Pluriassociative algebras I: The pluriassociative operad

Abstract

Diassociative algebras form a categoy of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $γ$ of diassociative algebras, called $γ$-pluriassociative algebras, so that $1$-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with $2γ$ associative binary operations satisfying some relations. We provide a complete study of the $γ$-pluriassociative operads, the underlying operads of the category of $γ$-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in $γ$-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.