In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Malcev algebra (or Maltsev algebra or
Moufang–
Lie
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be inter ...
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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
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 ...
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 In algebra, a Malcev-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Malcev algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include alternative algebras, Malcev algebra ...
Notes
References
*
*
*
Non-associative algebras
Lie algebras
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