**On Lie algebras all of whose minimal subalgebra...**152 KB, PDF-document

3/11/15

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 2004 |
---|---|

<mark>Journal</mark> | Communications in Algebra |

Issue number | 12 |

Volume | 32 |

Number of pages | 19 |

Pages (from-to) | 4515-4533 |

<mark>State</mark> | Published |

<mark>Original language</mark> | English |

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.

The final, definitive version of this article has been published in the Journal, Communications in Algebra, 32 (12), 2004, © Informa Plc