# flexible algebra

A non-associative algebra $A$ is 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:

1. 1.

$L^{1}(a)=a$,

2. 2.

$L^{n}(a)=a\cdot L^{n-1}(a)$.

Similarly, we can define the right power of $a$ as:

1. 1.

$R^{1}(a)=a$,

2. 2.

$R^{n}(a)=R^{n-1}(a)\cdot a$.

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.

Title flexible algebra FlexibleAlgebra 2013-03-22 14:43:30 2013-03-22 14:43:30 CWoo (3771) CWoo (3771) 11 CWoo (3771) Definition msc 17A20 Associator AlternativeAlgebra left power right power flexible