Quantum quasi-shuffle algebras II. Explicit formulas, dualization, and representations

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 structure 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 a formal power series whose terms come from a special representation of some kind of quasi-shuffle algebra and whose evaluation at 1 is the multiple $q$-zeta values.