# Malcev algebra

In mathematics, a **Malcev algebra** (or **Maltsev algebra** or **Moufang–Lie algebra**) over a field is a nonassociative algebra that is antisymmetric, so that

and satisfies the **Malcev identity**

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

- 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

## Notes

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

## 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) [1994], "Mal'tsev algebra", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Mal'cev, A. I. (1955), "Analytic loops",
*Mat. Sb. N.S.*(in Russian),**36**(78): 569–576, MR 0069190