Calabi-Yau Frobenius algebras

TL;DR

This paper introduces Calabi-Yau Frobenius algebras over arbitrary rings, explores their Hochschild cohomology with BV and Frobenius structures, and computes cohomology for various quiver-related algebras, linking to Ginzburg's theory.

Contribution

It defines Calabi-Yau and periodic Frobenius algebras over general rings and computes their Hochschild cohomology, establishing new connections with Ginzburg's Calabi-Yau algebras.

Findings

Hochschild cohomology of many quiver-related algebras is computed for the first time.

Stable Hochschild cohomology has a BV and Frobenius algebra structure.

Maps between extended Dynkin and Dynkin preprojective algebras are characterized.

Abstract

We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include (centrally extended) preprojective algebras of (generalized) Dynkin quivers, and group algebras of classical periodic groups. We use this theory to compute (for the first time) the Hochschild cohomology of many algebras related to quivers, and to simplify the description of known results. Furthermore, we compute the maps on cohomology from extended Dynkin preprojective algebras to the Dynkin ones, which relates our CY property (for Frobenius algebras) to that of Ginzburg (for algebras of finite Hochschild dimension).