Hochschild Cohomology,Nonsymmetric Operads And BV Structure

This doctoral dissertation has two parts.In the first theoretical part,firstly,we review a classic result of M.Gerstenhaber,which says that a nonsymmetric operad with multiplication gives rise to a Gerstenhaber algebra,and a result of M.Gerstenhaber and A.Voronov which is the version of the above result at complex level,that is,the complex given by nonsymmetric operad with multiplication is a Gerstenhaber-Voronov(abbre-viated as GV)algebra;we give a slight refinement by showing further that a nonsymmetric operad with multiplication and unit gives rise to a unital GV algebra.Secondly,in 2004,L.Menichi proves that the cohomology of the complex given by a nonsymmetric cyclic operad with multiplication and unit is a unital Batalin-Vilkovisky(abbreviated as BV)algebra,and the complex level version of this result is provided,that is,the cohomological complex of a nonsymmetric cyclic operad with multiplication and unit is a unital Quesney algebra.Finally,we recall a result of N.Kowalzig,which says that a cyclic opposite operad module over a nonsymmetric operad with multiplication and unit gives rise to a differential calculus structure;we define cyclic opposite operad module with pairing and show that the existence of such cyclic opposite operad module with pairing implies that the operad is a cyclic operad and hence its cohomological complex is a Quesney algebra.The second part is the part of examples.We calculate the BV structure on the Hochschild cohomology ring of exterior algebras;we show that the Hochschild cohomology and homology of A_?algebra admit a differential calculus structure,a result well know to the expert,but rarely found in the literature and usually with some sign problem;as a consequence,we give a new proof of a result of T.Tradler,that is,the Hochschild homology of a cyclic A_?algebra is a BV algebra.