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

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.