Rota-Baxter operators on cocommutative Hopf algebras

TL;DR

This paper extends the concept of Rota-Baxter operators to cocommutative Hopf algebras, establishing a correspondence with operators on groups and Lie algebras, thereby unifying these structures.

Contribution

It introduces Rota-Baxter operators on cocommutative Hopf algebras and proves their correspondence with operators on groups and Lie algebras.

Findings

Rota-Baxter operators on group algebras correspond to those on groups.

Rota-Baxter operators on universal enveloping algebras correspond to those on Lie algebras.

The paper generalizes Rota-Baxter theory to a broader algebraic context.

Abstract

We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group algebra of a group $G$ or $H=U(\mathfrak{g})$ the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, then we prove that Rota-Baxter operators on $H$ are in one to one correspondence with corresponding Rota-Baxter operators on groups or Lie algebras.