Nilpotent algebras, implicit function theorem, and polynomial quasigroups

Abstract

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.