Research On Some Problems Of Hopf (CO)Quasigroups

Hopf (co)quasigroups were firstly introduced by J. Klim and S. Majid in order to cap-ture the quasigroup features of the algebraic7-sphere. They are diferent from Hopf alge-bras for their possibly-non(co)associative (co)multiplications. But their (co)multiplicationssatisfy two conditions which are limited by the antipodes. Since the notions were proposed,they have attracted many mathematicians' attentiones. Up to now, their main work hasbeen to generalize the theories of Hopf algebras to Hopf (co)quasigroups. This paper isspecialized in this field of work.This paper discusses two major questions. One is the definition and properties of a qu-asitriangular Hopf coquasigroup. The other is the structural theorem of Hopf quasigroups'H ratand its application. Specific details go as follows.In Chapter1, we not only present the background and status quo of Hopf algebrasand Hopf (co)quasigroups, but also illustrate the methodologies to bring forth the topic.In Chapter2, in order to obtain the notion of a quasitriangular Hopf coquasigroup,we firstly introduce the notion of an almost cocommutative Hopf coquasigroup and ac-quire its two equivalent conditions using module actions. Next, by the notion of an almostcocommutative Hopf coquasigroup, we give the notion of a quasitriangular Hopf coquasi-group and prove its Hopf-like properties. Finally, inspired by2-cocycle of bialgebras, wepropose the definition of2-cycle for a Hopf coquasigroup. And we construct a new Hopfcoquasigroup from a Hopf coquasigroup with a2-cycle. What's more, when the new Hopf coquasigroup is triangular is showed.In Chapter3, we present the definition and properties of rational module of a Hopfcoquasigroup. And then the structural theorem of H ratis proven on the basis of the thestructural theorem of Hopf quasigroups' Hopf module. At last, as an application of thestructural theorem of H rat, we show that every finite dimensional Hopf quasigroup is leftco-Frobenius which is first defined in this paper.