskip

{\bf Corollary.} {\it The dimension of the real vector-space
of the hyperhermitian forms in $\HH^m$ equals}
$m+ 4m(m-1)/2 = m(2m-1)$.

\bigskip

{\bf Definition.} A {\it hyperunitary transformation}
of a hyperhermitian space is a quater\-nionic-linear
transformation, preserving the hyperhermitian structure.
(A quaternionic-linear operator is a linear operator $A$,
for which $A(qx) = qA(x)$ for any quaternion $q$).

\bigskip

The dimension of the hyperunitary group of the
hyperhermitian space $\HH^m$ equals $m(2m+1)$. This
group is denoted (unfortunatly) $Sp(m)$.

The hyperunitary transformations are defined by
the matricies, whose elements should be written
{\it from the right} of the components of the vectors, like
$$
A(q_1,q_2) = (q_1a_{1,1} + q_2a_{2,1}, q_1a_{1,2} + q_2a_{2,2} ).
$$

The condition that transformation is hyperunitary
(with respect to the standard hyperhermitian structure 
$F(q) = \sum q_i \bar {q_i}$ in $\HH$) takes for the matrix the form
$$
\sum \limits_j a_{i,j} \bar {a_{k,j}} = \delta_{i,k}.
$$
Hence, the hyperhermitian scalar product in $\HH^m$ (with
quaternionic values) defined by the formula
$$
\langle q, r \rangle = \sum q_i \bar r_i
$$
is invariant under the hyperhermitians transformations:
$$
\langle Aq, Ar \rangle = \langle q, r \rangle .
$$

\bigskip

{\bf Remark.} This follows also, of course, from the quaternionic
polarization formula, which is however too long to be reproduced
here.

\bigskip

{\bf Theorem.} {\it The variety of the ordered $n$-tuples of vectors in the
hyperhermitian space $\HH^{n-1}$, considered up to hyperunitary transformations
of this space, is the sphere $S^M, M=n(2n-1) - 2$.}

\bigskip

{\bf Example.} For $n=2$ we get the quaternionic Hopf
bundle, $S^4 = S^7/{S^3}.$ For $n=3$ we get $S^{13} = S^{23}/{Sp(2)}.$
This suggest some relation to the Caley projective plane.

\bigskip

The proof of the theorem, as well as that of its Hermitian
version, is based on the same reduction to the theory of the
hyperbolic cones which was used in Section 3 to prove the Euclidean
version of this theorem.

\bigskip

{\bf Definition.} The {\it Gram matrix} of a system of $n$ vectors
$q_i$ in the standard hyperhermitian (hermitian) space $\HH^m$ ($\CC^m$)
is the matrix of the hyperhermitian (Hermitian) form
$$
f(c) = F(c_1q_1+\dots+c_nq_n),\:\;\;\;\;\;\;\; c \in \HH^m (\CC^m).
$$

\bigskip

This form is  hyperhermitian (Hermitian) since the argument of $F$
is multiplied by $u$ from the left when all the $q_i$ are
multiplied by $u$ from the left. By the definition of the standard
hyperhermitian (Hermitian) structure $F$, we get
$$
f(s) = \sum c_i\langle q_i, q_j\rangle \bar c_j,\;\;\;\;
{\rm \mbox{whence}} \;\;\;\; f_{i,j}=\langle q_i, q_j\rangle.
$$
Thus the form $f$ is invariant under the hyperunitary (unitary)
transformations of the space $\HH^m$ ($\CC^m$).

The hyperhermitian (Hermitian) matrix $f_{i,j}$ defines the
orthogonalization of the $n$-tuple $q_i$. Hence this matrix
determines this $n$-tuple up to a hyperunitary (unitary)
transformations of the space $\HH^m$ ($\CC^m$).

\section{The equivariant Neuman-Wigner theorem on the 
nonintersections of electronic levels.}

As we have seen, the eigenvalues theories for quadratic,
Hermitian and hyperhermitian forms are theories of the
hyperbolic cones in the space of forms which are
invariant under some representation (of group $U(1)$ in
the Hermitian case and of group $SU(2)$ in the the
hyperhermitian case), namely under a representation
which is a multiple of an irreducible one. The
corresponding generalized von Neuman-Wigner theorems
(see \cite{6}, \cite{3}) claim in our present terminology that
the codimensions of the varieties of the singular points
on the cones of the degenerate points are equal,
in the real, complex and quaternionic cases, to $2$, $3$ and $5$
(these numbers are the codimensions of the one-dimensional
spaces of the diagonal forms of two variables in the
spaces of quadratic, Hermitian and hyperhermitian forms
of two variables).

One can replace here the standard irreducible
representations of groups $U(1)$ and $SU(2)$ by any irreducible
representation of any compact Lie group. The results,
describe below, show, that such a generalization {\it provides
no new hyperbolic cones:} all the spaces of quadratic 
forms, invariant under the multiples of the irreducible real
representations of compact Lie groups, are naturally isomorphic
either to the space of all real quadratic forms, or
to the space of the Hermitian forms, or to the space 
of the hyperermitian forms.

Fix a real irreducible representations of a compact 
Lie group $G$ by orthogonal transformations of the
Euclidean space $\RR^n$. 

\bigskip

{\bf Definition.} A {\it symmetry} of the representation is a
real linear operator, commuting with all the operators of
the representation: $Ag=gA$ (in other terms it is an operator
interwinning the representation with itself).

\bigskip

The symmetries of a given representation form an
(associative) algebra (a subalgebra of the algebra of
all the operators from $\RR^n$ to itself). We shall use the
following real version of the Shur's lemma:

\bigskip

{\bf Theorem.}{\it The symmetry algebra of any real irreducible representation
of a compact Lie group is isomorphic (as an algebra of
linear transformations of the Euclidean space) to one of
the following three algebras: }

$1)$ {\it algebra $\RR$ of real numbers, acting in $\RR^n$ as the scalar 
matricies;}

$2)$ {\it algebra $\CC$ of complex numbers, acting in $\CC^m$ ($n=2m$) as the 
scalar matricies;}

$3)$ {\it algebra $\HH$ of quaternions, acting in $\HH^k$ ($n=4k$) by the
multiplication of the component of a vector by a quaternion from the right.}

\bigskip

{\bf Remark.} This classical theorem provides the royal way to
quaternions, which appear here not as an uncomplete axiomatic
generalization of complex numbers, but as the solution of a
natural problem in real Euclidean geometry. All the ``axioms''
of quaternions are simply the necessary {\it properties} of the
solutions of this natural problem, and to discover them
one is not forced to use the spirits as employed by Hamilton.
Simultaneously one gets the classification of the associative
algebras with division (not only as of abstract objects but
also as operators algebras). The proof is so simple that
I shall give it below.

\bigskip

{\bf Proof.} Complexify $\RR^n$ to get $\CC^n \approx \RR^{2n}$. The
complexified representation operators $g$ and symmetries $A$ act
on $\CC^n$ as complex linear operators. The representation of $G$
in $\RR^{2n}$ that we obtain is reducible: it is the direct
sum of two copies of the original representation.

\bigskip

{\bf Lemma $1$.}{\it The complexified symmetry operator $A$ either is
the multiplication by a real number, or has
two complex-conjugate eigenvalues $\lambda = \alpha \pm \omega$ of
multiplicity $m = n/2$ each (the dimension of the
original real representation space being even).}

\bigskip

{\bf Proof.} Otherwize $A$ would have a nontrivial complex eigenspace
whose complex dimension either smaller or 
greater than $m.$ The first is impossible, since this invariant
space of the representation would have a nontrivial intersection with $\RR^n$,
in contradiction with the irreducibility. In the second case
the orthogonal complement to the eigenspace (which is
also invariant under the representation) would have
a nontrivial intersection with $\RR^n$,
in contradiction with the irreducibility.

Denote $I = (A-\alpha {\bf 1})/\omega$. One obviously get the
following

\bigskip

{\bf Lemma $2$.} {\it Symmetry operator $I$ is a complex structure
in $\RR^n$, i.e. $I^2 = -{\bf 1}$.}

\bigskip

{\bf Lemma $3$.} {\it Each complex structure, which is a symmetry
of an irreducible orthogonal representation in $\RR^n$,
preserves the Euclidean structure.}

\bigskip

Indeed, the nonnegative form $(Ix,Ix)$ is invariant,
since $(Igx,Igx) = (gIx,gIx) = (Ix,Ix)$, the
form $(y,y)$ being $g$-invariant. Hence one gets $(Ix,Ix) = \lambda (x,x)$
(othervize the eigenspace of this form would be a nontrivial $G$-invariant
subspace in $\RR^n$). Therefore we find, that $(x,x) = (I^2x,I^2x) = \lambda^2 
(x,x), \lambda^2 = 1$.
Since the form $(Ix,Ix)$ is nonnegative, $\lambda = 1$. Thus
every symmetry which is a complex structure is
orthogonal.

\bigskip

If the real linear combinations of ${\bf 1}$ and $I$ exhaust
the symmetry algebra, we get the case $2$ of the theorem.
Suppose there is one more symmetry in the algebra.
Replacing it by its linear combination with the identity ${\bf 1}$,
we construct a complex structure $B$. The symmetry
operator $IB + BI$ is symmetric. The irreducibility of
the original representation implies that any
symmetric symmetry operator is a scalar: $IB + BI = 2 \varepsilon \in \RR$.
Denote by $J$ the operator $J = (B + \varepsilon I)/ \sqrt{1-\varepsilon^2. }$ 
Here $\varepsilon^2 \le 1,$
since the orthogonal operator $C=IB$ is different
from $\pm{\bf 1}$, $B$ being independent of $I$.

\bigskip

{\bf Lemma $4$.} {\it Operator $J$ is a complex structure, anticommuting
with $I$.}

\bigskip

Indeed, 
$$
J^2 = (B + \varepsilon I)^2/ (1-\varepsilon^2) = -{\bf 1},\:\:
IJ+JI = (IB+BI + 2\varepsilon I^2)/\sqrt{1-\varepsilon^2 } = 0.
$$

\bigskip

{\bf Lemma $5$.} {\it Operator $K = IJ$ is also a complex 
structure; all the three complex structures (I,J,K) 
anticommute.}

\bigskip

Indeed, 
$$
K^2 = (IJ)(IJ) = IJ(-JI) = I^2 = -{\bf 1},
$$
$$
IK = I^2J, KI=-JI^2, JK=-J^2I, KJ = IJ^2.
$$
Therefore the vectors $(a, Ia, Ja, Ka)$, where $a$ is
any unite vector, are mutually orthogonal and generate $\RR^4$
on which the operators $(I,J,K)$ are acting as the
quaternions $(i,j,k).$ In the orthogonal complement to
$\RR^4$ we choose one more unite vector and construct one more $\RR^4$.
Continuing this way, we identify at the end the original
Euclidean space $\RR^n$ with $\HH^k, n = 4k$.

\bigskip

{\bf Lemma $6$.} {\it Any symmetry of the irreducible representation
is a real linear combination
of the four operators $({\bf 1}, I, J, K)$ that we have constructed.}

\bigskip

If there were one more symmetry, we would
construct, as above, its linear combination $L$ with ${\bf 1}$ and $I$
which would be, as $J$, a complex structure, anticommuting
with $I: IL + LJ = 0, L^* = -L.$

Operator $JL = LJ$ is symmetric. The irreducibility implies
that $JL = LJ = 2\xi{\bf 1}$ is a real number. Operator $X = K(L + \xi J)$
is symmetric 
$$
X^* = LK + \xi JK = LIJ + \xi I = -ILJ + \xi I = -I(2\xi - JL) + \xi I = - \xi I 
+ KL. 
$$
The irreducibility implies that $X = \mu \in \RR,$ whence
$L = -\xi J - \mu K$. This proves the Lemma and hence the 
Theorem.

\bigskip

Now suppose that the compact Lie group is represented
in $\RR^{Nn}$ and that this representation is the direct sum
of $N$ copies of the irreducible representation in $\RR^n$ whose
symmetry algebra ($\RR, \CC$ or $\HH$) has the real dimension
$d = 1,2$ or $4$.

\bigskip

{\bf Theorem.} {\it The quadratic forms in $\RR^{Nn}$ which are invariant
under this representation form a real vector-space of
dimension $N+dN(N-1)/2$, which is isomorphic to the
space of the quadratic (Hermitian, hyperhermitian) forms
in $\RR^N$ (in $\CC^N$, in $\HH^N$).

The determinant of the invariant quadratic form, considered
as a polynomial of the coefficients of the form in $\RR^{Nn}$,
is equal to the $\frac nd$-th power of the determinant of the corresponding
quadratic form in $\RR^N$ (in $\RR^{2N}$, in $\RR^{4N}$).}

\bigskip

{\bf Corollary.} {\it The cone of the degenerate invariant
quadratic forms on $\RR^{Nn}$ is sent by the isomorphism
mentioned in the theorem onto the cone of the
degenerate quadratic forms in $\RR^{N}$ (of the
degenerate Hermitian forms in $\RR^{2N}$, of the
degenerate hyperhermitian forms in $\RR^{4N}$).}

\bigskip

For instance, the codimensions of the varieties of
the singularities of these cones (and hence
the codimensions of the varieties of the forms with 
eigenvalues of nonminimal multiplicity) do not 
depend on the irreducible representation nor on
the group --- they only depend on the symmetry algebra
(being equal to $d+1$ on the cone of the degenerate
forms and $d+2$ in the space of forms).

Therefore the phenomenon of the repulsion of the
eigenvalues, discribed by the von Neuman-Wigner theorem,
has only three variants: the real one, the Hermitian
one and the hyperhermitian one. {\it To collide two
eigenvalues one needs $2,3$ or $5$ independent parameters,
correspondingly, whatever the compact group and its
representation, which is the direct sum of some copies of an
irreducible real representation one considers.}

\bigskip

{\bf Proof.} Decompose the representation space into
orthogonal irreducible invariant subspaces $\RR^n_i, i = 1, \dots, N.$
The quadratic form can be written, as in Section $5$, as the
sum of $N^2$ blocks of the sise $n \times n$:
$$
f(x) = \sum (A_{i,j}x_i, x_j),\;\;\;\;\;\;  A^*_{i,j}=A_{j,i}.
$$
The invariancy condition under the
$g$ of the group takes the form
$g^{-1}A_{i,j}g = A_{i,j},$   $g$ being orthogonal and
$(A_{i,j}x, y)$ being equal to $(x, A_{j,i}y)$.
Hence $A_{i,j}$ commutes with $g$. From the (real)
Shur lemma we know, that operator $A_{i,j}:\RR^n \mapsto \RR^n$
acts either as the multiplication by a real scalar,
or as the multiplication by a complex scalar in $\CC^{n/2}$,
or as the multiplication (from the right) by a quaternion in $\HH^{n/4}$.

Present the space $\RR^n (\CC^{n/2}, \HH^{n/4})$ as the
orthogonal sum of the spaces $\RR_i(\CC_i, \HH_i)$. We
thus reduce the matrix of our quadratic form to the
block-diagonal form from $n(n/2, n/4)$ identical
blocks, each block being the matrix of a quadratic
(Hermitian, hyperhermitian) form in the space $\RR^N (\CC^{N}, \HH^{N})$.
The Theorem follows.

\bigskip

{\bf Remark.} The characteristic polynomials of our invariant form in $\RR^{Nn}$
are the $\frac{n}{d}$-th power of the polynomials which, generically, have no
multiple roots. The forms for which the multiple root occur,
 form an algebraic submanifold of real codimension $d+1 = 2, 3, 5$ in the
 space of the quadratic (Hermitian, hyperhermitian) forms.
 
 \bigskip
 
 The discriminants of these polynomials with real roots do not
 change the sign. Probably these polynomials of the coefficients
 of the forms are sums of squares of several polynomials.

\bigskip
 
 {\bf Example.} For $N=2$ the number of squares is $2$ in the
 Euclidean case, $3$ -- in  the Hermitian case and $5$ in the 
 hyperhermitian case.

\bigskip
 
 {\bf Remark.} A simpler but also important example of an almost
 everywhere positive polynomial is provided by the attempt to complexify and to
 quaternionnise the determinants of real operators.
 
 A complex linear operator $A: \CC^n \rightarrow \CC^n$ defines a real
 linear operator $ ^{\RR}A: \RR^{2n} \rightarrow \RR^{2n},$ for which
 $ \det (^{\RR}A) = |\det A|^2.$
 Therefore, the determinant of the real operator $ ^{\RR}A $ is the sum of
 squares of two polynomials of the elements of its matrix ( as  it should be,
 the variety of the degenerate complex operators being of real codimension $2$).
 
 The variety of the quaternionic-degenerate operators $ A: \HH^n \rightarrow 
\HH^n$ 
 has the real codimension $4$ in the space of the quaternionic operators. Hence 
it is
 natural to consider the nonnegative polynomial
$\det\left( ^{\RR}A: \RR^{4n} \rightarrow \RR^{4n}\right)$ of the elements 
of the quaternions forming the matrix of $A$ and to try to represent it as the
sum of (at least four) squares of polynomials.

A quaternion can be represented by a complex $2\times 2$-matrix
$\left( \begin{array}{cc} 
             z & -\bar w \\
             w &  \bar z
             \end{array}\right).$
This operation transforms
a quaternionic operator into a complex operator
$^{\CC}\!A: \CC^{2n} \rightarrow \CC^{2n}$. Since $^{\RR}\!A= ^{\RR \CC}\!A$, we get
the formula
$$
\det (^{\RR}\!A) = |H|^2, \;\;\;\;{\rm {\mbox {where}}} \;\;\;\; H=\det(^{\CC}\!A).
$$             

The function $H$, which is defined as a complex-valued polinomial,
is in fact a (nonnegative) real polynomial of the
elements of the quaternions forming the matrix of $A$.
Therefore $\det (^{\RR}\!A) = H^2$ is not a sum of two
squares but just one square of a nonnegative polynomial
$H$, whose zeroes still form an algebraic variety of real
codimension $4$ in the space of the quaternionic matricies.

In the case $n=1$ this polynomial $H$ is the sum of 
the squares of the four components of the quaternion.
One might conjecture that it is still represantable
as the sum of at least four squares in the general
case.

For $n=2$ the polynomial $H$ has a simple expression
in terms of the minors of the $2\times 4$ complex matrix $(z, w)$:
$$
H = |M_{12}|^2 + |M_{14}|^2 + |M_{34}|^2 - 2{\rm {\mbox {Re}}}M_{13}{\bar 
M}_{24},
$$
where $M_{12}M_{34} + M_{14}M_{23} = M_{13}M_{24}.$

This formula is strangely similar (differing only by
the presence of the complex conjugation) to
the formula of the Klein representation of the space of
the lines in $\CC P^3$, which is quoted by
Atiyah \cite{10} in the description of the Penrose twistors space.


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Steklov Mathematical Institute  and Universit\'e Paris-Dauphine

\end{document}
