Characteristically nilpotent extensions of nilradicals of solvable rigid laws

Abstract

A Lie algebra g is called characteristically nilpotent if its algebra of derivations is nilpotent. The authors construct the examples of (2m+2)-dimensional characteristically nilpotent Lie algebras g2m+2 with characteristic sequence c(g2m+2) equal to (2m, 1, 1) (c(g) of a nilpotent Lie algebra g is maximum in a lexicographic ordering of the sequence of dimensions of the Jordan blocks of adX, X 2 g−[g, g]). The algebra g2m+2 is obtained by means of three consecutive one-dimensional central extensions e1(L2m−1), e1(e1(L2m−1)), g2m+2 of the filiform Lie algebra L2m−1. L2m−1 is defined by its basis e1, . . . , e2m−1 and commutation relations [e1, ei] = ei+1, 2 i 2m−2. On the other hand the semi-direct sum t(m,m−1) = Ce1(e1(L2m−1)) of Lie algebras is considered such that t(m,m−1) is a solvable, rigid, complete Lie algebra. Thus the algebra g2m+2 is a one-dimensional central extension of the nilradical of t(m,m−1).