Batalin-Vilkovisky algebras and the noncommutative Poincare duality of Koszul Calabi-Yau algebras

TL;DR

This paper proves a deep connection between the Hochschild cohomology of Koszul Calabi-Yau algebras and their Koszul duals, establishing a BV algebra isomorphism that confirms a conjecture by Rouquier.

Contribution

It demonstrates an isomorphism of BV algebras between Hochschild cohomologies of Koszul Calabi-Yau algebras and their duals, generalizing Rouquier's conjecture.

Findings

Hochschild cohomology rings are isomorphic as BV algebras

Confirms a conjecture of Rouquier in the context of Koszul Calabi-Yau algebras

Establishes a noncommutative Poincaré duality for these algebras

Abstract

Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a conjecture of R.~Rouquier.