Of Hopf Over Galois I Expansion Theory

Hopf Galois Coextensions TheoryAbstractThe main contents of this paper is Hopf Galois coextensions. Galois coex-tenaion can be viewed as the dual of Galois extensions. And many conclusions on Galois coextensions dualling to the Galois extensions have been got [2][6][7], In this paper, we mainly consider Hopf bigalois coextensions and twistings of Hopf Galois coextensions.In the first section, we introduce some definitions and conclusions about Galois coextensions. especially discuss the equivalence of Crossed coproducts and get1.2.3.Theorem: JJ-cocleft extensions D/C and D'/C are isomorphic ? (D/C} = (D'/C}. Then [D/CT ? (D/C} defines an one-one correspondence between the isomorphic classes of #-cocleft extension of C and equivalent classes of crossed cosystem of H over C.1.2.4.Theorem: Let D/C be .ff-cocleft extension, (p, a) be the corresponding crossed cosystem, then the fallowings are equivalent: i) D/C is H -smash coextension,ii) (D/C} is the equivalent classes of crossed ccsystem (p'.o/). where o/(c) = e-(c)l玪.iii) There exists u 6 Reg(C..ff) satisfying EH o u = ec- auch that for all where pIn the second section, we introduce Kcpf bigaloia ccextensicn. Firstly, we define a map f : H fcr the sake cf solving problems in the later part. and consider a few equations.2.1.3.Proposition: Assume that C/B is an //-Galcia coextension. the map r sat-isfies the following properties for all c 8 d, x ?y G CCBC\ h € fl, c € C,Secondly, when coalgebra C is faithfully coflat over k, we prove that a bialgebra that admits a Hopf Galoia coextension is a Hopf algebra, that is2.2.2.Tbeorem: Let H be a /:- bialgebra and C/B a right H -Galoia coextension such that C is faithfully coflat over k. Then H is a Hopf algebraIn the third part of section 2, for B = A:, given a faithfully coflat right H-Galois coextension C/k, we construct another Hopf algebra L such that C/k is an L-H-bigaloia coextenaions. That ia, C/k is both a left L-Galoia coextension and a right .ff- Galois coextension such that the two module structures make it a Zr/f-bimodule. We get2.3.7. Theorem: Let H be a Hopf algebra and C/k a faithfully coflat right /f-Galoia coextension. Then L = (C ?C}" is a Hopf algebra. For C = C/CH' = it, we denote by c the image in k of the element c € C. the multiplication , unit , comultiph'cation , counit and antipode of L are given byrespectively, and C/fc ia a L-//-bigaloia coextenaiona. For any bialgebra B and leftB-module structure 6 on C making C/k a B-//-bigalois coextensiona. there is a unique isomorphism / : D ?> L of bialgebra such that = 6'(f 8 C}.Defined the coequalizer and a category "H\ Objects are all commutative fc-Hopf algebras with bijective antipodes. The morphisms are isomorphic classes of faithfully coflat bigalois coextensions with the composition given by coequalizer. We conclude2. 3. 11. Theorem: Let L.H,Rbe. Hopf algebras, C/k a faithfully coflat L- //-bigalois coextension and D/k a faithfully coflat //-.fl-bigalois coextension. Then (C&n D}/k la a faithfully coflat L-fi-bigalois coextension.2.3.14.Theoerm: H is a groupoid.For a right fT-module coalgebra C, by [9], from an element r € Hon(C, // ?C), r(c) = ?c-i ?CD satisfying the normality condition, we can redefine a coalgebra structure on C, AT(c) = j ?ca.-i & ca.o, such that CT = (C, AT) is also a right /J-module coalgebra. we fall C* the twisting of C. In the third part, we discuss twisting and Hopf Galois coextension.At first, on twisting and left twisting, we have3.1.2.Proposition. For r e U(T(C}), a invertible twisting with inverse A, write thenDefine a twisted 2-cocyde a. then we can construct a twisting3. 1.4. Proposition: For an ff-module coalgebra C, if a : twisted 2-cocyck, then the map TQ : For all twisting CT. CA, we get3.1.5.Proposition: For T, A € T(C), write r(c) _o, -Me) = let i)