The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).

In the paper we describe class of all solvable extensions an infinite-dimensional filiform Leibniz algebra. The algebra is taken as a maximal pro-nilpotent ideal residually It proven that second cohomology group extension trivial.

The notion of n-ary algebras, that is vector spaces with a multiplication concerning n-arguments, n ≥ 3, became fundamental since the works of Nambu. Here we first present general notions concerning n-ary algebras and associative n-ary algebras. Then we will be interested in the notion of n-Lie algebras, initiated by Filippov, and which is attached to the Nambu algebras. We study the particular...

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable ...

We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We describe the connection and curvatures, and we investigate necessary and sufficient conditions for subgroups to be totally geodesic submanifolds. We also classify the one-parameter subgroups which are geodesics. Department of Mathematics, Wellesley College, 106 Central St., Wellesley, MA 02481-8203 m...

The exterior algebra over the centre of a Lie algebra acts on the cohomology of the Lie algebra in a natural way. Focusing on nilpotent Lie algebras, we explore the module structure afforded by this action. We show that for all two-step nilpotent Lie algebras, this module structure is non-trivial, which partially answers a conjecture of Cairns and Jessup [4]. The presence of free submodules ind...

We study the C∗-algebra An,θ generated by the Anzai flow on the n-dimensional torus T. It is proved that this algebra is a simple quotient of the group C∗-algebra of a lattice subgroup Dn of a (n + 2)-dimensional connected simply connected nilpotent Lie group Fn whose corresponding Lie algebra is the generic filiform Lie algebra fn. Other simple infinite dimensional quotients of C∗(Dn) are also...

This paper shows a new computational method to obtain filiform Lie algebras, which is based on the relation between some known invariants of these algebras and the maximal dimension of their abelian ideals. Using this relation, the law of each of these algebras can be completely determined and characterized by means of the triple consisting of its dimension and the invariants z1 and z2. As exam...

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a la...