Classifications And Representations Of 16-dimension Nonpoint Hopf Algebras

The mathematician Hopf introduced the concept of Hopf algebra while he studied the topological properties of Lie groups.An algebraic system over the field K is called a Hopf algebra if it has both K-algebra structure and K-coalgebra structure that satisfy certain compatible conditions.The invariant is an important tool to study the structure of algebras.It is well-known that Hopf algebra has many invariants,such as the group of automorphisms of a Hopf algebra.The group of automorphisms of an algebra reflects some symmetric structure of an algebra and plays an important role in understanding the structure of algebra.The group of automorphisms of Hopf algebra is an interesting field of Hopf algebra.It is also an crucial tool to study the structure and classification of finite dimensional Hopf algebras.In the thesis,we classify a family of 16-dimension Hopf algebras,which are noncommutative,non-cocommutative,non-point,and non-semisimple.We also describe the group of automorphisms of those Hopf algebras.Then their irreducible modules and indecomposable projective modules are constructed.Furthermore,we describe their projective class rings by generators and generating relations.More precisely,firstly,we construct a family of Hopf algebras which are isomorphic to the given Hopf algebras.The classification of such Hopf algebras is given and their automorphism groups are described.Secondly,all the non-isomorphic irreducible modules and indecomposable projective modules are characterized.Furthermore,the decomposition formulas of the tensor products of the irreducible modules and indecomposable projection modules are established.Finally,we give the projective class rings of those Hopf algebras.