Group Of Hopf Algebra On Complementary Module Algebra And The Properties Of General Cleft Of Dual Complementary Module Algebra

This Master degree thesis is devoted to two aspects:one is the property of a comodule algebra over a group Hopf algebra, proving that under a weakened condition it is a Galois extension of its invariant space; the other is about the property of a class of special module coalgebra, which can be considered as a dual result of the property of a cleft comodule algebra.Let A be a Hopf algebra, B a right A-comodule algebra, and B0the invariant space of B. If the map is bijective, we say B/B0is a Galois extension. When A is finite-dimensional, if β is surjective then β is bijective([6, p.123]. Y. Doi([4, theorem2.5])pointed out when A is commutative and there exists a total integral map, which maps A to the center of B, the surjectiveness of β implies that B/B0is a Galois extension. We prove that when A is a group Hopf algebra, the surjectiveness of β also implies that B/B0is a Galois extension but there is no need to argue that A is finite-dimensional or commutative, and no need for the existence of the total integral map.The last part of this thesis proves the equivalence of three sayings about an A-module coalgebra:(1) There exists an A-module isomorphism φ∈Hom(C, A), and φ is convolution-invertible;(2) Given an arbitrary [C,A]-Hopf module N, the map is a [C,A]-Hopf module isomorphism, and there exists a map from C(?)A to C, which is a right A-module isomorphism and also a left C-comodule isomorphism;(3) The map is a [C,A]-Hopf module isomor-phism, and there exists a map from C(?)A to C, which is a right A-module isomorphism and also a left C-comodule isomorphism.This result can be considered dual to the one given by Y. Doi and M. Takeuchi([5, theorem9]), which said there are three equivalent sayings about a cleft module algebra.