Completable filiform Lie algebras. (Q1873706)

A Lie algebra is said to be complete if it is centerless and has only inner derivations. A nilpotent finite-dimensional complex Lie algebra \({\mathfrak n}\) is said to be completable if there exists a maximal abelian Lie algebra \({\mathfrak t}\) of derivations of \({\mathfrak n}\) such that the semidirect product \({\mathfrak t}\oplus{\mathfrak n}\) is a complete Lie algebra. With this terminology, the main point of the paper under review is to determine all filiform Lie algebras that are completable (Theorem~4). As a consequence, a filiform Lie algebra is completable provided it is isomorphic to the nilradical of a solvable rigid Lie algebra (Corollary~1). To explain the results that conclude the paper, we note that a Lie algebra \({\mathfrak g}\) is complete if and only if for the cohomology groups of its adjoint representation we have \(H^0({\mathfrak g},{\mathfrak g})=H^1({\mathfrak g},{\mathfrak g})=\{0\}\). Now, the last statement of the paper (Corollary~2) says that there exist complete Lie algebras \({\mathfrak g}\) with arbitrarily large dimension of \(H^2({\mathfrak g},{\mathfrak g})\).