quadratic algebra
A nonassociative algebra $A$ (with unity ${1}_{A}$) over a commutative ring $R$ (with unity ${1}_{R}$) is called a quadratic algebra if $A$ admits a quadratic form^{} $Q:A\to R$ such that

1.
$Q({1}_{A})={1}_{R}$,

2.
the quadratic equation ${x}^{2}b({1}_{A},x)x+Q(x){1}_{A}=0$ is satisfied by all $x\in A$, where $b$ is the associated symmetric bilinear form^{} given by $b(x,y):=Q(x+y)Q(x)Q(y)$.
Title  quadratic algebra 

Canonical name  QuadraticAlgebra 
Date of creation  20130322 15:11:38 
Last modified on  20130322 15:11:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 17A45 
Related topic  QuadraticLieAlgebra 