Rota–Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota–Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota–Baxter algebras and braided dendriform algebras, under the commutativity condition. We further generalize the notion of dendriform Hopf algebra to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative Rota–Baxter algebras of braided commutative dendriform algebras are obtained.

Rota-Baxter 代数和密切相关的树状代数具有重要的物理应用，特别是在量子场论的重整化方面。编织结构提供了有效的量化方式，例如量子群。继续最近关于这两种结构的研究，本文考虑了编织环境中的 Rota-Baxter 代数和树状代数。应用量子混洗和量子准混洗产品，我们在交换律条件下构造了编织Rota-Baxter代数和编织树状代数类别中的自由对象。我们进一步将树枝状 Hopf 代数的概念推广到编织上下文，并表明量子混洗代数给出了编织树枝状 Hopf 代数。