# Weak composition quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras

## TL;DR

This paper introduces a new Hopf algebra of weak composition quasi-symmetric functions, extending classical quasi-symmetric functions, and connects it to Rota-Baxter algebras, addressing a question posed by G.-C. Rota.

## Contribution

It constructs the Hopf algebra WCQSym, generalizes transformation formulas, and links it to free Rota-Baxter algebras, expanding the algebraic framework of quasi-symmetric functions.

## Key findings

- QSym is a Hopf subalgebra of WCQSym
- WCQSym is isomorphic to a free Rota-Baxter algebra
- Transformation formulas extend classical quasi-symmetric functions

## Abstract

We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra WCQSym of weak composition quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C.~Rota relating symmetric type functions and Rota-Baxter algebras. We provide the transformation formulas between the weak composition monomial and fundamental quasi-symmetric functions, which extends the corresponding results for quasi-symmetric functions. Moreover, we show that QSym is a Hopf subalgebra and a Hopf quotient algebra of WCQSym. Rota's question is addressed by identifying WCQsym with the free commutative unitary Rota-Baxter algebra of weight 1 on one generator, which also allows us to equip this algebra with a Hopf algebra structure.