A Pair Of Dual Hopf Algebras On Permutations

Hopf algebra is one of the important research subject in algebraic combinatorics because it has compatible algebraic and coalgebraic structures with an antipode mapping.In particular,its dual space is an important tool to study a Hopf algebra.It is known that a graded connected bialgebra is a graded Hopf algebra.According to this property,we first define two different bialgebras on the vector space spanned by permutations,and then get two different Hopf algebras.It is known that the image of an element in a graded Hopf algebra under the antipode mapping can always be obtained by graded iteration,but sometimes it is difficult to figure out a closed formula of an antipode.Here we also give and prove the corresponding closed-formulas of the antipodes respectively when we get the graded Hopf algebras.Finally,according to the definition of graded dual Hopf algebras,we prove that these two graded Hopf algebras are dual to each other.In Chapter 1,we preliminarily introduce the research background,research status and the main work of this paper.In Chapter 2,the definition and theorems of Hopf algebras and the knowledge of permutations are introduced.In Chapter 3,we define the new shuffle product on the vector space spanned by permutations,and get a graded Hopf algebra.Then,the closed-formula of antipode mapping is given and proved.In Chapter 4,we propose the concatenation product and the draw coproduct on the vector space spanned by permutations,and get a new Hopf algebra.Then,the closed-formula of antipode is also given.In Chapter 5,we prove that these two graded Hopf algebras are dual to each other through the definition of graded dual Hopf algebras.In Chapter 6,we apply shuffle product,concatenation product,deconcatenation coproduct and draw coproduct on different objects and get some other pairs of dual graded Hopf algebras and link some of them to dual graded graphs.In addition,some open problems about ungraded bialgebras are proposed.