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{{for|the Lie algebras or groups |Malcev Lie algebra}}
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In [[mathematics]], a '''Malcev algebra''' (or '''Maltsev algebra''' or '''[[Ruth Moufang|Moufang]]–[[Sophus Lie|Lie]] algebra''') over a field is a [[nonassociative algebra]] that is antisymmetric, so that
 
:<math>xy = -yx\ </math>
 
and satisfies the '''Malcev identity'''
 
:<math>(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.\ </math>
 
They were first defined by [[Anatoly Maltsev]] (1955).
 
==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''&nbsp;&minus;&nbsp;''yx''.
*The imaginary [[octonions]] form a 7-dimensional Malcev algebra by defining the Malcev product to be ''xy''&nbsp;&minus;&nbsp;''yx''.
 
==See also==
 
*[[Malcev-admissible algebra]]
 
==References==
* Alberto Elduque and Hyo C. Myung ''Mutations of alternative algebras'',  Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
*{{springer|id=M/m062170|author=V.T. Filippov|title=Mal'tsev algebra}}
*{{Citation | last1=Mal'cev | first1=A. I. | title=Analytic loops | id={{MathSciNet | id = 0069190}} | year=1955 | journal=Mat. Sb. N.S. | volume=36 | pages=569–576|language=Russian | issue=78}}
 
{{algebra-stub}}
[[Category:Non-associative algebras]]
[[Category:Lie algebras]]

Latest revision as of 14:07, 28 July 2014

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