Abstract
We introduce notions of đȘ-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of RotaâBaxter operators. The invertible đȘ-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 YangâBaxter equation in these algebras can be understood as the đȘ-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.
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