Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

Abstract We investigate Hopf algebras with non-zero integral from a coalgebraic point of view. Categories of Doi–Koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of Doi–Koppinen modules.

[1]  C. Menini,et al.  Strongly rational comodules and semiperfect Hopf algebras over QF rings , 2001 .

[2]  L. Grunenfelder,et al.  Constructing Pointed Hopf Algebras by Ore Extensions , 2000 .

[3]  D. Bulacu.  Injective modules graded by g-sets , 1999 .

[4]  Freddy Van Oystaeyen,et al.  The Categories of Yetter—Drinfel’d Modules, Doi—Hopf Modules and Two-sided Two-cosided Hopf Modules , 1998, Appl. Categorical Struct..

[5]  L. Grunenfelder,et al.  Finiteness Conditions, Co-Frobenius Hopf Algebras, and Quantum Groups , 1998 .

[6]  G. Militaru,et al.  Crossed modules and doi-hopf modules , 1997 .

[7]  C. Menini,et al.  Equivalence Theorems and Hopf)Galois Extensions , 1997 .

[8]  S. Dascalescu,et al.  Gradings of finite support. Application to injective objects , 1996 .

[9]  M. Koppinen.  Variations on the smash product with applications to group-graded rings , 1995 .

[10]  C. Nastasescu,et al.  Quasi-co-Frobenius Coalgebras , 1995 .

[11]  D. Ştefan.  The uniqueness of integrals , 1995 .

[12]  S. Caenepeel,et al.  Induction Functors for the Doi-Koppinen Unified Hopf Modules , 1995 .

[13]  Susan Montgomery,et al.  Hopf algebras and their actions on rings , 1993 .

[14]  Yukio Doi.  Unifying Hopf modules , 1992 .

[15]  S. Raianu,et al.  Modules graded byG-sets , 1990 .

[16]  T. Albu,et al.  Infinite group-graded rings, rings of endomorphisms, and localization , 1989 .

[17]  M. Takeuchi.  A correspondence between Hopf ideals and sub-Hopf algebras , 1972 .

[18]  P. Gabriel,et al.  Des catégories abéliennes , 1962 .