Pre-Lie algebra

Summary

A pre-Lie algebra, also called a left-symmetric algebra, is a nonassociative algebra whose associator is symmetric in two of its three arguments. Its commutator $xy - yx$ is always a Lie bracket, so every pre-Lie algebra gives a Lie algebra. Pre-Lie algebras arise from flat torsion-free connections and from the composition of vector fields on affine space. They also arise from rooted trees in the study of numerical integrators, and in deformation theory.