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.
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
- Nagy, Peter T. (1992). "Moufang loops and Malcev algebras" (PDF). Seminar Sophus Lie. 3: 65–68. CiteSeerX 10.1.1.231.8888.
References
- Elduque, Alberto; Myung, Hyo C. (1994), Mutations of alternative algebras, Kluwer, ISBN 0-7923-2735-7
- Filippov, V.T. (2001) , "Mal'tsev algebra", Encyclopedia of Mathematics, EMS Press
- Mal'cev, A. I. (1955), "Analytic loops", Mat. Sb., New Series (in Russian), 36 (78): 569–576, MR 0069190
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