PlanetMath: solvable Lie algebra

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solvable Lie algebra

(Definition)

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

$\displaystyle \mathcal{D}_1 \mathfrak{g}\supset \mathcal{D}_2 \mathfrak{g}\sups... ..._3 \mathfrak{g}\supset \cdots \supset \mathcal{D}_k \mathfrak{g}\supset \cdots$

of $\mathfrak{g}$ , inductively defined for every natural number

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

$\displaystyle \mathcal{D}^1 \mathfrak{g}\supset \mathcal{D}^2 \mathfrak{g}\sups... ...^3 \mathfrak{g}\supset \cdots \supset \mathcal{D}^k \mathfrak{g}\supset \cdots$

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 . 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.

"solvable Lie algebra" is owned by djao.

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Also defines:

nilpotent Lie algebra, solvable, nilpotent, lower central series, upper central series

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Cross-references: terms, right, ideals, natural number, subalgebras, filtration, Lie algebra
There are 15 references to this entry.

This is version 1 of solvable Lie algebra, born on 2002-05-29.
Object id is 2964, canonical name is SolvableLieAlgebra.
Accessed 9849 times total.

Classification:

AMS MSC:

17B30 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Solvable, nilpotent algebras)

Pending Errata and Addenda

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