# Hyporeductive and Pseudoreductive Hopf algebras

## TL;DR

This paper extends Lie's fundamental theorems to hyporeductive and pseudoreductive loops by translating Sabinin's geometric results into an algebraic framework using non-associative Hopf algebras.

## Contribution

It introduces an algebraic formulation of hyporeductive and pseudoreductive loops through non-associative Hopf algebras, generalizing Sabinin's geometric results.

## Key findings

- Established algebraic identities for hyporeductive Hopf algebras
- Connected loop identities with non-associative Hopf algebra structures
- Extended Lie's theorems to broader classes of loops

## Abstract

In his generalization of reductive homogeneous spaces, Lev Sabinin showed that Lie's fundamental theorems hold for local analytic hyporeductive and pseudoreductive loops. We derive Sabinin's results in an algebraic context in terms of non-associative Hopf algebras that satisfy the analog of the hyporeductive and pseudoreductive identities for loops.