Infinite Type Hopf Pi - Duality Of Algebra

In the research field of mathematics, Hopf algebra has been extensively studied. It not only limited in theoretical study of algebraic structures, but also closely related to the other fields of mathematics. As the extension of Hopf algebra, the notion of Hopf π-coalgebra, where π is a multiplicative group, is introduced to construct π-category by V. G. Turaev. Hopf π-coalgebra was used by A. Virelizier to construct Hennings-like and Kuperberg-like invariants of principal π-bundles over link complements and over3-manifolds based on the characters of the Hopf π-coalgebra. The concept of Hopf algebra is also introduced and studied in the same time [1-5]. On this basis, the dual space of infinite-type Hopf π-algebra is discussed in this paper. We have given and proved that the duality of infinite-type Hopf π-algebra is a Hopf π-coalgebra. The dual spaces of Hopf π-ideal and Hopf π-subalgebra in Hopf π-algebra are also discussed. Some conclusions about Hopf algebra are given as corollaries.This paper is organized as follows. In section1, as preparations for the following contents, some notions we need are introduced. For instance, we introduce the concepts of π-algebra, π-coalgebra, Hopf π-algebra, Hopf π-coalgebra. In section2, we given and proved that the duality of infinite-type Hopf π-algebra is a Hopf π-coalgebra. In section3, the concepts of Hopf π-subcoalgebra and Hopf π-ideal are introduced. We also give the relations between Hopf π-ideal of Hopf π-algebra and Hopf π-subcoalgebra of Hopf π-coalgebra. In section4, we introduced the concepts of Hopf π-subalgebra and Hopf π-coideal. We also give the relations between Hopf π-subalgebra of Hopf π-algebra and Hopf π-coideal of Hopf n-coalgebra.