flexible algebra

A non-associative algebra A is flexible if [a,b,a]=0 for all a,bA, where [,,] is the associator on A. In other words, we have (ab)a=a(ba) for all a,bA. Any associative algebra is clearly flexible. Furthermore, any alternative algebraMathworldPlanetmath with characteristic 2 is flexible.

Given an element a in a flexible algebra A, define the left power of a iteratively as follows:

  1. 1.


  2. 2.


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

  1. 1.


  2. 2.


Then, we can show that Ln(a)=Rn(a) for all positive integers n. As a result, in a flexible algebra, one can define the (multiplicative) power of an element a as an unambiguously.

Title flexible algebra
Canonical name FlexibleAlgebra
Date of creation 2013-03-22 14:43:30
Last modified on 2013-03-22 14:43:30
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 17A20
Related topic Associator
Related topic AlternativeAlgebra
Defines left power
Defines right power
Defines flexible