Operads for $n$-ary algebras – calculations [4pt] and conjectures

Martin Markl and Elisabeth Remm

Mathematical Institute of the Academy, Žitná 25, 115 67 Prague 1, The Czech Republic
Laboratoire de Mathématiques et Applications, Université de Haute Alsace, Faculté des Sciences et Techniques, 4, rue des Frères Lumière, 68093 Mulhouse cedex, France


Abstract: 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.

AMSclassification: primary 18D50; secondary 55P48.

Keywords: operad, Koszulity, minimal model.