# solvable Lie algebra

Let $\U0001d524$ be a Lie algebra^{}. The lower central series^{} of $\U0001d524$ is the filtration^{} of subalgebras^{}

$${\mathcal{D}}_{1}\U0001d524\supset {\mathcal{D}}_{2}\U0001d524\supset {\mathcal{D}}_{3}\U0001d524\supset \mathrm{\cdots}\supset {\mathcal{D}}_{k}\U0001d524\supset \mathrm{\cdots}$$ |

of $\U0001d524$, inductively defined for every natural number^{} $k$ as follows:

${\mathcal{D}}_{1}\U0001d524$ | $:=$ | $[\U0001d524,\U0001d524]$ | ||

${\mathcal{D}}_{k}\U0001d524$ | $:=$ | $[\U0001d524,{\mathcal{D}}_{k-1}\U0001d524]$ |

The upper central series of $\U0001d524$ is the filtration

$${\mathcal{D}}^{1}\U0001d524\supset {\mathcal{D}}^{2}\U0001d524\supset {\mathcal{D}}^{3}\U0001d524\supset \mathrm{\cdots}\supset {\mathcal{D}}^{k}\U0001d524\supset \mathrm{\cdots}$$ |

defined inductively by

${\mathcal{D}}^{1}\U0001d524$ | $:=$ | $[\U0001d524,\U0001d524]$ | ||

${\mathcal{D}}^{k}\U0001d524$ | $:=$ | $[{\mathcal{D}}^{k-1}\U0001d524,{\mathcal{D}}^{k-1}\U0001d524]$ |

In fact both ${\mathcal{D}}^{k}\U0001d524$ and ${\mathcal{D}}_{k}\U0001d524$ are ideals of $\U0001d524$, and ${\mathcal{D}}^{k}\U0001d524\subset {\mathcal{D}}_{k}\U0001d524$ for all $k$. The Lie algebra $\U0001d524$ is defined to be nilpotent if ${\mathcal{D}}_{k}\U0001d524=0$ for some $k\in \mathbb{N}$, and solvable if ${\mathcal{D}}^{k}\U0001d524=0$ for some $k\in \mathbb{N}$.

A subalgebra $\U0001d525$ of $\U0001d524$ is said to be nilpotent or solvable if $\U0001d525$ is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of $\U0001d524$, since every ideal of $\U0001d524$ is also a subalgebra.

Title | solvable Lie algebra |
---|---|

Canonical name | SolvableLieAlgebra |

Date of creation | 2013-03-22 12:41:06 |

Last modified on | 2013-03-22 12:41:06 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 17B30 |

Defines | nilpotent Lie algebra |

Defines | solvable |

Defines | nilpotent |

Defines | lower central series |

Defines | upper central series |