SolvableLieAlgebra 2002-05-29 05:06:12 2002-05-29 05:06:12 Definition nilpotent Lie algebra solvable nilpotent lower central series upper central series % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\g}{\mathfrak{g}} \newcommand{\D}{\mathcal{D}} \newcommand{\h}{\mathfrak{h}} Let $\g$ be a Lie algebra. The {\em lower central series} of $\g$ is the filtration of subalgebras $$ \D_1 \g \supset \D_2 \g \supset \D_3 \g \supset \cdots \supset \D_k \g \supset \cdots $$ of $\g$, inductively defined for every natural number $k$ as follows: \begin{eqnarray*} \D_1 \g & := & [\g,\g] \\ \D_k \g & := & [\g, \D_{k-1} \g] \end{eqnarray*} The {\em upper central series} of $\g$ is the filtration $$ \D^1 \g \supset \D^2 \g \supset \D^3 \g \supset \cdots \supset \D^k \g \supset \cdots $$ defined inductively by \begin{eqnarray*} \D^1 \g & := & [\g,\g] \\ \D^k \g & := & [\D^{k-1} \g, \D^{k-1} \g] \end{eqnarray*} In fact both $\D^k \g$ and $\D_k \g$ are ideals of $\g$, and $\D^k \g \subset \D_k \g$ for all $k$. The Lie algebra $\g$ is defined to be {\em nilpotent} if $\D_k \g = 0$ for some $k \in \mathbb{N}$, and {\em solvable} if $\D^k \g = 0$ for some $k \in \mathbb{N}$. A subalgebra $\h$ of $\g$ is said to be {\em nilpotent} or {\em solvable} if $\h$ is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of $\g$, since every ideal of $\g$ is also a subalgebra.