ABSTRACT

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 ≤ 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 ≤ 7 which separates all algebras except some algebras having a null weight or a continuous series and some of its ”limit points”.

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.

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.

The aim of this book is to provide the reader with complete cohomological data for indecomposable complex Lie algebras of dimension ≤ 7, which can also be used for further calculations.

We consider only trivial and adjoint cohomology:

* Trivial cohomology because of the so-called ”Riemann Hypothesis” on the spectrality of the maximal torus and the interest in the Poincaré polynomial.

* Adjoint cohomology because of its relevance to deformation theory and the amount of structural informations it contains.

To be precise , this book (which is self-contained) gives the following data which do not appear elsewhere in the litterature:

Trivial cohomology:

coboundaries, Betti numbers, characters of each cohomological space for the action of the maximal torus, and bases of eigenspaces, reduction over ZZ of the Poincaré polynomial, test of the Riemann Hypothesis. These computations are performed in two distinct ways: the first way doesn’t make use of Poincaré Duality , the second uses harmonic cohomology and Poincaré Duality since we explain in the Introduction why harmonic cohomology is needed in order to get explicitely the dual basis via Poincaré Duality.

Adjoint cohomology:

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.

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.

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 G_{7,1.3(i_{λ})} and G_{7,1.3(iii)}. It is also a fact that Riemann Hypothesis holds for Lie algebras of dimension ≤ 7 under restrictions that are precisely formulated in the Introduction.

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 be used to test new results.