\documentstyle[12pt]{article}

\def\eop{{ \vrule height5pt width5pt depth0pt}\par\bigskip }

\newenvironment{prf}{\noindent{\it Preuve:} }{\eop}
\newtheorem{theorem}{Th\'eor\`eme}
\newtheorem{definition}{D\'efinition}
\newtheorem{lemma}{Lemme}
\newtheorem{proposition}{Proposition}

\def\build#1_#2^#3{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}}

\newcommand{\rf}[1]{(\ref{#1})}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\beqn}{\begin{eqnarray}}
\newcommand{\eeqn}{\end{eqnarray}}
\newcommand{\beqaa}{\begin{eqnarray*}}
\newcommand{\eeqaa}{\end{eqnarray*}}
\newcommand{\bitem}{\begin{itemize}}
\newcommand{\eitem}{\end{itemize}}

\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}

\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}

\font\twelve=cmbx10 at 15pt
\font\ten=cmbx10 at 12pt
\font\eight=cmr8



\begin{document}


\begin{titlepage}

\begin{center}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

{\ten Centre de Physique Th\'eorique\footnote{
Unit\'e Propre de Recherche 7061} - CNRS - Luminy, Case 907}
{\ten F-13288 Marseille Cedex 9 - France }

\vspace{2 cm}

{\twelve DUAL PAIRS OF QUANTUM SPACES}

\vspace{0.3 cm}

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

{\bf Daniel KASTLER}

\vspace{3 cm}

{\bf Abstract}

\end{center}

We discuss the notion of ``dual riemannian quantum space''
corresponding to the ``algebraic condition'' in  Alain Connes'
definition of non-commutative Poincar\'e duality. We show that the
DeRham complex of a dual riemannian quantum space is the homomorphic
image of a skew tensor product, this leading to a natural definition
of ``biconnection''.

\vspace{3,5 cm}

\noindent Key-Words : non-commutative geometry

\bigskip

\noindent July 1994

\noindent CPT-94/P.3056

\bigskip

\noindent anonymous ftp or gopher: cpt.univ-mrs.fr

\end{titlepage}



\centerline{\bf Dual pairs of quantum spaces}

\medskip

\centerline{Daniel Kastler}

\bigskip

Alain Connes initially defined the ``non-commutative differential
geo\-me\-try'' of a (cohomologically finite-dimensional) algebra $A$
by specifying a $d^+$-summable $K$-cycle $(H,D)$ of $A$ [1][2] - a
procedure generalizing to the non-commutative frame the classical
situation where classical differential geo\-me\-try is coded in the
Dirac operator. The couples $(A, (H, D))$ of an algebra and a
$d^+$-summable $K$-cycle do not however fully deserve to be considered
as ``non-commutative riemannian manifolds'' because they still lack
the important feature of ``Poincar\'e duality''\footnote{ 
We propose below to call these couples ``riemannian quantum
spaces''.
} generalized to the non-commutative frame - a concept which
was (astonishingly!) discovered whilst looking at the standard  model
[3] - and which arises by considering  couples $(A', A'')$
of algebras and $A' - A''$-bimodules (in other terms
$A' \otimes A''$-modules) under particular conditions
(generalizing the  classical situation) ensuring the isomorphim of
the homology of $A'$ and the cohomology of $A''$ -
see~{\bf 1}~(iii) below. In this note we study the structure
(rebaptized dual riemannian quantum space - see~{\bf 1}~(ii) below)
corresponding to the ``algebraic condition'' in Alain Connes'
definition of Poincar\'e duality. The ``topological condition''
recalled under (3), below will not be used in what follows.

\bigskip

\noindent{\bf 1. Definitions. Terminology.}

\begin{itemize}

\item[(i)] A {\it riemannian quantum space
} is a couple $(A, {\bf H})$ of a unital$^{\star}$-algebra $A$ and a
$K$-cyle ${\bf H} = (H, D, \chi)$ over $A$. The quantum space $(A,
{\bf H})$ is $d$-dimensional whenever the
$K$-cyle ${\bf H}$ is $d^+$-summable.

\item[(ii)] A ($d$-dimensional) {\it dual riemannian quantum space
} is a riemannian quantum space $(A' \otimes A'', {\bf H})$, where $A'
\otimes A''$ is the algebraic tensor product of the
unital$^{\star}$-algebras $A',\ A''$ (with respective units
$1\!\!1'$ and
$1\!\!1''$); and where ${\bf H} = (H,D,\chi)$ is a ($d^+$-summable)
$K$-cyle of $A' \otimes A''$ fullfilling the algebraic condition:
\beq
\left[ [ D,\, \underline{a' \otimes 1\!\!1 } ],\ \underline{1\!\!1
\otimes a'' } \right] = 0 \quad,\ a' \in A',\ a'' \in A'',
\eeq 
(In order to alleviate notation, we shall consider $A'$ and
$A''$ as acting on $H$ by commuting  representations $a'
\to \underline{a'} = \underline{a' \otimes 1\!\!1 }$ and $a'' \to
\underline{a''} = \underline{1\!\!1 \otimes a'' }$, then
reading
\beq
\left. 
\left[ [ D,\, \underline{a'} ],\ \underline{a''} \right] = 0 \quad,\ a'
\in A',\ a'' \in A'' \right). 
\eeq

\item[(iii)] A d-dimensional {\it riemannian quantum manifold
} is a dual riemannian quantum space
$(A' \otimes A'', {\bf H})$ whose $d^+$-summable $K$-cyle ${\bf H} =
(H,D,\chi)$ has its ``Hochschild obstruction'' vanishing under the
operator B, i.e. 
\beq
Tr_{\omega} \left\{
\chi \underline a_0 [ D, \underline a_1 ] \dots [ D, \underline a_k ]
| D |^{-k} \right\} = 0 \quad,\ a_0,a_1, \dots, a_k \in A.
\eeq

\item[(iv)] The quantum  space $(A,{\bf H}),\ {\bf H} = (H,D,\chi)$,
is {\it self-dual
} whenever the operator algebra $\{ \underline a, a
\in A \}$ is commutative, and one has:
\beq
\left[ [ D,\, \underline a ],\ \underline b \right] = 0 \quad,\ a, b
\in A. 
\eeq
\end{itemize}

The {\it gauge group of the dual riemannian quantum space
} $(A' \otimes A'', {\bf H})$ is by definition the product ${\cal
G}' \times {\cal G}''$ of the respective gauge group ${\cal G}' {\cal
G}''$ (=groups of unitaries of
$A'$, resp. $A''$).

\bigskip

\noindent{\bf 2. Remarks}

\begin{itemize}

\item[(a)] There is a bijection between riemannian quantum spaces
$(A' \otimes A'', {\bf H})$, and pairs $(A', {\bf
H}), (A'', {\bf H})$ of riemannian quantum spaces with a common
$d^+$-summable $K$-cycle ${\bf H} = (H, D, \chi)$ for which $\ul{a'}$ 
and $\ul{a''}$ commute for all $a' \in A'$ and $a'' \in 
A''$. The bijection is effected via the correspondence $\ul{a'
\otimes a''} \leftrightarrow \ul{a'} \cdot \ul{a''}$, in accordance
with the former  identifications $\ul{a'} = \ul{a' \otimes 1\!\!1''}$
and
$\ul{a''} = \ul{1\!\!1' \otimes a''}$.

\item[(b)] With $(A' \otimes A'', {\bf H})$ a dual riemannian quantum
space we have that: 
\beq
\left[ \pi_D ( \Omega A' ),\ \pi_D ( A'' ) \right] = 
\left[ \pi_D ( \Omega A'' ),\ \pi_D ( A' ) \right] = 0,
\eeq

\item[(c)] There is a bijection between self-dual riemannian quantum
spaces $(A, {\bf H})$ and riemannian quantum spaces $(A \otimes A,
{\bf H})$ with $A$ commutative.

\item[(d)] The couple of the algebra of smooth functions over a
compact spin$^c$ riemannian manifold, and its Dirac $K$-cycle is (the
archetype of) a self-dual quantum space.

\end{itemize}

\bigskip

\noindent \ul{Proof:}

\begin{itemize}

\item[(a):] obvious.

\item[(b):] follows from (1a) via the definition of $\pi_D$.

\item[(c):] obvious.

\item[(d):] follows from the fact that the a commute with the
$\ul{\gamma} ({\bf d} a)$.

\end{itemize}

\qquad Our first problem is the definition of connections of quantum
manifolds (or, for that matter of dual riemannian quantum spaces).
The passage from riemannian quantum spaces involving one algebra to
dual quantum spaces involving a commuting\footnote{
in the representation $\pi_D$.
} couple of algebras $(A', A'')$ (i.e. their tensor
pro\-duct
$A = A' \otimes A''$ together with its tensorial
splitting) raises questions as to the maintenance in this
enlarged\footnote{
enlarged in view of Remark see.
} frame of the previous philosophy pertaining to riemannian quantum
spaces (and of the latter's implications on connections, the quantum
Yang-Mills algorithm, etc.). Taking the tensor product $A = A' \otimes
A''$ as the new ``basic'' algebra, one is at first tempted to
consider $(\Omega A, \delta)$ as the ``basic'' differential algebra.
However the latter does not contain the information of the
commutativity of $A'$ and $A''$ and of the vanishing of commutators
(1a) in the  representation $\pi_D$. This information is incorporated
in the following definitions:

\bigskip

\noindent{\bf 3. Definitions.} Let $A'$ and $A''$ be unital
*-algebras with $A = A' \otimes A''$ and with $(\Omega A, \delta)$ the
unital differential envelope of $A$; and consider the following 
subsets of $\Omega A^1$:\footnote{ 
Note that ${\bf S}' = - {\bf S}''$ because $\delta \left[ 1\!\!1'
\otimes a'', a' \otimes 1\!\!1'' \right] = 0 = \left[ \delta (1\!\!1'
\otimes a''),\ a' \otimes 1\!\!1'' \right] + \left[ 1\!\!1'
\otimes a'', \delta (a' \otimes 1\!\!1'') \right].$ 
}
\beq
{\bf S}' = \left\{
\left[ \delta (a' \otimes 1\!\!1''),\ 1\!\!1' \otimes a'' \right]
\in \Omega A^1 ; a' \in A', a'' \in A''
\right\} = - {\bf S}''
\eeq
resp.
\beq
{\bf S}'' = \left\{
\left[ \delta (1\!\!1' \otimes a''),\ a' \otimes 1\!\!1'' \right]
\in \Omega A^1 ; a' \in A', a'' \in A''
\right\}.
\eeq
We denote by ${\bf j}_P$ the ideal of $\Omega A$ generated by ${\bf
S}'$ (or, for that matter, by ${\bf S}''$), and define the
{\it Poincar\'e ideal
} as:
\beq
{\bf k}_P = {\bf j}_P + \delta {\bf j}_P .
\eeq

This definition implies:

\bigskip

\noindent{\bf 4. Lemma.} {\it With $(A, {\bf H}),\ A = A' \otimes
A'',\ {\bf H} = (H, D, \chi),\ {\bf j}_P$, and
${\bf k}_P$ as in the previous definitions we have that: 
}

\begin{itemize}

\item[(i):] ${\bf j}_P$ {\it is a graded ideal of $\Omega A$. 
} 

\item[(ii):] ${\bf k}_P$ {\it is a graded differential ideal of
$\Omega A$, with ${\bf k}_P^n = {\bf j}_P^n + \delta {\bf j}_P^{n-1}$,
hence $\Omega A / {\bf k}_P$ is a {\bf N}-graded differential algebra,
with differential $d$ obtained from that of $\Omega A$ by passage to
the  quotient through the ideal ${\bf k}_P$. 
}

\item[(iii):] {\it One has $\left[ \delta (a' \otimes 1\!\!1''),
\delta (1\!\!1' \otimes a'') \right] \in {\bf k}_P^2,\ a' \in A', a''
\in A''$. 
}

\item[(iv):] {\it The ideal ${\bf k}_P$ is generated by
the elements $\left[ \delta (a' \otimes 1\!\!1''),
1\!\!1' \otimes a'' \right]$ and\break
 $\left[ \delta (a' \otimes
1\!\!1''), \delta (1\!\!1' \otimes a'') \right],\ a' \in A', a'' \in
A''$. 
}

\item[(v):] {\it If in addition $(A, {\bf H}),\ A = A' \otimes
A'',\  {\bf H} = (H,D,\chi)$, is a dual riemannian quantum space, the
quantum DeRham complex $\Omega A_D$ is a quotient of $\Omega A / {\bf
k}_P$ as a {\bf N}-graded differential algebra. 
}

\end{itemize}

\bigskip

\noindent \ul{Proof:}

\begin{itemize}

\item[(i):] ${\bf j}_P$ is indeed generated by homogeneous elements.

\item[(ii):] cf.~[1].

\item[(iii):] We have, since $\delta^2 = 0:$
\beq
\delta \left[ 1\!\!1' \otimes a'',\ \delta (a' \otimes 1\!\!1'')
\right] = \left[ \delta (1\!\!1' \otimes a''),\ \delta (a' \otimes
1\!\!1'')
\right].
\eeq

\item[(iv):] ${\bf j}_P$ consists of linear combinations of elements
of the type $\omega \left[ \delta (a' \otimes
1\!\!1''), \right.$ $\left. 1\!\!1' \otimes a'' \right] \Psi,\ a' \in
A',\ a''
\in A'',\ \omega, \Psi \in \Omega A$, thus ${\bf k}_P =
{\bf j}_P + \delta {\bf j}_P$ consists of linear combinations of
elements of the latter type plus elements of the type
\beq
\begin{array}{ll}
(\delta \omega) \left[ \delta (a' \otimes 1\!\!1''), 1\!\!1' \otimes
a'' \right] & \Psi^{\pm} \omega \left[ \delta (a' \otimes 1\!\!1''),\
\delta (a' \otimes 1\!\!1'') \right] \\
\\
& \Psi^{\pm} (\delta \omega)
\left[ \delta (a' \otimes 1\!\!1''), 1\!\!1' \otimes a'' \right] \Psi,
\end{array}
\eeq
whence the result by (iii). 

\item[(v):] The definition of a dual riemannian quantum space entails
that the ideal ${\bf j}_P$ is contained in the kernel ${\bf K}^*$ of
$\pi_D$, hence the graded differential ideal ${\bf k}_P$ is contained
in the graded differential ideal ${\bf J} = {\bf K}^* + \delta {\bf
K}^*$. Thus $\Omega A_D$ is a quotient of $\Omega A / {\bf
k}_P$ as a ${\bf N}$-graded differential algebra.

\end{itemize}

The fact which enlightens the issues raised above is now the
isomorphism $(\Omega A / {\bf k}, \delta) \cong (\Omega A_{\otimes},\
\delta_{\otimes})$ - cf.~${\bf 6.}$ - and the ensuing fact that the
skew tensor product $\Omega A' \wh{\otimes} \mu
\Omega A''$ ``sits above'' $\Omega_D A$ - cf.~${\bf 7.}$ -
and thus is a natural receptacle for connections. The relationship
between $\Omega A' \wh{\otimes},\ \Omega A''$ and
the differential envelope $\Omega A$ of $A'
\wh{\otimes},\ A''$ has been studied in [4], we recall the
results here:

\bigskip

\noindent{\bf 5. Reminder} (differential envelope of tensor
products cf.~[4]). Let $A'$ and $A''$ be unital (real or complex)
*-algebras, with respective units $1\!\!1'$ and $1\!\!1''$, and with
respective unital differential envelopes $(\Omega A', \delta')$
and $(\Omega A'', \delta'')$. Let $A = A'
\otimes A''$, with unit $1\!\!1$ and unital differential
envelope $(\Omega A, \delta)$, be the algebraic
tensor-product of $A'$ and $A"$. Set $\Omega A_{\otimes} = \Omega A'
\wh{\otimes} \Omega A''$, with differential $\delta_{\otimes}$,
for the skew tensor product of
$\Omega A'$ and $\Omega A''$: i.e. we have,
for $\omega', \psi' \in A', \omega'', \psi'' \in A''$:
\beq
\left\{
\begin{array}{l}
\Omega A_{\otimes}^n = \sum_{p + q = n} \Omega A'^p \otimes 
\Omega A''^q, \\
\\
(\omega' \otimes \omega'') (\psi' \otimes \psi'') = \omega' \psi'
\otimes \omega'' \psi'' \\
\\
\delta_{\otimes} (\omega' \otimes \omega'') = \delta' \omega' \otimes
\omega'' + \omega' \otimes \delta \omega''
\end{array}
\right. .
\eeq
We then have a split short exact sequence $0 \to {\bf k} \to \Omega
A \build{\to}_{ \build{J}_{\gets}^{} }^{\ol i}
\Omega A_{\otimes} \to 0$, where:
\beq
\begin{array}{l}
 \ol i \left[ \left( a'_0 \wh{\otimes} a''_0 \right) \delta \left(
a'_1
\wh{\otimes} a''_1 \right) \dots \delta \left( a'_n \wh{\otimes} a''_n
\right) \right] \\
\\
 = \left( a'_0
\wh{\otimes} a''_0 \right)
\delta_{\otimes} \left( a'_1 \wh{\otimes} a''_1 \right) \dots
\delta_{\otimes} \left( a'_n \wh{\otimes} a''_n \right) \\
\\
 = \left( a'_0 \wh{\otimes} a''_0 \right) \left[ \delta' a'_1
\wh{\otimes} a''_1 + a'_1 \wh{\otimes} \delta'' a''_1 \right] \dots
\left[ \delta' a'_n \wh{\otimes} a''_n + a'_n \wh{\otimes} \delta''
a''_n \right], \\
\end{array}
\eeq
and
\beq
J \left( \omega' \wh{\otimes} \omega'' \right) = \ol{j'} (\omega')
\ol{j''} (\omega'') , \quad \omega' \in \Omega A',\ \omega''
\in \Omega A'',
\eeq
where:
\beq
\begin{array}{l}
 \ol{j'} \left( a'_0 \delta' a'_1 \dots \delta' a'_n \right) \\
\\
 = \left(
a'_0 \wh{\otimes} 1\!\!1'' \right) \delta \left( a'_1 \wh{\otimes}
1\!\!1'' \right) \dots \delta \left( a'_n \wh{\otimes}
1\!\!1'' \right), \quad a'_0, a'_1, \dots, a'_n \in A',
\end{array}
\eeq
\beq
\begin{array}{l}
 \ol{j''} \left( a''_0 \delta'' a''_1 \dots \delta'' a''_n \right) \\
\\
 = \left( 1\!\!1' \wh{\otimes} a''_0 \right) \delta \left( 1\!\!1'
\wh{\otimes} a''_1 \right) \dots \delta \left( 1\!\!1'
\wh{\otimes} a''_n \right) , \quad \ a''_0, a''_1, \dots, a''_n \in
A'',
\end{array}
\eeq
hence, for $\omega' \in \Omega A',\ \omega'' \in \Omega A''$:
\beq
\left\{\normalbaselineskip=18pt\matrix{
\ol i \circ \ol{j'} (\omega') = \omega' \wh{\otimes} 1\!\!1'' \hfill &
\cr
\ol i \circ \ol{j''} (\omega'') = 1\!\!1' \wh{\otimes} \omega'' & \cr
} \right.
\eeq
Here $\ol i,\ \ol{j'}$ and $\ol{j''}$ are homomorphisms of bigraded
differential algebras, whilst $J$ is simply linear injective (neither
multiplicative, nor intertwining differentials). As a consequence, 
the kernel ${\bf k}$ of $\ol i$ is a bigraded differential ideal of 
$\Omega A$, with
\beq
\Omega A_{\otimes}^n = \Omega A^n / {\bf k}^n,
\eeq
and one has
\beq
{\bf k}^n = ( i d - J \ol i ) \Omega A_{\otimes}^n .
\eeq

\bigskip

\noindent {\bf 6. Proposition.} {\it The Poincar\'e ideal ${\bf k}_P$
of Definition 1 coincides with the kernel ${\bf k}$ of $\ol i$: 
\beq
{\bf k}_P^n = {\bf k}^n , \qquad n \in {\bf N} ,
\eeq
Consequently we have the following isomorphism of ${\bf N}$-graded
differential algebras: 
\beq
( \Omega A / {\bf k}, \delta ) \cong \left( \Omega A_{\otimes},\
\delta_{\otimes} \right).
\eeq
}

\bigskip

\noindent {\bf 7. Corollary.} {\it The canoni\-cal map $\psi : \Omega
A \to \Omega A_D$ factors through the canoni\-cal map $\phi : \Omega
A \to \Omega A_{\otimes}$ as $\psi = \xi \circ \phi$,
where $\xi : \Omega A_{\otimes} \to \Omega A_D$ is a
homomorphism of ${\bf N}$-graded differential algebras.
}

\bigskip

\noindent \ul{Proof of the Corollary:} We recall that we have, with
${\bf K}^*$ the homogeneized kernel of $\pi_D$:
\beq
\Omega A_D = \Omega A / {\bf J} = \Omega A / ( {\bf
K}^* + \delta {\bf K}^* ) \, .
\eeq
Now requirement (1) entails that $\pi_D$ vanishes on ${\bf S}'$, thus
on ${\bf j}_P$, whence ${\bf j}_P \subset {\bf K}^*,\ \delta {\bf j}_P
\subset \delta {\bf K}^*$ and ${\bf k}_P \subset {\bf J}$, whence the
claim.

\bigskip

\noindent \ul{Proof of the Proposition:} We first prove the
inclusion
$\subset$ in (19). Owing to [4](vi) it suffices to check $[ \delta
(a' \otimes 1\!\!1''),\ 1\!\!1' \otimes a'' ], \ [ \delta
(a' \otimes 1\!\!1''),\ \delta (1\!\!1' \otimes a'') ] \in {\bf k}$,
now we have, by (21):
\beq
\begin{array}{ll}
\ol i \left\{ [ \delta
(a' \otimes 1\!\!1''),\ 1\!\!1' \otimes a'' ] \right\} & = \left[ \ol i
\delta (a' \otimes 1\!\!1''),\ \ol i (1\!\!1' \otimes a'') \right] \\
\\
& = [ \delta a' \otimes 1\!\!1'',\ 1\!\!1' \otimes a'' ] = 0,
\end{array}
\eeq
but $\ol i$ then vanishes also on $[\delta (a' \otimes 1\!\!1''),\
\delta (1\!\!1' \otimes a'') ] = \delta [ \delta (a' \otimes
1\!\!1''),\ 1\!\!1' \otimes a'' ]$ because it commutes with
$\delta$. 

We now prove the inclusion $\subset$ in (19). According to
(18) it is enough to check that $(i d - J \ol i) \omega \in {\bf
K}_P^n$, for all $\omega$ of the form:
\beq
\begin{array}{ll}
\omega & = \left( a'_0 \wh{\otimes} a''_0 \right) \delta \left( a'_1
\wh{\otimes} a''_1 \right) \dots \delta \left( a'_n \wh{\otimes} a''_n
\right) \\
\\
& = \left( a'_0 \wh{\otimes} a''_0 \right) \left[ \delta \left( a'_1
\wh{\otimes} 1\!\!1 \right) \left( 1\!\!1' \wh{\otimes} a''_1 \right)
+ \left( a'_1 \wh{\otimes} 1\!\!1 \right) \delta \left( 1\!\!1'
\wh{\otimes} a''_1 \right) \right] \dots \\
\\
& \hphantom{
= \left( a'_0 \wh{\otimes} a''_0 \right)
} \dots \left[ \delta \left( a'_n \wh{\otimes} 1\!\!1 \right) \left(
1\!\!1' \wh{\otimes} a''_n \right) + \left( a'_n \wh{\otimes} 1\!\!1
\right) \delta \left( 1\!\!1' \wh{\otimes} a''_n \right) \right] \\
\end{array}
\eeq
for $a'_i \in A',\ a''_i \in A,\ i=1, \dots, n$. We
compute $J \ol i \omega$: we have:
\beq
\ol i \omega = \left( a'_0 \wh{\otimes} a''_0 \right) \left[ \delta'
a'_1 \wh{\otimes} a''_1 + a'_1 \wh{\otimes} \delta'' a''_1 \right]
\dots \left[ \delta' a'_n \wh{\otimes} a''_n + a'_n \wh{\otimes}
\delta'' a''_n \right],
\eeq
to which we apply $J$ as in (13): for this we must effect the products 
(23) so as to get a sum of terms of the form
$\omega' \wh{\otimes} \omega''$, where the $\omega'$ are products of
primed symbols $a'_i$ and $\delta' a'_i$, whilst the $\omega''$ are
products of double-primed symbols $a''_j$ and $\delta'' a''_i$. The
product $\ol{j'} (\omega') \ol{j''} (\omega'')$ is then obtained from
$\omega' \omega''$ by effecting the changes $a'_i \to a'_i
\wh{\otimes} 1\!\!1'',\ \delta' a'_i \to \delta \left( a'_i
\wh{\otimes} 1\!\!1'' \right),\ a''_j \to 1\!\!1' \wh{\otimes} a''_j,\
\delta'' a''_i \to \delta \left( 1\!\!1' \wh{\otimes} a''_i \right)$,
ending up with an expression which can be turned into the expression
in the second line (23) by commutations of primed with double-primed
terms: the difference $(i d - J \ol i) \omega$ is thus a sum of
products each of which  contains a commutator generator of the
Poincar\'e ideal ${\bf k}_P$.

\qquad We are now in a position to discuss the issue of
``connections'' of the dual riemannian quantum space $(A = A' \otimes
A'',\ {\bf H}),\ {\bf H} = (H, D, \chi)$. The quantum forms of $A$, if
we forget about its tensorial decomposition $A = A' \otimes A''$, are
the elements of the  quantum DeRham complex $(\Omega A_D, {\bf d})$,
its connections are thus the grade-one ${\bf d}$-derivations $\nabla$
of
$\Omega A_D$. Now, utilizing the homomorphism $\xi : \Omega
A_{\otimes} \to \Omega A_D$ (cf.~{\bf 7.}), the latter can be
considered as
$\xi$-images of $\delta$-derivations of $\Omega A_{\otimes}$
(``preconnexions''), this according to the scheme:

\bigskip

\noindent{\bf 8. Lemma.} {\it Let $\phi : (\Omega, \delta) \to
(\widetilde{\Omega},
\widetilde{\delta})$ be an epimorphism of {\bf N}-graded
diffe\-ren\-tial algebras, and denote by $Der_{\delta} \Omega$ the set
of grade-one
$\delta$-derivations of $\Omega$ considered as a right
$\Omega$-module (resp. by $Der_{ \widetilde{\delta} }
\widetilde{\Omega}$ the set of $\widetilde{\delta}$-derivations of
$\widetilde{\Omega}$ considered as a right
$\widetilde{\Omega}$-module). Then, given
$\widetilde{\nabla} \in Der_{ \widetilde{\delta} }
\widetilde{\Omega}$, there is $\nabla \in Der_{\delta} \Omega $
such that $\phi$ intertwines $\nabla$ and
$\widetilde{\nabla} : \phi \circ \nabla = \widetilde{\nabla} \circ
\phi$. Specifically, for $\widetilde{\nabla} = \widetilde{\delta} +
\widetilde{\rho}, \rho' \in \widetilde{\Omega}^1$, one has $\nabla =
\delta + \rho$, picking $\rho \in \Omega^1$ such that $\phi \rho =
\widetilde{\rho}$.\footnote{
In the expression $\nabla = \delta + \rho$ the symbol $\rho$ has to
be interpreted as usual as denoting multiplication from the left by
$\rho$. 
} The respective
curvatures $\theta$ and $\widetilde{\theta}$ of $\nabla$ and
$\widetilde{\nabla}$ are related by $\widetilde{\theta} = \phi
\theta$. 
}

\bigskip

\noindent \ul{Proof:} The surjectivity of $\phi$ insures the
existence of $\rho$. One has then, for $\omega \in \Omega$:
\beq
\begin{array}{ll}
(\phi \circ \nabla) \omega & = \phi (\delta \omega + \rho \omega) =
(\phi \circ \delta) \omega + (\phi \rho) (\phi \omega) \\
\\
& = (\wt{\delta} \circ \phi) \omega + \wt{\rho} (\phi \omega) =
(\wt{\nabla} \circ
\phi) \omega.
\end{array}
\eeq
One has furthermore:
\beq
\phi \theta = \phi (\delta \rho + \rho^2) = \wt{\delta}
(\phi \rho) + (\phi \rho)^2 = \wt{\delta} \wt{\rho} +
\wt{\rho}^2 = \theta'.
\eeq
	
Our problem was to select, amongst all connexions of $A$, the ones
matching the tensorial decomposition $A = A' \otimes A''$. Looking at
the preconnections acting on $\Omega A_{\otimes} = \Omega A'
\widehat{\otimes} \Omega A''$, one is now naturally led
to the following notion of ``biconnection''.

\bigskip

\noindent{\bf 9. Definitions.} Let $(A = A' \otimes A'',\ {\bf H})$,
with ${\bf H} = (H, D, \chi)$, be a dual riemannian quantum space,
and consider the above ${\bf N}$-graded differential algebras
$(\Omega A_{\otimes}, \delta)$ and $(\Omega A_D,
{\bf d})$.

\begin{itemize}

\item[(i):] a {\it preconnection of
} $(A, {\bf H})$ is a
grade-one graded $\delta$-derivation $\nabla$ of the right $\Omega
A_{\otimes}$-module $\Omega A_{\otimes}$,
with {\it curvature
} is $\theta = \nabla^2$. The {\it preconnection-one form
} $\rho \in \Omega A_{\otimes}^1$ arises by asking $\nabla = \delta +
\rho$ (one has thus $\theta = \delta
\rho + \rho^2$). The action $\nabla \to \nabla^{\bf u}$ of the element
{\bf u} of the gauge group ${\cal G}$ of $A$ on the preconnection
is by definition:\footnote{
${\cal G}$ is the group of unitaries of $A = \Omega
A_{\otimes}^0$. 
}
\beq
\nabla^{\bf u} \omega = {\bf u} \nabla ({\bf u}^{\star} \omega)
\eeq
(one has then $\nabla^{\bf u} = \delta + \rho^{\bf
u}$ with
\beq
\rho^{\bf u} = {\bf u} \rho {\bf u}^{\star} + {\bf u} \delta {\bf
u}^{\star}).
\eeq

\item[(ii):] a {\it biconnection
} is a preconnection of the form
$\nabla = \nabla' \widehat{\otimes} id + id \widehat{\otimes}
\nabla''$, where $\nabla'$ and $\nabla''$ are respective connections
of $A'$ and $A''$.\footnote{
Thus $\nabla'$ is a grade-one $\delta'$-derivation of $\Omega A'$,
with $\nabla' = \delta' + \rho', \rho' \in \Omega A_{\otimes}^{' 1}$,
and $\nabla''$ is a grade-one
$\delta''$-derivation of $\Omega A''$ with $\nabla'' = \delta'' +
\rho'', \rho'' \in
\Omega A_{\otimes}^{'' 1}$.
} (one has thus $\rho = \rho' \otimes 1\!\!1'' + 1\!\!1' \otimes
\rho''$ for the preconnection-one form $\rho$ of $\nabla$) 

\item[(iii):] a {\it connection of the dual riemannian quantum
space
} $(A = A' \otimes A'', {\bf H})$ is a
grade-one graded
$\delta$-derivation $\nabla$ of the right $\Omega A_D$-module
$\Omega A_D$ which is the image of a  biconnection in the
homomorphism $\xi : \Omega A_{\otimes} \to \Omega A_D$.

\end{itemize}

We check that $\nabla = \nabla' \widehat{\otimes} id + id
\widehat{\otimes} \nabla''$ is a preconnection with preconnection-one 
form $\rho = \rho' \otimes 1\!\!1'' + 1\!\!1' \otimes
\rho''$: for homogeneous $\omega', \psi' \in \Omega A',
\omega'',
\psi'' \in \Omega A''$, we have:
\beq
\begin{array}{ll}
\nabla (\omega' \widehat{\otimes} \omega'') & = \nabla' \omega'
\widehat{\otimes} \omega'' + (-1)^{\partial \omega'} \omega'
\widehat{\otimes} \nabla'' \omega'' \\
\\
& = (\delta' + \rho') \omega' \widehat{\otimes} \omega'' +
(-1)^{\partial \omega'} \omega' \widehat{\otimes} (\delta'' + \rho'')
\omega'' \\
\\
& = \delta (\omega' \widehat{\otimes} \omega'') + (\rho' \otimes
1\!\!1'' + 1\!\!1' \otimes \rho'') (\omega' \widehat{\otimes}
\omega'').
\end{array}
\eeq

We then have the following satisfying set of facts:

\bigskip

\noindent {\bf 10. Proposition.} {\it Let $\nabla = \nabla'
\widehat{\otimes} id + id
\widehat{\otimes} \nabla'' = \delta + \rho,\ \rho = \rho'
\widehat{\otimes} 1\!\!1'' + 1\!\!1' \widehat{\otimes} \rho''$, be a
biconnection  of $(A = A' \otimes A'', {\bf H})$. Then 
}

\begin{itemize}

\item[(i):] {\it The curvature $\theta$ of $\nabla$ is the sum of the
curvatures $\theta'$ and $\theta''$ of $\rho'$ and $\rho''$ in the 
following sense: one has: 
}
\beq
\nabla^2 = \nabla'^2 \widehat{\otimes} id + id \widehat{\otimes}
\nabla''^2,
\eeq
\beq
\theta = \theta' \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes}
\theta'',
\eeq
\beq
\rho^2 = \rho'^2 \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes}
\rho''^2.
\eeq

\item[(ii):] {\it The set of biconnections is stable under the gauge
group ${\cal G}' \times {\cal G}''$ of the dual riemannian quantum
space $(A = A' \otimes A'', {\bf H})$ one has, for ${\bf u}' \in
{\cal G}', {\bf u}'' \in {\cal G}'', {\bf u} = {\bf u}'
\widehat{\otimes} {\bf u}''$: 
}

\end{itemize}
\beq
\nabla^{\bf u} = \nabla'^{\bf u'} \widehat{\otimes} id + id
\widehat{\otimes} \nabla''^{\bf u''},
\eeq
\beq
\rho^{\bf u} = \rho'^{\bf u'} \widehat{\otimes} 1\!\!1'' + 1\!\!1'
\widehat{\otimes} \rho''^{\bf u''}.
\eeq

\bigskip

\noindent \ul{Proof:} Check of (32): we have:
\beq
\begin{array}{ll}
\nabla^2 & = \left( \nabla' \wh{\otimes} i d + i d \wh{\otimes}
\nabla'' \right) \left( \nabla' \wh{\otimes} i d + i d \wh{\otimes}
\nabla'' \right) \\
\\
& = \nabla^{'2} \wh{\otimes} i d + \nabla'
\wh{\otimes} \nabla'' - \nabla' \wh{\otimes} \nabla'' + i d
\wh{\otimes} \nabla^{''2} \\
\\
& = \nabla^{'2} \wh{\otimes} i d + i d
\wh{\otimes} \nabla^{''2} \\
\\
& = \: \hbox{(multiplication from the left
by)}\ \theta' \otimes 1\!\!1'' + 1\!\!1' \otimes \theta'' \\
\end{array}
\eeq
(one can also deduce (33) from (34) via $\theta = \delta \rho +
\rho^2$. Check of (34):
\beq
\begin{array}{ll}
2 \rho^2 = [\rho, \rho] & = \left[ \rho' \wh{\otimes} 1\!\!1'' +
1\!\!1' \wh{\otimes} \rho'',\ \rho' \wh{\otimes} 1\!\!1'' + 1\!\!1'
\wh{\otimes} \rho'' \right] \\
\\
& = \left[ \rho' \wh{\otimes} 1\!\!1'',\ \rho' \wh{\otimes}
1\!\!1'' \right] + \left[ \rho' \wh{\otimes} 1\!\!1'',\ 1\!\!1'
\wh{\otimes} \rho'' \right] \\
\\
& \hphantom{=} + \left[ 1\!\!1' \wh{\otimes} \rho'',\
\rho' \wh{\otimes} 1\!\!1'' \right] + \left[ 1\!\!1' \wh{\otimes}
\rho'',\ 1\!\!1' \wh{\otimes} \rho'' \right], \\
\\
& = [\rho' \rho'] \wh{\otimes} 1\!\!1'' + 1\!\!1' \wh{\otimes}
[\rho'' \rho''] = 2 \rho^{'2} \wh{\otimes} 1\!\!1'' + 2 1\!\!1'
\wh{\otimes} \rho^{''2} ) . \\
\end{array}
\eeq
Check of (35): one has:
\beq
\begin{array}{ll}
\nabla^{\bf u} \left( \omega' \wh{\otimes} \omega'' \right) & = \left(
{\bf u}' \wh{\otimes} {\bf u}'' \right) \nabla \left[ \left(
{\bf u}^{'*} \wh{\otimes} {\bf u}^{''*} \right) \left( \omega'
\wh{\otimes} \omega'' \right) \right] \\
\\
& = \left( {\bf u}' \wh{\otimes} {\bf u}'' \right) \left(
\nabla' \wh{\otimes} i d + i d \wh{\otimes} \nabla'' \right)
\left[ \left(
{\bf u}^{'*} \omega' \wh{\otimes} {\bf u}^{''*} \omega'' \right)
\right] \\
\\
& = \left( {\bf u}' \wh{\otimes} {\bf u}'' \right) \left[
\nabla' \left( {\bf u}^{'*} \omega' \right) \wh{\otimes} {\bf
u}^{''*} \omega'' + (-1)^{\partial \omega'} {\bf u}^{'*} \omega'
\wh{\otimes} \nabla'' \left( {\bf u}^{''*} \omega'' \right) \right] \\
\\
& = \left( \nabla^{' {\bf u}} \omega' \right) \wh{\otimes} \omega'' +
(-1)^{\partial \omega'} \omega' \wh{\otimes} \nabla^{'' {\bf u}}
\omega'' \\
\\
& = \left( \nabla^{' {\bf u}'} \wh{\otimes} i d + i d
\wh{\otimes} \nabla^{'' {\bf u}''} \right) \left( \omega'
\wh{\otimes} \omega'' \right) \\
\end{array}
\eeq
(other proof)
\beq
\begin{array}{ll}
\rho^{\bf u} & = \left( {\bf u}' \wh{\otimes} {\bf u}'' \right) \left(
\rho' \wh{\otimes} 1\!\!1'' + 1\!\!1' \wh{\otimes} \rho'' \right)
\left( {\bf u}^{'*} \wh{\otimes} {\bf u}^{''*} \right) \\
\\
& \hphantom{=} + \left( {\bf u}' \wh{\otimes} {\bf u}'' \right) \left(
\delta' {\bf u}^{'*} \wh{\otimes} {\bf u}^{''*} + {\bf u}^{'*}
\wh{\otimes}
\delta'' {\bf u}^{''*} \right) \\
\\
& = \left( {\bf u}' \rho' {\bf u}^{'*} \wh{\otimes} 1\!\!1'' +
1\!\!1' \wh{\otimes} {\bf u}'' \rho'' {\bf u}^{''*} \right) + \left(
{\bf u}' \delta' {\bf u}^{'*} \wh{\otimes} 1\!\!1'' +
1\!\!1' \wh{\otimes} {\bf u}'' \delta'' {\bf u}^{''*} \right) \\
\\
& = \rho^{' {\bf u}'} \wh{\otimes} 1\!\!1'' +
1\!\!1' \wh{\otimes} \rho^{'' {\bf u}'} \\
\end{array}
\eeq

Our next concern is a discussion of  duality versus tensor products,
defined as follows:

\bigskip

\noindent{\bf 11. Definitions.} With $(A_1, {\bf
H}_1),\ (A_2, {\bf H}_2)$ riemannian quantum spaces, the {\it (tensor)
product of
} $(A_1, {\bf H}_1)$ {\it and
} $(A_2, {\bf H}_2)$ is the
riemannian quantum space $(A_1 \otimes A_2,\ {\bf H}_1
\otimes {\bf H}_2)$, where ${\bf H}_1 \otimes {\bf H}_2$ is the tensor
product of the
$K$-cycles ${\bf H}_1$ and ${\bf H}_2$: one has thus, with
${\bf H}_1 = (H_1, D_1, \chi_1),\ {\bf H}_2 = (H_2, D_2, \chi_2)$,
and ${\bf H}_1 \otimes {\bf H}_2 = (H, D, \chi),\ a_1 \in A_1,\ a_2
\in A_2$:
\beq 
\left\{ 
\begin{array}{l} H = H_1
\otimes H_2 \\ 
{}~~ \\
\chi = \chi_1 \otimes \chi_2 \\
{}~~ \\
\underline{a_1 \otimes a_2} = {\underline a_1} \otimes {\underline
a_2} \\ 
{}~~ \\
\!\!\begin{array}{cl}
D & = D_1 \hat{\otimes} i d_2 + i d_1 \hat{\otimes} D_2 \\ 
& \\
& = D_1 \otimes i d_2 + \chi_1 \otimes D_2 
\end{array} 
\end{array} \right.
\eeq

\bigskip

\noindent {\bf 12. Proposition.} {\it The (tensor) product of two dual
riemannian quantum spaces $(A'_1 \otimes A''_1, H_1), (A'_2 \otimes
A''_2, H_2)$ is a dual riemannian quantum space $( (A'_1 \otimes
A'_2) \otimes (A''_1 \otimes A''_2), H_1 \otimes H_2 )$. 
}

\bigskip

\noindent \ul{Proof:} With $(A_1 = A'_1 \otimes A''_1, {\bf H}_1),\
{\bf H}_1 = (H_1,D_1,\chi_1),\ (A_2 = A'_2 \otimes A''_2, {\bf H}_2)
{\bf H}_2 = (H_2,D_2,\chi_2)$, dual riemannian quantum spaces, we
consider the riemannian quantum space $(A_1 \otimes A_2, H_1 \otimes
H_2)$ tensor product of $(A_1,\ H_1)$ and $(A_2,\ H_2)$ cf.~[4].

With $a'_1 \in A'_1,\ a''_1 \in A''_1,\ a'_2 \in A'_2,\ a''_2 \in
A''_2$, we want to check that one  has:
\beq
\left[ \left[ D, \ul{a'_1} \otimes \ul{a'_2} \right],\ 
\ul{a''_1} \otimes \ul{a''_2} \right] = 0
\eeq
We have, since $\ul{a'_1}$ is even:
\beq
\left( D_1 \otimes id + \chi_1 \otimes D_2 \right)
\left( \ul{a'_1} \otimes \ul{a'_2} \right) = 
D_1 \ul{a'_1} \otimes \ul{a'_2} + 
\chi_1 \ul{a'_1} \otimes D_2 \ul{a'_2},
\eeq
\beq
\left( \ul{a'_1} \otimes \ul{a'_2} \right)
\left( D_1 \otimes id + \chi_1 \otimes D_2 \right) = 
\ul{a'_1} D_1 \otimes \ul{a'_2} + 
\chi_1 \ul{a'_1} \otimes \ul{a'_2} D_2,
\eeq
hence
\beq
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right] = 
\left[ D_1, \ul{a'_1} \right] \otimes \ul{a'_2} + 
\chi_1 \ul{a'_1} \otimes 
\left[ D_2, \ul{a'_2} \right],
\eeq
and further
\beq
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right] 
\left( \ul{a''_1} \otimes \ul{a''_2} \right) = 
\left[ D_1, \ul{a'_1} \right] \ul{a''_1} \otimes \ul{a'_2 a''_2}
+ \chi_1 \ul{a'_1 a''_1} \otimes
\left[ D_2, \ul{a'_2} \right] \ul{a''_2},
\eeq
\beq
\left( \ul{a''_1} \otimes \ul{a''_2} \right) 
\left[ D, \ul{a'_1} \otimes \ul{a'_2} \right] = 
\ul{a''_1} \left[ D_1, \ul{a'_1} \right] \otimes  
\ul{a''_2 a'_2} + 
\chi_1 \ul{a''_1 a'_1} \otimes \ul{a''_2}
\left[ D_2, \ul{a'_2} \right] ,
\eeq
hence, since $\ul{a'_1}$ and $\ul{a''_1}$
commute:
\beq
\left[ \left[ D, \ul{a'_1} \otimes \ul{a'_2} \right],\ 
\ul{a''_1} \otimes \ul{a''_2} \right] = 
\left[ \left[ D_1, \ul{a'_1} \right],\ \ul{a''_1} \right]
\otimes \ul{a'_2 a''_2}
+ \chi_1 \ul{a'_1 a''_1} \otimes
\left[ \left[ D_2, \ul{a'_2} \right],\ \ul{a''_2} \right],
\eeq
vanishing by the duality property of the systems 1 and 2.

A first type of dual riemannian quantum spaces is obtained as follows.

\bigskip

\noindent {\bf 13. Proposition.} {\it The (tensor) product $(A_1
\otimes A_2,\ H_1 \otimes H_2)$ of two riemannian quantum spaces
$(A_1, H_1)$ and $(A_2, H_2)$ is a dual riemannian quantum spaces. 
}

\bigskip

\noindent \ul{Proof:} We have
\beq
\begin{array}{ll}
\left[ D, \ul{a_1 \otimes 1\!\!1_2} \right] & = \left[ D_1 \otimes i
d_2 + \chi_1 \otimes D_2, \ul{a_1} \otimes i d_2 \right] \\
& \\
& = \left[
D_1, \ul{a_1} \right] \otimes i d_2 + \left[ \chi_1, \ul{a_1} \right]
\otimes D_2 \\
& \\
& = \left[
D_1, \ul{a_1} \right] \otimes i d_2 \\
\end{array}
\eeq
hence
\beq
\begin{array}{ll}
\left[ \left[ D, \ul{a_1 \otimes 1\!\!1_2} \right], \ul{1\!\!1_2
\otimes a_1} \right] & = \left[ \left[ D_1, \ul{a_1} \right] \otimes i
d_2, i d_1 \otimes \ul{a_2} \right] \\
& \\
& = \left[
D_1, \ul{a_1} \right] \otimes \ul{a_2} - \left[
D_1, \ul{a_1} \right] \otimes \ul{a_2} = 0
\end{array}
\eeq

Other examples of dual riemannian quantum spaces abund in the theory
of Von Neumann algebras whose ``bilaterality'', as displayed on their
standard representations, is essentially an expression of the fact
that (at least in a supersymmetric situation~[5]) they are
(non-commutative) self dual riemannian quantum spaces. We will return
to this question in a later publication.


\begin{thebibliography}{99}

\bibitem{1} A. Connes, The metric aspect of non-commutative
geometry, Coll\`ege de France preprint (1991).

\bibitem{2} A. Connes, Non-commutative geometry and physics, Les
Houches Lecture Notes (1992).

\bibitem{3} A. Connes, Non-commutative geometry, Book in
press, Oxford University Iron.

\bibitem{4} D. Kastler and D. Testard, Tensor products of quantum
forms, Comm. Math. Phys., {\bf 155} (1993) 135.

\bibitem{5} D. Kastler, Cyclic Cocycles from Graded KMS Functionals,
Comm. Math. Phys., {\bf 121} (1989) 345-350.

\end{thebibliography}


\end{document}