Splitting of operations for di-associative algebras and tri-associative algebras

Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic $K$-theory. The purpose of this paper is to study the splittings of operations of di-associative algebras and tri-associative algebras. First, we introduce the notion of a quadri-dendriform algebra, which is a splitting of a di-associative algebra. We show that a relative averaging operator on dendriform algebras gives rise to a quadri-dendriform algebra. Conversely, a quadri-dendriform algebra gives rise to a dendriform algebra and a representation such that the quotient map is a relative averaging operator. Furthermore, any quadri-dendriform algebra can be embedded into an averaging dendriform algebra. Finally, we introduce the notion of six-dendriform algebras, which are a splitting of tri-associative algebras, and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.