Pre-Lie algebras and the rooted trees operad (Q2725069)

In the paper under review the authors study a class of nonassociative algebras which they call pre-Lie algebras. A pre-Lie algebra is a vector space with multiplication satisfying the identity \((xy)z-x(yz)=(xz)y-x(zy)\). Pre-Lie structures have appeared in different contexts, including Vinberg's work on convex homogeneous cones, in the study of affine manifolds, and Hochschild cohomology. In the literature pre-Lie algebras have been given different names, e.g. left and right symmetric algebras and Vinberg algebras. Rooted trees are related with vector fields, numerical analysis, quantum field theory.NEWLINENEWLINEIn the paper the authors define the underlying operad of pre-Lie algebras in terms of rooted trees. Then they define the operadic homology of pre-Lie algebras and prove that the operad associated to pre-Lie algebras is a Koszul operad. In particular, they show that the free pre-Lie algebra is a free module of the enveloping algebra of the underlying Lie algebra.