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

We devote to the calculation of Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krahmer'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 Podles quantum spheres, and the Batalin-Vilkovisky algebra structures for them are described completely.