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 QSymN˜ 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 QSymN˜. Rota's question is addressed by identifying QSymN˜ with the free commutative unitary Rota–Baxter algebra Ш(x) of weight 1 on one generator x, which also allows us to equip Ш(x) with a Hopf algebra structure.