Gerstenhaber Algebraic Structure On The Hochschild Cohomologies Of Two Classes Of Beilinson Algebras

The Hochschild homology and cohomology of associative algebra play an important role in the representation theory of algebras,noncommutative geometry theory and many other fields.Moreover,there are abundant algebraic structures on the Hochschild cohomology of finite di-mensional associative algebras.In this dissertation,we mainly study the Gerstenhaber algebraic structure on the Hochschild cohomologies of two kinds of Beilinson algebras.The first kind of algebra considered in this paper is Beilinson algebra of three-dimensional quantum polynomial algebra =K〈x,y,z〉/(yz-αzy,zx-βxz,xy-γyx).The second kind of algebra is the Beilinson algebra of binary quantum outer algebra B’=K〈x,y〉/(x~2,xy+qyx,y~2).For these two classes of algebras,we first construct a minimal projective resolution,using parallel paths,by analyzing the parameters q,α,β,γ in detail,a base of each degree of cohomology is given in detail,the cup product of cohomologies on Hochschild of these two classes of algebras are described,the structure of Hochschild cohomology rings of these two kinds of algebras are described.Finally,we construct a chain map between their minimal pro-jective bimodular resolution and reduced Bar resolution by using weak self-homotopy,and give the Gerstenhaber algebra structure on Hochschild cohomologies of these two kinds of algebras.