Malcev algebra
From Wikipedia, the free encyclopedia
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)
