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'flexible algebra'
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| Title of object: |
flexible algebra |
| Canonical Name: |
FlexibleAlgebra |
| Type: |
Definition |
| Created on: |
2004-10-10 14:34:05 |
| Modified on: |
2006-01-27 00:24:39 |
| Classification: |
msc:17A20 |
Revision comment (for changes between this and next version):
| Changes for correction #10349 ('emphasis on defined terms and add to defines list'). |
Preamble:
% 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,amscd}
\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 |
Content:
A non-associative algebra $A$ is \emph{flexible} if $[\ a,b,a\ ]=0$ for all $a,b\in A$, where $[\ , , ]$ is the associator on $A$. In other words, we have $(ab)a=a(ba)$ for all $a,b\in A$. Any associative algebra is clearly flexible. Furthermore, any alternative algebra with characteristic $\neq 2$ is flexible.
Given an element $a$ in a flexible algebra $A$, define the left power of $a$ iteratively as follows:
\begin{enumerate}
\item $L^1(a)=a$,
\item $L^n(a)=a\cdot L^{n-1}(a)$.
\end{enumerate}
Similarly, we can define the right power of $a$ as:
\begin{enumerate}
\item $R^1(a)=a$,
\item $R^n(a)=R^{n-1}(a)\cdot a$.
\end{enumerate}
Then, we can show that $L^{n}(a)=R^{n}(a)$ for all positive integers $n$. As a result, in a flexible algebra, one can define the (multiplicative) power of an element $a$ as $a^n$ unambiguously. |
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