Generalization Of Twisted Smash Product And Co-cleft Module Coalgebra For Bialgebra

The main contents of this paper is generalization of twisted smash prod-uct(coproduct)and biproduct, We know smash product can be bialgebra on some conditions.We can get some new conclusions when we change its alge-bra(coalgebra)structures.In the first section,we mainly discuss generalization of twisted smash product.Proposition!. 1 Let H be a Hopf algebra,A and B bialgebra,let A be a H-bimodule algebra,B a H-bicomodule algebradefine A',B as follows:(l)as K-space A',B - .4g>B(2)multiplication is given by ,then A',B be an algebra with its unit 101.Propl.2 A.B.A B as mentioned in propl.l,if define Q(a ?x) = Y.(a\ ?then .4:5 be a bialgebra be a coalgebra morphism. Furthermore. be algebra morphism. If .4 and B are Hopf algebra , then A',B is //op/algebra, its antipode is given by Theoreml.l H,A,X,.4;*A' as mentioned,B is a bialgebra, if a : .4 5, 3 : X -+ B are bialgebra morphism,then there exists bialgebra morphism such that F Theorem1.2 H,A,X,A X as mentioned,if A X is coquasitriangular,then A, X are coquasitriangular,and there exists *-invertible skew pairing T : K such thatis the same as Theoreml.3 H,A,X,Aj}X as mentioned, then M is leftAftX module o M is left A module,leftX module such that In the second section, we mainly discuss dual of generalization of twisted smash product.Prop2.1 Let H be a Hopf algebra, A and B bialgebra,let A be a H-bicomodule coalgebra, B a H-bimodule coalgebra, define AQB as follows: (l)as K-space AQB ?.4?(2)comultiplication is given by ,then AQB be an coalgebra .Prop2.2 H. A, X,.4QA' as mentioned, if define multiplication (a畑)(b%>y) = ,then. is bialgebra be algebra morphism.If A and X are Hopf algebra, then AQX is Hopf algebra such that antipode is given by Prop2.3 H.A.X,.4QA' as mentioned. if quasitriangular. then A. X are quasitriangular,and there exists weak R matrix I A B such that is the same as Doublecrossed coproduct.Theorem2.1 as mentioned. B is a bialgebra, if a : B ?> A. 3 : are bialgebra morphism.then there exists bialgebra morphism F : B -> .40.A such that L^oF = a, Theorem2.2 H.A,X,.4;A' as mentioned, then M is left comodule M is left A comodule, left X comodule such that In the third section, we mainly discuss generalization of biproduct.PropS.l Let H be bialgebra, Aleft H module algebra,left H comodule coalgebra ,Xleft H comodulo algebra, left. H module coalgebra,defme space = A?X, defined to be smash product as algebra, that is xoy, defined to be smash coproduct as coalgebra,that is X2, if A is module coalgebra,comodule algebra , EX are algebra morphism,A/i(l) = 1?1, Ax(l) = 101, if following conditions hold, then are bialgebra:if .4, A" are Hopf algebra, then .4<>A"is Hop/ algebra. antipode is given byTheorein3.1 H.A.X.-40-V as mentioned. B is a bialgebra, if a : B. 3 : are bialgebra morphism.then there exists bialgebra morphism such that F Theorem3. 2 H.A.X,, as mentioned. then M is left. 40 A' module M is left A module, left X module such that In the fourth section, we mainly discuss generalization of biproduct for bimodule.Prop4.1 Let H be bialgebra, A be H bimodule algebra, H bicomodule coalgebra ,X be H bicomodule algebra, H bimodule coalgebra,define AVX&s K-space 桝 0 AT, denned to be generalization of twisted smash product as algebra,that is ( ,defined to be generalization of twisted smash coproduct as coalgebra,that is A(a's?x) =,if A is bimodule coalgebra,bicomodulea,lgebra,EA ,ex are algebra morphism,A/i(l) = 1 <8> 1, Ax(l) = 1 ?1, If following conditions hold, then AVX are bialgebra:if A and X are Hopf algebra, then A<\>B is Hopf algebra, antipode is given by In the fifth section, we mainly discuss cocleft module coalgebra for a bialgebra.Prop5.1 Let A be a bialgebra, B be a A-module coalgebra ,if there exists *-invertible cointegral , then (1)L2 is *-invertible,(2)10(3)if there exists from K to B coalgebra map , then unit map is *-invertible.Prop5.2 Let A be a bialgebra,B be a A-module coalg...