Classification of all non-split central extensions of naturally graded 2-filiform non-Lie Leibniz algebras

Abstract

This paper deals with the naturally graded 2-filiform non-Lie Leibniz algebras over the field of complex numbers. There are two finite series of such algebras, defined in every dimension starting from five, and two further algebras which occur only in dimension five. For each of these algebras we classify, up to isomorphism, all of its non-split central extensions, of every possible dimension. The one-dimensional central extensions are already known from an earlier work, so the main part of the paper is devoted to the extensions of dimension two and higher. As usual, the classification is reduced to the description of the orbits of the automorphism group of the algebra on the subspaces of its second cohomology space with trivial coefficients. For each algebra and each dimension we give the complete list of these orbits together with the multiplication tables of the corresponding algebras, and we point out the extensions whose annihilator coincides with the added central ideal. Among the algebras obtained there are 2-filiform Leibniz algebras which are not naturally graded; this class of algebras has not been systematically studied so far.