Interpolation in Species and a Lift of the Hopf Algebra of r-Quasisymmetric Functions

TL;DR

This paper extends the theory of r-quasisymmetric functions into Hopf monoids, introducing a method to interpolate between different types of Hopf monoids, enriching the algebraic framework.

Contribution

It presents a novel approach to lift r-quasisymmetric functions to Hopf monoids and introduces a general interpolation method between key Hopf monoids.

Findings

Developed a lifting method for r-quasisymmetric functions to Hopf monoids

Provided a general interpolation technique between free monoid and free commutative monoid Hopf structures

Enhanced the algebraic understanding of Hopf monoids in combinatorics

Abstract

In this manuscript we lift the theory of r-quasisymmetric functions to the theory of Hopf monoids. We provide a general method of interpolating between two Hopf monoids, one being the free monoid on a positive comonoid and the other being the free commutative monoid on a positive comonoid.