A Pair Of Dual Hopf Algebras On Posets

Hopf algebra is a bialgebra with an antipode,where bialgebra is a vector space with compatible algebra and coalgebra structures.Hopf algebra is one of the important research contents in algebraic combinatorics.A lot of combinatorial objects have Hopf algebra structures and posets are important combinatorial objects.The dual spaces of finite-dimensional graded Hopf algebras have corresponding dual Hopf algebra structures.In this thesis,we prove that the vector space spanned by posets with the stack product and the unshuffle coproduct is a graded connected Hopf algebra.Then gives the definition of the super-shuffle product and the cut coproduct on posets,and proves that the vector space with these is a graded connected Hopf algebra.Futhermore,the respective antipode formulas are given and proved.Finally,according to the conditions that should be satisfied by the dual Hopf algebras,it is proved that the two graded Hopf algebras are dual to each other.In Chapter 1,we briefly introduces the research background,research status and main work of this thesis.In Chapter 2,we introduce some knowledge about algebra,coalgebra,bialgebra,Hopf algebra,antipode and dual Hopf algebras.In Chapter 3,we introduce the definitions of the stack product and the unshuffle coproduct on the vector space spanned by posets,and then we prove the compatibility of them.Furthermore,we obtained a graded Hopf algebra structure on posets given by stack product.At the same time,we give the closed formula of its antipode.In Chapter 4,we introduce the definitions of the super-shuffle product and the cut coproduct on the vector space spanned by posets,and then we prove the compatibility of them.Furthermore,we obtain a graded Hopf algebra structure on posets given by super-shuffle product.At the same time,we give the closed formula of its antipode.In Chapter 5,we prove that the Hopf algebra given by stack product on posets is dual to the Hopf algebra given by the super-shuffle product on posets.In Chapter 6,we summarize the main results of this thesis and give some open problems,considering whether the product and coproduct mentioned in this thesis can be extended to(0,1)-matrices,or even to other combinatorial objects,and give the corresponding Hopf algebra structure and its dual.