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\centerline{\bf{ABSTRACT}}
%centerline{\it{(back cover and inside back cover)}}  \normalfont
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 Due to the lack of a good invariant, a general classification of complex
finite-dimensional Lie algebras seems to be, for the time being, hopeless. However several
classifications have been computed "by hand" in different ways for Lie algebras
of dimension $\leq 7.$ In particular, the weight system for the action of a maximal torus
is an invariant of the Lie algebra under consideration and gives rise to a classification
of Lie algebras  of dimension $\leq 7$ which separates all algebras except some
algebras having a null weight or a continuous series and some of its
"limit points".
\par
Then an obvious question is how going to weight systems on the cohomology would
refine this classification, i.e. separate algebras which are not separated by weights
on the algebra itself.
\par
But very few results are known on the cohomology of nilpotent Lie algebras, except for
the very special case of the nilradical of a minimal parabolic subalgebra of a semisimple
Lie algebra. Anyone interested in computing by hand cohomology , even in dimensions
as low as 4,5,6, will be discouraged by the amount of calculation. This is naturally
the domain of such a computer algebra  system like REDUCE.
\par
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The aim of this book is to provide the reader with complete cohomological data
for indecomposable complex Lie algebras of dimension $\leq 7$, which can
also be used for further calculations.
\par
We consider only trivial and adjoint cohomology:
\begin{itemize}
\item[*] Trivial cohomology because of the so-called "Riemann Hypothesis" on the
spectrality of the maximal torus and the interest in the Poincar\'e polynomial.
\item[*] Adjoint cohomology because of its relevance to deformation theory and the
amount of structural informations it contains.
\end{itemize}
\par
To be precise , this book (which is self-contained) gives the following data
which do not appear elsewhere in the litterature:
\medskip
\par {\bf{Trivial cohomology:}}
\par
coboundaries, Betti numbers, characters of each cohomological space for the action
of the maximal torus, and bases of eigenspaces, reduction over $Z \hskip -5pt Z$
of the Poincar\'e polynomial, test of the Riemann Hypothesis.
\\
These computations are performed in two distinct ways: the first way doesn't make use of
Poincar\'e Duality , the second uses harmonic cohomology and Poincar\'e Duality since
we explain in the Introduction why harmonic cohomology is needed in order to get
explicitely the dual basis via Poincar\'e Duality.
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\par {\bf{Adjoint cohomology}:}
\par
Structure of the derivation algebra and the 1-cohomology algebra (bases and commutation
relations), adjoint Betti numbers, characters of all cohomological spaces, explicit
basis for the 2-cohomological space consisting of eigenvectors for the action of the
maximal torus.
\\  % \it{(continued inside back cover)}  \normalfont
\par
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Continuous series are carefully studied and "singular values" of the parameter (i.e.
values which give rise to a cohomology different from the generic one) are computed.
\par
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%{\bf{3}}.
From these data it is readily seen that trivial cohomology doesn't refine the
classification but adjoint cohomology does except in one case , namely the algebras
${\cal G}_{7,1.3(i_{\lambda})}$ and
${\cal G}_{7,1.3(iii)}$.
It is also a fact that Riemann Hypothesis holds for Lie algebras of dimension $\leq 7$
under restrictions that  are precisely formulated  in the Introduction.
\par
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%{\bf{4}}.
This book may be of interest to people working in Lie algebra cohomology, deformation
theory , classification or structure of Lie algebras, since the data it provides can
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be used to test new results.
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