Operads for $n$-ary algebras – calculations and conjectures

Markl, Martin ; Remm, Elisabeth

Archivum mathematicum, Tome 47 (2011) no. 5, pp. 377-387.

Résumé

In [8] we studied Koszulity of a family ${t\mathcal{A}\it ss}^n_d$ of operads depending on a natural number $n \in \mathbb{N}$ and on the degree $d \in \mathbb{Z}$ of the generating operation. While we proved that, for $n \le 7$, the operad ${t\mathcal{A}\it ss}^n_d$ is Koszul if and only if $d$ is even, and while it follows from [4] that ${t\mathcal{A}\it ss}^n_d$ is Koszul for $d$ even and arbitrary $n$, the (non)Koszulity of ${t\mathcal{A}\it ss}^n_d$ for $d$ odd and $n \ge 8$ remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.

Détail

Classification : 18D50, 55P48
Keywords: operad; Koszulity; minimal model