Some classes of flexible Lie-admissible algebras

Abstract

Let A \mathfrak {A} be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic ≠ 2 \ne 2 . Suppose that A − {\mathfrak {A}^ - } has a split abelian Cartan subalgebra H \mathfrak {H} which is nil in A \mathfrak {A} . It is shown that if every nonzero root space of A − {\mathfrak {A}^ - } for H \mathfrak {H} is one-dimensional and the center of A − {\mathfrak {A}^ - } is 0, then A \mathfrak {A} is a Lie algebra isomorphic to A − {\mathfrak {A}^ - } . This generalizes the known result obtained by Laufer and Tomber for the case that A − {\mathfrak {A}^ - } is simple over an algebraically closed field of characteristic 0 and A \mathfrak {A} is power-associative. We also give a condition that a Levi-factor of A − {\mathfrak {A}^ - } be an ideal of A \mathfrak {A} when the solvable radical of A − {\mathfrak {A}^ - } is nilpotent. These results yield some interesting applications to the case that A − {\mathfrak {A}^ - } is classical or reductive.