On the Leibniz (co)homology of the Lie algebra of the Euclidean group

Loday showed that the Leibniz homology of any simple Lie algebra (thus of the orthogonal Lie algebra) is trivial. We show that the (co)homology of the Lie algebra hn of the Euclidean group, which is an abelian extension of the orthogonal Lie algebra, is highly non trivial. The purpose of this work is the computation of this Leibniz (co)homology. We deal at once with hn for orthogonal Lie algebras of type Bl and Dl as classified by Humphreys [6, p.2]. We use the tensor algebra on hn to find many new invariants for Euclidean geometry that are not detectable from the exterior algebra on hn . Finally we use these invariants in several spectral sequences to .show that the Leibniz cohomology of hn is up to isomorphism, a Zienbiel algebra generated by a ( n -- 1)-fold tensor and an n-fold tensor.