What is meant by $A^{+}$ and $A^{-}$ in algebra? I read it in Jordan Algebra $A^+$
2026-02-23 12:33:44.1771850024
What is meant by $A^{+}$ and $A^{-}$ in algebra?
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If $A$ is an associative algebra over a field of characteristic $\ne2$, it has two related nonassociative algebras, $A^+$ and $A^-$; in the former the multiplication is defined by $$ x\circ y=\frac{1}{2}(xy+yx) $$ and it makes $A^+$ into a Jordan algebra.
In the latter the multiplication is defined by $$ [xy]=xy-yx $$ and it makes $A^-$ into a Lie algebra.
Note that addition and multiplication by scalars remains the same as in $A$. Notation may vary, though.