# Koszul calculus of preprojective algebras

## TL;DR

This paper investigates the Koszul calculus of preprojective algebras, revealing vanishing properties, dualities, and a generalized Calabi-Yau condition, especially for ADE Dynkin graphs, with explicit calculations and a duality theorem.

## Contribution

It introduces the concept of Koszul complex Calabi-Yau (Kc-Calabi-Yau) algebras, generalizes Calabi-Yau properties, and provides explicit calculations for ADE Dynkin graphs.

## Key findings

- Koszul calculus vanishes in degrees p>2 for certain preprojective algebras
- Isomorphism between (co)homological calculus via degree exchange p and 2-p
- Preprojective algebras of ADE Dynkin graphs are Kc-Calabi-Yau of dimension 2

## Abstract

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A$\_1$ and A$\_2$, vanishes in any (co)homological degree $p>2$. Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees $p$ and $2-p$, and we prove a generalised version of the 2-Calabi-Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi-Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi-Yau property and we say that they are Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension $2$. For Kc-Calabi-Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincar\'e Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.