In
mathematics, a Malcev algebra (or Maltsev algebra or
Moufang–
Lie algebra) over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is a
nonassociative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...
that is antisymmetric, so that
:
and satisfies the Malcev identity
:
They were first defined by
Anatoly Maltsev Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, an ...
(1955).
Malcev algebras play a role in the theory of
Moufang loop Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
s that generalizes the role of
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
in the theory of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s. 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 In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is o ...
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
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
. 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
*
*
*
Non-associative algebras
Lie algebras
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