The Smash Biproduct For Hopf (CO) Quasigroups

As the generalizations of Hopf algebras, Hopf quasigroups and Hopf coquasigroupswere introduced by J.Klim and S.Majid in order to capture the quasigroup features of thealgebraic7-sphere. In this paper, we mainly discuss the necessary and sufcient conditionsfor Hopf quasigroups to construct a smash biproduct and the dual case. In the final, thealgebra deformation for Hopf quasigroups is studied.In Chapter1, we introduce the background of Hopf quasigroups and Hopf coquasi-groups, and clarify the origin of the questions investigated in this paper. In addition, thenotions and properties which will be used in this paper are reviewed.In Chapter2, we first review some related definitions and results about the smashproduct Hopf quasigroup. Then we introduce the definitions of quasi H-comodule Hopfquasigroups and smash coproduct Hopf quasigroups, and discuss the necessary and suf-cient conditions for Hopf quasigroups to construct a smash coproduct. Based on this, wediscuss the necessary and sufcient conditions for Hopf quasigroups to construct a smashbiproduct.In Chapter3, the dual cases of the results of Chapter2are discussed.In Chapter4, we first introduce the related definitions and properties about2-cocycle,then discuss how to construct a new Hopf quasigroup by changing the algebraic structure ofthe original Hopf quasigroup. At last, the relation between the coquasitriangular structureof the new Hopf quasigroup and that of the original one is investigated.