Quasi-triangular and factorizable dendriform D-bialgebras

In this paper, we introduce the notions of quasi-triangular and factorizable dendriform D-bialgebras. A factorizable dendriform D-bialgebra leads to a factorization of the underlying dendriform algebra. We show that the dendriform double of a dendriform D-bialgebra naturally enjoys a factorizable dendriform D-bialgebra structure. Moreover, we introduce the notion of relative Rota–Baxter operators of nonzero weights on dendriform algebras and find that every quasi-triangular dendriform D-bialgebra can give rise to a relative Rota–Baxter operator of weight [Formula: see text]. Then we introduce the notion of quadratic Rota–Baxter dendriform algebras as the Rota–Baxter characterization of factorizable dendriform D-bialgebras, and show that there is a one-to-one correspondence between factorizable dendriform D-bialgebras and quadratic Rota–Baxter dendriform algebras. Finally, we show that a quadratic Rota–Baxter dendriform algebra can give rise to an isomorphism from the regular representation to the coregular representation of a Rota–Baxter dendriform algebra.