Nilpotent algebras, implicit function theorem, and polynomial quasigroups

TL;DR

This paper explores the relationship between finite-dimensional nilpotent algebras and quasigroups, establishing an implicit function theorem and extending classical correspondences to nonassociative algebraic structures.

Contribution

It introduces an implicit function theorem for finite-dimensional nonassociative algebras and links these algebras to quasigroups, expanding classical algebraic correspondences.

Findings

Established the implicit function theorem for nonassociative algebras

Created a correspondence between nilpotent algebras and quasigroups

Analyzed properties of nilpotent groups and related algebraic structures

Abstract

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.