Quantum quasi-shuffle algebras II

Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra which is dual to the quantum quasi-shuffle algebra. We provide representations of quantum quasi-shuffle algebras on commutative braided Rota-Baxter algebras. As an application, we establish formal power series whose terms come from a special representation of the quasi-shuffle algebra on polynomial algebra and whose evaluations at 1 are the multiple q-zeta values.