Malcev algebra

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).

Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop.[1]

Examples

See also

Notes

  1. Peter T. Nagy (1992). "Moufang loops and Malcev algebras" (PDF). Seminar Sophus Lie 3: 65–68.

References


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