Bowman, Kevin and Towers, David A. and Varea, Vicente R. (2004) *On Lie algebras all of whose minimal subalgebras are lower modular.* Communications in Algebra, 32 (12). pp. 4515-4533. ISSN 0092-7872

## Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.

Item Type: | Article |
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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, 32 (12), 2004, © Informa Plc |

Uncontrolled Keywords: | Lie algebras ; Lattice of subalgebras ; Modular subalgebra |

Subjects: | Q Science > QA Mathematics |

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

ID Code: | 1660 |

Deposited By: | Dr David A. Towers |

Deposited On: | 15 Feb 2008 14:47 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 21 Sep 2017 03:09 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/1660 |

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