Sabinin Algebras and Nonassociative Lie Theory

Ivan Shestakov

Abstract:
Lie algebras naturally apear from and are related with various associative
objects: associative algebras, Lie groups, Hopf algebras, groups with
central series, etc. There were many attempts to extend some of these
concepts and results to nonassociative setting. We show that the so called
hyperalgebras or Sabinin algebras provide a good substitution for Lie
algebras in what might be called a "nonassociative Lie Theory":  the Lie
correspondence between loops and Sabinin algebras, the nonassociative
Poincare-Birkhoff-Witt theorem, the Milnor-Moore Theorem for
nonassociative Hopf algebras, the Magnus imbedding for loops with central
series give an evidence to this assertion.