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

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 QSym((N) over tilde) of weak 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 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 QSym((N) over tilde). Rota's question is addressed by identifying QSym((N) over tilde) with the free commutative unitary Rota-Baxter algebra III (x) of weight 1 on one generator x, which also allows us to equip III(x) with a Hopf algebra structure. (C) 2018 Elsevier Inc. All rights reserved.