# nLab alternative algebra

A magma $A$ is (left and right) alternative if its product satisfies

• left alternativity $x(x y)=(x x)y$ for all $x,y\in A$
• right alternativity $(x y)y=x(y y)$ for all $x,y\in A$

A nonassociative algebra $(A,\cdot,+)$ is alternative if its underlying magma $(A,\cdot)$ alternative.

In particular, every associative algebra is alternative. The properly nonassociative algebra of octonions is also alternative. The reason is that it is obtained by a doubling procedure from the algebra of quaternions which is associative; and the double of any associative $\mathbf{R}$-algebra with involution is alternative.

Revised on September 8, 2011 19:22:31 by Zoran Škoda (161.53.130.104)