The Operad Quad is Koszul

Abstract

The purpose of this paper is to prove the koszulity of the operad $\mathcal{Q}uad$, governing quadri-algebras. That $\mathcal{Q}uad$ is Koszul was conjectured by Aguiar and Loday. The operad $\mathcal{D}end$, governing dendriform algebras, is known to be Koszul, by work of Loday, and $\mathcal{Q}uad$ is its second black square power. We find a new complex, based on the associahedron, which captures the structure of $\mathcal{D}end$. This complex behaves well with respect to the black squaring process, and allows us to conclude. Also, this proves koszulity of higher powers of $\mathcal{D}end$.

14 pages; exposition clarified, notation updated and typos corrected

Cited by in corpus (1)

• On Products and Duality of Binary, Quadratic, Regular Operads