Wreath products of cocommutative Hopf algebras

TL;DR

This paper introduces wreath products for cocommutative Hopf algebras, establishing their universal property for classifying cleft extensions and connecting them to group rings and universal enveloping algebras.

Contribution

It defines wreath products for cocommutative Hopf algebras and demonstrates their universal property, extending classical group and Lie algebra extension classifications.

Findings

Wreath products of group rings equal the wreath product of groups.

Universal enveloping algebra of a wreath product of Lie algebras is the wreath product of their enveloping algebras.

Classifies group and Lie algebra extensions via substructures of wreath products.

Abstract

We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property of classifying cleft extensions, analogous to the Kaloujnine-Krasner theorem for groups. We show that the group ring of a wreath product of groups is the wreath product of their group rings, and that (with a natural definition of wreath products of Lie algebras) the universal enveloping algebra of a wreath product of Lie algebras is the wreath product of their enveloping algebras. We recover the aforementioned result that group extensions may be classified as certain subgroups of a wreath product, and that Lie algebra extensions may also be classified as certain subalgebras of a wreath product.