Ennea-algebras

A generalisation of a recent work of M. Aguiar and J.-L. Loday on quadrialgebras called t-ennea-algebras constructed over dendriform trialgebras is proposed. Such algebras allow the construction of nested dendriform trialgebras and are related to pre-Lie algebras, t-infinitesimal bialgebras and t-Baxter operators. We also show that the augmented free t-ennea-algebra has a structure of connected Hopf algebra. In the last part, we use Baxter operators to produce formal deformations of dendriform algebras, quadrialgebras and ennea-algebras. Examples and relations of this work to combinatorial objects are given.