On Hopf 2-algebras

TL;DR

This paper categorifies the relationships among groups, Lie algebras, and Hopf algebras by interpreting them as 2-objects, leading to new constructions like the enveloping algebra of the string Lie algebra.

Contribution

It introduces a framework for understanding Hopf 2-algebras via crossed modules, extending classical functors to the 2-categorical setting.

Findings

Constructed an enveloping algebra of the string Lie algebra of Baez-Crans.

Clarified the passage from crossed modules of Hopf algebras to Hopf 2-algebras.

Demonstrated compatibility of standard functors with crossed modules.

Abstract

Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.