nLab
alternative algebra

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

  • left alternativity x(xy)=(xx)yx(x y)=(x x)y for all x,yAx,y\in A
  • right alternativity (xy)y=x(yy)(x y)y=x(y y) for all x,yAx,y\in A

A nonassociative algebra (A,,+)(A,\cdot,+) is alternative if its underlying magma (A,)(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 R\mathbf{R}-algebra with involution is alternative.

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