Malcev algebra

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

In mathematics, a Malcev algebra (or Maltsev algebra or MoufangLie 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[edit]

  • 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[edit]

References[edit]