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

TL;DR

This paper investigates the algebraic structure of the dioperad alV^{(d)} and proves its Koszul property, revealing duality relations and contrasting properties with the associated properad.

Contribution

It establishes the Koszulness of alV^{(d)} and describes its quadratic dual, providing new insights into its algebraic and homological properties.

Findings

Quadratic dual of alV^{(d)} is alV^{(-d)}.

alV^{(d)} is proven to be Koszul.

The associated properad is not Koszul contractible.

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.