The Quantum Symmetry Algebra Hochschild Cohomology

This dissertation consists of four sections. We shall investigate Hochschild homology and cohomology of quantum symmetric algebra and skew group algebra, and show the corresponding examples.The first and second sections are introduction and preliminaries respectively.In the third section,we use quantum differential operators to define a combinatorial converter map T,which does not depend on choices of bases,and give a special chain maps from bar resolution to free resolution.In the forth section,we deduce that combinatorial converter map T defines automor-phisms of Hochschild cohomology and use cup product to obtain some propositions of Hochschild cohomology, and present similar automorphisms of Hochschild homology.