On almost locally solvable Lie algebras with null Jacobson radical of a locally nilpotent radical for Lie algebras

O. A. Pikhtilkova ; E. V. Meshcherina ; A. A. Gorelik

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 284-297

Résumé

In paper proves an analogue of the theorem of F. Kubo [1] for almost locally solvable Lie algebras with zero Jacobson radical. The first section aims to clarify some aspects of the homological description of the Jacobson radical. We prove a theorem generalizing E. Marshall's theorem to the case of almost locally solvable Lie algebras, the consequence of which is an analogue of Kubo's theorem. In the second section, we investigate some properties of a locally nilpotent radical of a Lie algebra. Primitive Lie algebras are considered. Examples are given to show that infinite-dimensional commutative Lie algebras are primitive over any fields; finite-dimensional Abelian algebra, dimensions greater than 1, over an algebraically closed field is not primitive; an example of a non-Artin noncommutative Lie algebra being primitive. It is shown that for special Lie algebras over the characteristic field, the zero $PI$-irreducibly presented radical coincides with the locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, examples of primitive Lie algebras and a non-primitive Lie algebra are given.

Détail

Keywords: Lie algebra, primitive Lie algebra, special Lie algebra, irreducible $PI$-representation, Jacobson radical, locally nilpotent radical, reductive Lie algebra, almost locally solvable Lie algebra.