Hopf π-Crossed Biproduct And Braided Monoidal Category Over π-Crossed Product

In this paper, we mainly discuss the following questions: one is the structure togetherwith the related conclusions over Hopf π-crossed biproduct A×πH; the other gives theconditions for the categoryAσπHM of the left π-comodules over Hopf π-crossed productHopf π-coalgebra AσπH to be a braided monoidal category.The dissertation is orgenized as follows:In chapter one, we introduce the background of Hopf π-coalgebra and present somebasic properties as well as facts about Hopf π-coalgebra. In addition, we introduce howthe question investigated in this disseriation is produced.In chapter two, frstly, we prove the necessary and sufcient condition of a Hopfπ-coalgebra’s crossed product and a π-Smash coproduct to be a semi-Hopf π-coalgebraA×πH, which is called Hopf π-crossed biproduct. Secondly, we construct the antipodeover Hopf π-crossed biproduct A×πH to make it to be a Hopf π-coalgebra.In chapter three, we give the coquasitriangular structure over Hopf π-crossed productHopf π-coalgebra AσπH. Then we obtain the necessary and sufcient condition for the cat-egoryAσπHM of the left π-comodule over Hopf π-crossed product Hopf π-coalgebra AσπHto be a braided monoidal category.