Malcev algebra
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In mathematics, a Malcev algebra (or Maltsev algebra) over a field is a (nonassociative) algebra that is antisymmetric (xy = −yx) and satisfies the Malcev identity (xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y. They are named after Anatoly Maltsev.
[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
- Mutations of alternative algebras, by Alberto Elduque and Hyo C. Myung, Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
