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 $\gamma$ of diassociative algebras, called $\gamma$-pluriassociative algebras, so that $1$-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with $2\gamma$ associative binary operations satisfying some relations. We provide a complete study of the $\gamma$-pluriassociative operads, the underlying operads of the category of $\gamma$-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 $\gamma$-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.Comment: 38 pages. Published version of a part of arXiv:1502.00835 [math.CO

Pluriassociative algebras II: The polydendriform operad and related operads

“…} forms a basis of Free G Dendr γ (2) and then, generates Free G Dendr γ as an operad. This change of basis from Free G Dendr γ to Free(G ′ Dendrγ ) is similar to the change of basis from Free(G ′ Diasγ ) to Free G Diasγ introduced in Section 2.3.6 of [Gir16]. Let us now express a presentation of Dendr γ through the family G ′ Dendrγ .…”

“…showing thatH Dias γ (t) andH Dendr γ (t) are the inverses for each other for series composition. Now, since by Theorem 2.3.1 and Proposition 2.1.1 of [Gir16], Dias γ is a Koszul operad and its Hilbert series is H Dias γ (t), and since Dendr γ is by definition the Koszul dual of Dias γ , the Hilbert series of these two operads satisfy (1.1.3). Therefore, (2.1.4) implies that the Hilbert series of Dendr γ is H Dendr γ (t).…”

“…The main reason is that Relations (2.1.1f) and (2.1.1g) of the first presentation involve a nonconstant number of terms, while all relations of this second presentation always involve only two or three terms. As a very remarkable fact, it is worthwhile to note that the presentation of Dendr γ provided by Theorem 2.1.4 can be directly obtained by considering the Koszul dual of Dias γ over the K-basis (see Sections 2.3.5 and 2.3.6 of [Gir16]). Therefore, an alternative way to establish this presentation consists in computing the Koszul dual of Dias γ seen through the presentation having R ′ Dendrγ as space of relations, which is made of the relations of Dias γ expressed over the K-basis (see Proposition 2.3.8 of [Gir16]).…”

“…It follows immediately that As γ is a set-operad and that it provides a generalization of the associative operad. The algebras over As γ are the γ-multiassociative algebras introduced in Section 3.3.1 of [Gir16].…”

“…We use the approach consisting in considering the diassociative operad Dias [Lod01], the Koszul dual of the dendriform operad Dendr, rather that focusing on Dendr. For this, we rely on the definition of a generalization Dias γ on a nonnegative integer parameter γ of the diassociative operad introduced by the author in [Gir16]. These operads, called γ-pluriassociative operads, satisfy several properties and are among other set-operads and Koszul operads.…”