On the structure of maximal solvable extensions and of Levi extensions of nilpotent Lie algebras. (English) Zbl 1231.17004

Šnobl, L.

J. Phys. A, Math. Theor. 43, No. 50, Article ID 505202, 17 p. (2010).

As it is well-known, the problem of classification of all solvable (including nilpotent) Lie algebras in an arbitrary large finite dimension is presently unsolved and is generally believed to be unsolvable. This is due to that the problem stems from an obvious fact that the number of solvable Lie algebras in higher dimensions increases drastically, and infinite parametrized families of such non-isomorphic algebras arise already in very low dimensions. As a possible stopgap solution, the idea of the classification of solvable extensions of certain particular classes of nilpotent Lie algebras, that is, of all solvable non-nilpotent algebras with the given nilradical of arbitrary large dimension emerged. In this paper, the author establishes and improves the upper estimate on the dimension of any solvable Lie algebra \(\mathfrak s\) with its nilradical isomorphic to a given nilpotent Lie algebra \(\mathfrak n\). Next the author considers Levi decomposable algebras with a given nilradical \(\mathfrak n\) and investigates restrictions on possible Levi factors originating from the structure of characteristic ideals of \(\mathfrak n\). Indeed, the author presents a new perspective on Turkowski’s classification of Levi decomposable algebras up to dimension 9. However, as the improved bound cannot give a precise estimate on the maximal dimension of a solvable extension in all cases, the author finishes the paper by formulating as a conjecture, that the maximal solvable extension of the given nilradical over the field of complex numbers turns out to be unique up to isomorphism.

Reviewer: Juan Núñez Valdés (Sevilla)

MSC:

17B30	Solvable, nilpotent (super)algebras
17B05	Structure theory for Lie algebras and superalgebras

Keywords:

Noncommutative algebras; Theorem of Campbell, Baker, Hausdorff and Dynkin