Maltsev Algebra

In mathematics, a Malcev algebra (or Maltsev algebra or Ruth Moufang, Moufang–Sophus Lie, Lie algebra) over a field (mathematics), 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 group (mathematics), 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.

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 7-sphere may be given the structure of a smooth Moufang loop by identifying it with the unit octonions. The tangent space of the identity of this Moufang loop may be identified with the 7-dimensional space of imaginary octonions. The imaginary octonions form a Malcev algebra with the Malcev product ''xy'' − ''yx''.

See also

*Malcev-admissible algebra

Notes

References

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Non-associative algebras
Lie algebras

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