Koszuality of the \(\mathcal{V}^{(d)}\) dioperad (Q2002822)

Classical operads parametrize operations with multiple inputs and a single output. Dioperads are a generalization of operads that parametrize operations that allow multiple outputs as well as inputs. This paper investigates the dioperad \({\mathcal V}^{(d)}\), whose algebras are associative algebras equipped with a symmetric and invariant co-inner product of degree \(d\). Here a co-inner product on an algebra \(A\) means, in practice, a choice of an element of \(A\otimes A\). Such a choice is considered as an operation with zero inputs and two outputs. The main result of the paper is that the dioperad \({\mathcal V}^{(d)}\) is Koszul.