Towers, David A. and Varea, Vicente R. (2013) *Further results on elementary Lie algebras and Lie A-algebras.* Communications in Algebra, 41 (4). pp. 1432-1441. ISSN 0092-7872

## Abstract

A finite-dimensional Lie algebra L over a field F of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in a previous paper. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $\Phi$ the classes of A-algebras, almost algebraic algebras, E-algebras, elementary algebras and $\phi$-free algebras respectively, then it is shown that: \mathcal{L} \subset \Phi \subset \mathcal{G} \mathcal{L} \subset \mathcal{A} \subset \mathcal{E} \mathcal{G} \cap \mathcal{A} = \mathcal{L}. It is also shown that if L is a semisimple Lie algebra all of whose minimal parabolic subalgebras are $\phi$-free then L is an A-algebra, and hence elementary. This requires a number of quite delicate properties of parabolic subalgebras. Finally characterisations are given of $E$-algebras and of Lie algebras all of whose proper subalgebras are elementary.

Item Type: | Journal Article |

Journal or Publication Title: | Communications in Algebra |

Additional Information: | The final, definitive version of this article has been published in the Journal, Communications in Algebra, 41 (4), 2013, © Informa Plc |

Uncontrolled Keywords: | Lie algebra ; elementary ; E-algebra ; A-algebra ; almost algebraic ; ad-semisimple ; parabolic subalgebra |

Subjects: | ?? qa ?? |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 27159 |

Deposited By: | Dr David A. Towers |

Deposited On: | 05 Oct 2009 15:09 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 14 Apr 2018 00:04 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/27159 |
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