Extending Enveloping Algebras via Anti-Cocommutative Elements

TL;DR

This paper explores extending universal enveloping algebras of Lie algebras by incorporating anti-cocommutative elements, analyzing their properties and implications for connected Hopf algebras.

Contribution

It introduces a novel approach to extend enveloping algebras using anti-cocommutative elements, expanding understanding of their structure and properties.

Findings

Characterization of Lie algebras with anti-cocommutative elements

Construction methods for extended enveloping algebras

Results on the antipode properties of connected Hopf algebras

Abstract

Anti-cocommutativity was introduced by Wang, Zhuang, Zhang (2013) in their paper Coassociative Lie algebras. Since universal enveloping algebras of Lie algebras are connected Hopf algebras, we extend enveloping algebras using the notion of anti-cocommutative elements. This concept is the main focus of this thesis. We separate the results (Chapter 4) into four parts: 1) Lie algebras containing anti-cocommutative elements; 2) Extending enveloping algebras using anti-cocommutative elements; 3) A global dimension problem for connected Hopf algebras; 4) Properties of the antipode of some connected Hopf algebras.