Batalin--Vilkovisky algebra structures on the Hochschild cohomology of generalized Weyl algebras

TL;DR

This paper computes Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras, extending to quantum weighted projective lines and Podle's quantum spheres.

Contribution

It establishes Van den Bergh duality at the complex level and applies existing methods to explicitly determine BV algebra structures for these algebras.

Findings

BV algebra structures are explicitly computed for generalized Weyl algebras.

Results are applied to quantum weighted projective lines.

Complete description of BV structures for Podle's quantum spheres.

Abstract

This paper is devoted to the calculation of Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras. We firstly establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Kr\"ahmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin-Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podle\'s quantum spheres, and the Batalin-Vilkovisky algebra structures for them are described completely.