# quadratic algebra

A non-associative 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 $Q\colon A\to R$ such that

1. 1.

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

2. 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 QuadraticAlgebra 2013-03-22 15:11:38 2013-03-22 15:11:38 CWoo (3771) CWoo (3771) 4 CWoo (3771) Definition msc 17A45 QuadraticLieAlgebra