Rota-Baxter TD Algebra and Quinquedendriform Algebra

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Motivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD (RBTD) operator, an operator combining the Rota-Baxter operator and TD operator, and coming from a recent study of Rota’s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.