Coassociative Lie algebras

TL;DR

This paper introduces coassociative Lie algebras, exploring their structure and enveloping algebras, which serve as coalgebraic deformations of traditional universal enveloping algebras, enriching the theory of Hopf algebras.

Contribution

It defines coassociative Lie algebras and studies their enveloping algebras, providing new examples of noncommutative, noncocommutative Hopf algebras and classifying certain connected Hopf algebras.

Findings

Enveloping algebras are coalgebraic deformations of universal enveloping algebras.

New examples of noncommutative, noncocommutative Hopf algebras are constructed.

Classification of connected Hopf algebras of Gelfand-Kirillov dimension four is achieved.

Abstract

A coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of noncommutative and noncocommutative Hopf algebras and leads to a classification of connected Hopf algebras of Gelfand-Kirillov dimension four in [WZZ].