Extensions of Enveloping Algebras by Anti-Cocommutative Elements

TL;DR

This paper explores extensions of universal enveloping algebras using anti-cocommutative elements, examining their Hopf structures and comparing global dimensions with Lie algebra dimensions.

Contribution

It introduces a new method to extend enveloping algebras via anti-cocommutative elements and analyzes their Hopf algebra properties.

Findings

Connected Hopf algebras can be characterized as enveloping algebras under certain conditions.

The global dimension of these Hopf algebras relates closely to the dimension of their primitive Lie algebras.

The paper establishes criteria for when such extensions preserve enveloping algebra structures.

Abstract

Anti-cocommutative elements were introduced by Wang, Zhang, Zhuang (2013) in their paper Coassociative Lie Algebras. We use this notion to extend universal enveloping algebras of Lie algebras with regards to their Hopf structure, and see if these connected Hopf algebras are enveloping algebras. Furthermore, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements. Title has been renamed to "GK-Dimension of some Connected Hopf Algebras".