Pluriassociative algebras II: The polydendriform operad and related operads

Samuele Giraudo (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform op-erad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a non-negative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.

Domaines: Combinatoire [math.CO]; Algèbres quantiques [math.QA]

Samuele Giraudo. Pluriassociative algebras II: The polydendriform operad and related operads. Advances in Applied Mathematics, 2016, 77, pp.43--85. ⟨10.1016/j.aam.2016.02.004⟩. ⟨hal-01283859⟩