Zero product determined Jordan algebras, I

Abstract

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {., .} from 𝒮 × 𝒮 into a vector space X satisfies {x, y} = 0 whenever x ○ y = 0, then there exists a linear map T : 𝒮 → X such that {x, y} = T(x ○ y) for all x, y ∈ 𝒮 (here, x ○ y = xy + yx).

Keywords:

• zero product determined Jordan algebra
• symmetric matrix
• transpose involution
• symplectic involution
• bilinear map
• linear map

AMS Subject Classifications::

• 17C20
• 15A86