[[Algebra theory MOC]]
# Jordan algebra

A **Jordan algebra** $J$ over $\mathbb{K}$ is commutative non-associative [[K-algebra|algebra]] with a symmetric bilinear product $\{ -,- \} : J \times J \to J$ satisfying the **Jordan identity** #m/def/falg
$$
\begin{align*}
\{ \{ x,y \},\{ x,x \} \}= \{ x,\{ y,\{ x,x \} \} \}
\end{align*}
$$
The quintessential example is the [[Anticommutator]] of a [[K-monoid]],
usually renormalized so that $\{ x,x \} = x^2$.
We denote the anticommutator algebra by $A^{+}$ and the renormalized one as $A^{+ 1/2}$.
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