Hopf Algebras On Simple Graphs

Hopf algebra is a bialgebra with an antipode,which is widely used in combinatorics.There are many Hopf algebras with combinatorial significance in vector space generated by many combinatorial objects.Among them,the research of Hopf algebras on permutations is one of the most important directions.There are many classical product operations on permutations,such as shuffle product,conjunction product,super-shuffle product,etc.In this thesis,we first generalize the supershuffle product from permutations to labeled simple graphs.We prove that the super-shuffle product generates a Hopf algebra on the vector space spanned by labeled simple graphs,and we give the closure-formula of its antipode.Then through the dual condition,we prove the its dual Hopf algebra and we prove their duality of them.Finally,we prove that there is a Hopf homomorphism from permutations to labeled simple graphs.In Chapter 1,we briefly describe the development history of Hopf algebra and its application in combinatorics.In Chapter 2,we briefly introduce the basic concepts and definitions related to Hopf algebra.In Chapter 3,by generalizing the super-shuffle product on permutations to the labeled simple graphs,we get a Hopf algebra on the vector space spanned by labeled simple graphs,and give a closed-formula of antipode of this Hopf algebra.In Chapter 4,we give new product and coproduct on the labeled simple graphs,and prove that it can get a Hopf algebra on the vector space spanned by labeled simple graphs with this operations.In Chapter 5,we prove the duality between the above these two Hopf algebras.In Chapter 6,we prove that a linear map from permutations to labeled simple graphs is a Hopf homomorphism.