Crossed homomorphisms and Cartier-Kostant-Milnor-Moore theorem for difference Hopf algebras

TL;DR

This paper extends classical theorems relating Hopf algebras to Lie algebras and groups by incorporating crossed homomorphisms and difference operators, providing new structural insights and classifications.

Contribution

It introduces a strengthened relationship involving crossed homomorphisms and difference operators within the Milnor-Moore framework, including a new structure theorem for difference Hopf algebras.

Findings

Graph characterization of compatible crossed homomorphisms

Relationship between derived actions and crossed homomorphisms

Classification of difference operators for Hopf algebras

Abstract

The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close relationships of a connected and a pointed cocommutative Hopf algebra with its Lie algebra of primitive elements and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. This paper shows that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be strengthened to include crossed homomorphisms and differenece operators. We give a graph characterization of Hopf algebra crossed homomorphisms which are also compatible with the Milnor-Moore relation. We further investigate derived actions from crossed homomorphisms on groups, Lie algebras and Hopf algebras, and establish their relationship. Finally we obtain a Cartier-Kostant-Milnor-Moore type structure theorem for pointed cocommutative difference Hopf algebras. Examples and classifications of difference operators are also provided for several Hopf algebras.