Hellenica World

# .

In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that

$$xy = -yx\$$

and satisfies the Malcev identity

$$(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.\$$

They were first defined by Anatoly Maltsev (1955).

Examples

Any Lie algebra is a Malcev algebra.
Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy − yx.
The imaginary octonions form a 7-dimensional Malcev algebra by defining the Malcev product to be xy − yx.