Batalin-Vilkovisky Structures On The Hochschild Cohomology Of Generalized Weyl Algebras

As an important,part of noncommutative geometry,Hochschild cohomology plays a fundamental role in different branches of mathematics,such as algebra,geometry,and topology.In contemporary mathematics,it is prevailing to investigate the structures on Hochschild cohomology for all kinds of mathematical objects.This Master thesis is devoted to the calculation of Batalin-Vilkovisky structures on the Hochschild coho-mology of skew Calabi-Yau generalized Weyl algebras.In the first part,we present an unbounded free resolution F of any generalized Weyl algebra A as an Ac-module,which was constructed by different authors.This enables us to establish a Van den Bergh duality in the homotopy category for any skew Calabi-Yau generalized Weyl algebra A,and thus it builds a bridge between the Hochschild homology and cohomology of A.In the second part,we focus on the skew Calabi-Yau generalized Weyl algebra A of quantum type.Based on the results of Solotar et al.,we rephrase the Hochschild cohomology HH*(A)in terms of F,and then find out Hochschild cochains as the repre-sentatives of basis of all nontrivial Hochschild cohomological groups.In particular,the center,the outer derivation group,and first order deformation space of A are presented explicitly.We apply Kowalzig and Krahmer's method to the Hochschild homology of A,and translate the homological information into cohomological one by virtue of the Van den Bergh duality,obtaining the Batalin-Vilkovisky structure on the Hochschild cohomology of A.In addition,we figure out the fundamental class for A,which recovers the Van den Bergh duality via the cap product.Finally,we apply our results to quan-tum weighted projective lines and quantum 2-spheres,which arise from mathematical physics.The Batalin-Vilkovisky structures on the Hochschild cohomology of the two kinds of algebras,as well as their fundamental classes,are described completely.