Malcev algebra

For the Lie algebras or groups, see Malcev Lie algebra.

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.

References

• Alberto Elduque and Hyo C. Myung Mutations of alternative algebras, Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
• V.T. Filippov (2001), "Mal'tsev algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/M/m062170.htm 
• Mal'cev, A. I. (1955), "Analytic loops" (in Russian), Mat. Sb. N.S. 36 (78): 569–576, MR0069190