Extending Enveloping Algebras via Anti-Cocommutative Elements

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.