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${\mathcal O}$-operators of Loday algebras and analogues of the classical Yang-Baxter equation

We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and necessary condition on the existence of the $2^{n+1}$ operations on an algebra with the $2^{n}$ operations in an associative cluster. The analogues of the classical Yang-Baxter equation in these algebras can be understood as the $\mathcal O$-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.