Koszuality of the ${\mathcal{V}}^{(d)}$ dioperad

Abstract

Define a ${\mathcal{V}}^{(d)}$-algebra as an associative algebra with a symmetric and invariant co-inner product of degree $d$. Here, we consider ${\mathcal{V}}^{(d)}$ as a dioperad which includes operations with zero inputs. We show that the quadratic dual of ${\mathcal{V}}^{(d)}$ is $({\mathcal{V}}^{(d)}{)}^{!}={\mathcal{V}}^{(-d)}$ and prove that ${\mathcal{V}}^{(d)}$ is Koszul. We also show that the corresponding properad is not Koszul contractible.