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Using the concept of mixable shuffles, we formulate explicitly the quantumquasi-shuffle product. We also provide a desirable description of thesubalgebra generated by the set of primitive elements of the quantumquasi-shuffle bialgebra. A braided coalgebra structure which is dual to thequantum quasi-shuffle in some sense is constructed as well. We use quantumquasi-shuffle algebras to provide examples of Rota-Baxter algebras andtridendriform algebras.

Explicit results concerning quantum quasi-shuffle algebras and their applications