Malcev 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.

See also

Malcev-admissible algebra

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, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Mal'cev, A. I. (1955), "Analytic loops", Mat. Sb. N.S. (in Russian) 36 (78): 569–576, MR 0069190

Mathematics Encyclopedia