Malcev algebra

In mathematics, a Malcev algebra (or Maltsev algebra or MoufangLie 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.[1]

  1. ^ Nagy, Peter T. (1992). "Moufang loops and Malcev algebras" (PDF). Seminar Sophus Lie. 3: 65–68. CiteSeerX 10.1.1.231.8888.