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Properties of Injective Comodules
Junseok Park,Keun Chang Lee 湖西大學校 基礎科學硏究所 2013 기초과학연구 논문집 Vol.21 No.1
Let C be a coalgebra. We investigate injective C-comodulesand the properties they have in common with, projective odules. We find a dual of Baer Criterion in the category Mc where C is finite dimensional with (I ⊥)* ⊆ I for all left ideal I of C* and find a few equivalent conditions for right C-comoduie M.
BIPRODUCT BIALGEBRAS WITH A PROJECTION ONTO A HOPF ALGEBRA
Junseok Park 충청수학회 2013 충청수학회지 Vol.26 No.1
Let (D;B) be an admissible pair. Then recall that B £LHD À¼DiDD are bialgebra maps satisfying ¼D ± iD = I: We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipodesD and be a left H-comodule algebra and a left H-module coalgebra over a¯eld k: Let A be a bialgebra over k: Suppose A À¼ i D are bialgebra maps satisfying ¼ ± i = ID: Set ¦ = ID ¤ (i ± sD ± ¼);B = ¦(A) and j : B ! A be the inclusion. Suppose that ¦ is an algebra map. We show that (D;B)is an admissible pair and B ¿¦ j A À¼ i D is an admissible mapping system and that the generalized biproduct bialgebra B £LH D is isomorphic to A as bialgebras.
Junseok Park,Keun Chang Lee 호서대학교 기초과학연구소 2012 기초과학연구 논문집 Vol.20 No.1
We prove a few properties of A°. Let A be a reflexive algebra. If pM : M →M ⓧ A° is the comodule structure map of the right A° -comodule M, and M is a rational A°* - module and J is a two sided ideal of A such that JM 二 0, then pm(M) ⊆ M ⓧ ( J ⊥ ∩A°), i.e. M is a right comodule over the subcoalgebra J ⊥ (∩A° of A°. Let A be a finite dimensional algebra. If M is a right A°-comodule with comodule structure map φ : M →M ⓧ A° and B = (ann A (M)) ⊥ , then B∩A° is the smallest subcoalgebra of A° such that φ (M) ⊆ Mⓧ (B∩A°).
HOPF ALGEBRA STRUCTURE OVER GENERALIZED BIPRODUCTS
Junseok Park 호서대학교 기초과학연구소 2009 기초과학연구 논문집 Vol.17 No.1
The biproduct bialgebra has been generalized to generalized biproduct bialgebra in [5]. Let (D, B) be an admissible pair and that D is a bialgebra. We show that if generalized biproduct bialgebra BxLH D is a Hopf algebra with antipode 5, then D is a Hopf algebra and the identity idB has an inverse in the convolution algebra Homk(B,B). We show that if D is a Hopf algebra with antipode SD and sb ?? Homk(B,B) is an inverse of idB then BxLH D is a Hopf algebra w ith antipode s described by s(bxLH d) =∑(1bx LHsD)(b-1˙d ))(sB(bo)xLH 1D)This generalizes the corresponding results in [6]