# solvable Lie algebra

Let $\mathfrak{g}$ be a Lie algebra. The 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 SolvableLieAlgebra 2013-03-22 12:41:06 2013-03-22 12:41:06 djao (24) djao (24) 4 djao (24) Definition msc 17B30 nilpotent Lie algebra solvable nilpotent lower central series upper central series