solvable Lie algebra

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

\mathcal{D}_{1}\mathfrak{g}\supset\mathcal{D}_{2}\mathfrak{g}\supset\mathcal{D% }_{3}\mathfrak{g}\supset\cdots\supset\mathcal{D}_{k}\mathfrak{g}\supset\cdots

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

	$\displaystyle\mathcal{D}_{1}\mathfrak{g}$	$\displaystyle:=$	$\displaystyle[\mathfrak{g},\mathfrak{g}]$
	$\displaystyle\mathcal{D}_{k}\mathfrak{g}$	$\displaystyle:=$	$\displaystyle[\mathfrak{g},\mathcal{D}_{k-1}\mathfrak{g}]$

The upper central series of $\mathfrak{g}$ is the filtration

\mathcal{D}^{1}\mathfrak{g}\supset\mathcal{D}^{2}\mathfrak{g}\supset\mathcal{D% }^{3}\mathfrak{g}\supset\cdots\supset\mathcal{D}^{k}\mathfrak{g}\supset\cdots

defined inductively by

	$\displaystyle\mathcal{D}^{1}\mathfrak{g}$	$\displaystyle:=$	$\displaystyle[\mathfrak{g},\mathfrak{g}]$
	$\displaystyle\mathcal{D}^{k}\mathfrak{g}$	$\displaystyle:=$	$\displaystyle[\mathcal{D}^{k-1}\mathfrak{g},\mathcal{D}^{k-1}\mathfrak{g}]$

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

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