The Convolution Of The Algebra About Pointed Hopf Algebra

In recent decades, the study of Pointed Hopf algebras has been one of important research subjects in algebra, its theories have been widely applied. The aim of this thesis is to describe the structure of the group Alg(H,A) of the convolution algebra Hom(H,A)(when A is a commutative algebra) of Pointed Hopf algebra H, to discuss the relationships between the convolution algebra Hom(kG,kG) of the cyclic group algebra kG and the convolution matrix algebra, to give the classification of algebra homomorphism and coalgebra homomorphism from H4to kS3in the unit group of the convolution algebra Hom(HA,kS3).Explicitly, in the first part, we introduce the research background of Pointed Hopf algebra and the unit group, and further lead to the main object of this thesis.In the second part, we introduce some related definitions, notations and theorems.In the third part, we mainly discuss the structure of the group Alg(H,A) of the convolution algebra Hom(H,A) when A is a commutative algebra and H is specific Pointed Hopf algebra. Main conclusions are:Theorem3.1.1Let H be Sweedler’s4-dimensional Hopf algebra HA, A be a commutative algebra over a field k (char(k)≠2), we have a group isomorphism σ:Alg(H,A)â†'N={a|a∈A,a2=1} fâ†'α where f(g)=b, f(x)=a.Theorem3.1.2Let H be Sweedler’s4-dimensional Hopf algebra H4, A be a commutative algebra over a field k (char(k)=2), we have a group isomorphism)â†'MxN={(a,b)|a,beA,a2=0,b2=1} fâ†'(a, b) where f(g)=b, f(x)=a.Theorem3.2.1Let H be Taft algebra H2n,A be a commutative algebra over a field k (char(k)≠2), we have a group isomorphism σ:Alg(H,A)σ N={a|a∈A,an=1} fâ†'a where f(g)=a,f(x)=0.Theorem3.2.2Let H be Taft algebra H2n,A be a commutative algebra over a field k (char(k)=2),we have a group isomorphism φ:Alg(H,A)â†'M×N={(a,b)a,b∈A,a2=0,bn=1} fâ†'(a,b) where f(g)=b,f(x)=a.Theorem3.3.1Let Uq be quanttum group Uq(sl2))(g≠±1),A be a commutative algebra,we have a group isomorphism φ:Alg(Uq,A)â†'N={a|a∈A,a2=1} fâ†'a where f(K)=a, f(K-1)=a-1, f(E)=f(F)=0.Theorem3.3.2Let U be quantUm group Uq(fm(K))(g≠±1, m is a positive integer),A be a commutative algebra,we have:a group isomorphism φ:Alg(U,A)â†'N={a∈A|a2m=1} fâ†'a where f(K)=a, f(K-1)=a-1, f(E)=f(F)=0.In the fourth part,we mainly discuss the relationships between the convolution algebra Hom(kG,kG) of the cyclic group algebra kG and the matrices,and define convolution multiplication of the2×2一matrices,that is where A is(aη)2.2,A’ is (a’η)2.2,and study some properties of the convolution invertible element of the convolution algebra Hom(kG,kG) of the cyclic group algebra kG in the view of the matrix.Similarly,we define the convolution multiplication of the higher order matrices and study the related properties.Main conclusions are:Theorem4.2A=(aη)nxn.is convolution invertible if and only if[A]≠0.Then where Dη is the matrix that makes A1’si-th column change intoIn the last part, we mainly discuss the structure of the unit group of the convolution algebra Hom(H4, kS3) in the view of algebra homomorphism and coalgebra homomorphism and clearly give the classification of the algebra homomorphism and coalgebra homomorphism from H4to kS3.