Akivis algebra

Summary

An Akivis algebra is a vector space with an antisymmetric bilinear bracket and a trilinear operation, linked by the Akivis identity. The identity says that the Jacobi expression of the bracket equals an alternating sum of the trilinear operation. Every nonassociative algebra gives an Akivis algebra, with the commutator as bracket and the associator as trilinear operation. This parallels the way every associative algebra gives a Lie algebra. Akivis algebras arose in the differential geometry of webs. The tangent space at the identity of a local smooth loop is an Akivis algebra. This parallels the Lie algebra of a Lie group.