Hopf-like Algebras and Extended P-B-W Theorems

TL;DR

This paper introduces new algebraic structures called Hopf-like algebras, extends P-B-W theorems to novel representations of Lie and Leibniz algebras, and generalizes Hopf algebra structures on their enveloping algebras.

Contribution

It presents new representations for Lie and Leibniz algebras and extends classical theorems and structures to these frameworks.

Findings

Extended P-B-W theorems for new algebraic representations

Introduction of Hopf-like algebra structures

Generalization of Hopf algebra structures on enveloping algebras

Abstract

Based on invariant algebras, we introduce representations$^{6-th}$ of Lie algebras and representations$^{< 4-th>}$ of Leibniz algebras, give the extended P-B-W Theorems in the context of the new representations of Lie algebras and Leibniz algebras, and generalize the Hopf-algebra structure on the enveloping algebras of Lie Algebras.