Sage computations of \(\mathfrak{sl}_{2}(k)\)-Levi extensions. (English) Zbl 1306.17004

Benito, Pilar; de-la-Concepción, Daniel

Tbil. Math. J. 5, No. 2, 3-17 (2012).

It is well-known that Levi’s theorem decomposes any arbitrary Lie algebra of characteristic zero as a semidirect sum of a semisimple Lie algebra, known as Levi factor, and its solvable radical. At this respect, given a solvable Lie algebra \(\mathcal{R}\), a semisimple Lie algebra \(\mathcal{S}\) is said to be a Levi extension of \(\mathcal{R}\) if a Lie structure can be defined on the vector space \(\mathcal{S} \oplus \mathcal{R}\). It was already proved by Šnobl that the classes of characteristically nilpotent or filiform Lie algebras do not admit these extensions. In this paper, the authors carry on this research and show computational examples of series of nilpotent Lie algebras in arbitrary dimension not being abelian or Heisenberg and allowing Levi extensions by using Sage software. In the case of nilpotent Lie algebras admitting \(\mathfrak{sl}_{2}(k)\) as Levi factor special constructions are given by means of Sage routines based on transvections over \(\mathfrak{sl}_{2}(k)\)-irreducible modules.

Reviewer: Juan Núñez Valdés (Sevilla)

MSC:

17B30	Solvable, nilpotent (super)algebras
17-04	Software, source code, etc. for problems pertaining to nonassociative rings and algebras
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

Levi factor; nilpotent Lie algebra; simple Lie algebra; semisimple Lie algebra; representation; transvection

Citations:

Zbl 1231.17004

Software:

SageMath