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

x

y

=

−

y

x

{\displaystyle xy=-yx}

and satisfies the Malcev identity

(

x

y

)

(

x

z

)

=

(

(

x

y

)

z

)

x

+

(

(

y

z

)

x

)

x

+

(

(

z

x

)

x

)

y

.

{\displaystyle (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y.}

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.