Averaging operators on groups and Hopf algebras

TL;DR

This paper introduces averaging groups and Hopf algebras, explores their relationships with Rota-Baxter operators, and constructs free averaging groups, advancing the understanding of algebraic structures related to averaging operators.

Contribution

It defines averaging groups and Hopf algebras, studies their connections with Rota-Baxter operators, and explicitly constructs free averaging groups on a set.

Findings

Averaging groups induce disemigroup and rack structures.

Established relationships between averaging groups, Hopf algebras, and Lie algebras.

Constructed explicit free averaging groups on a set.

Abstract

Rota-Baxter operators on groups were studied quite recently. Motivated mainly by the fact that weight zero Rota-Baxter operators and averaging operators are Koszul dual to each other, we propose the concepts of averaging group and averaging Hopf algebra, and study relationships among them and the existing averaging Lie algebras. We also show that an averaging group induces a disemigroup and a rack, respectively. As the free object is one of the most significant objects in a category, we also construct explicitly the free averaging group on a set.