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Rota-Baxter algebras and the closely related dendriform algebras haveimportant physics applications, especially to renormalization of quantum fieldtheory. Braided structures provide effective ways of quantization such as forquantum groups. Continuing recent study relating the two structures, this paperconsiders Rota-Baxter algebras and dendriform algebras in the braided contexts.Applying the quantum shuffle and quantum quasi-shuffle products, we constructfree objects in the categories of braided Rota-Baxter algebras and braideddendriform algebras, under the commutativity condition. We further generalizethe notion of dendriform Hopf algebras to the braided context and show thatquantum shuffle algebra gives a braided dendriform Hopf algebra. Envelopingbraided commutative Rota-Baxter algebras of braided commutative dendriformalgebras are obtained.

Braided Rota-Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras