Maximal solvable extensions of a pure non-characteristically nilpotent Lie algebra

Abstract

In the present paper we prove that a complex maximal extension of a complex finitedimensional nilpotent Lie algebra is isomorphic to a semidirect sum of nilradical and its maximal torus with a product of natural action of the torus. We prove that such solvable Lie algebras are complete. In fact, we prove that among all complex solvable Lie algebras only maximal extensions of nilpotent algebras have not outer derivations. In addition, we allocate a subclass of maximal solvable extensions of nilpotent Lie algebras that have trivial cohomology group. The comparisons with a several existing results are given, as well.

On some solvable Leibniz algebras and their completeness

“…, x k } forms an abelian subalgebra of R. Therefore, we get a right action of Q on the nilradical N , which decomposes N into one-dimensional root subspaces. In fact, similarly to Lemma 3.2 in [1] we can prove that this is nothing but a torus of the right action on N (recall that the torus is an abelian subalgebra of the Lie algebra Der(N ) consisting of diagonalizable derivations). Since in non-Lie Leibniz algebras the anti-commutativity property is longer true the action of Q on N gives another root decomposition, which need not be diagonal action (the root subspaces with respect to the left action of Q on N happen to be non one-dimensional); • If among the non generator basis elements of the nilradical N there are no elements of the same structure, then…”

“…• In [1] the following result on completeness of solvable Lie algebras has been given which we make use.…”

“…This is the essential difference between maximal solvable Lie and non-Lie Leibniz extensions for the case of dimQ < dim(N/N 2 ) with non-split N . The uniqueness of complex maximal solvable Lie extensions of nilpotent algebras was proved in [1].…”