Malcev algebra

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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).

[edit] 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.

[edit] 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, Kluwer Academic Publishers, ISBN 978-1556080104 
• A.I. Mal'tsev, Analytic loops Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)