The Notion of Lower Central Series for Loops

Abstract The commutator calculus is one of the basic tools in group theory. However, its extension to the nonassociative context, based on the usual definition of the lower central series of a loop, is not entirely satisfactory. Namely, the graded abelian group associated to the lower central series of a loop is not known to carry any interesting algebraic structure. In this note we construct a new generalization of the lower central series to arbitrary loops that is tailored to produce a set of multilinear operations on the associated graded group.

Key words: loop, commutator-associator filtration, lower central series