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quadratic algebra (Definition)

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. $ 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)$.



"quadratic algebra" is owned by CWoo.
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See Also: quadratic Lie algebra
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Cross-references: symmetric bilinear form, quadratic equation, quadratic form, commutative ring, unity, non-associative algebra

This is version 1 of quadratic algebra, born on 2005-04-12.
Object id is 6950, canonical name is QuadraticAlgebra.
Accessed 2076 times total.

Classification:
AMS MSC: 17A45 (Nonassociative rings and algebras :: General nonassociative rings :: Quadratic algebras )
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