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Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of 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.