Crossed modules and doi-hopf modules

Caenepeel, Stefaan; Militaru, Gigel; Zhu, Shanfeng

doi:10.1007/bf02773642

Cited by 54 publications

(72 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Takeuchi has observed that entwined modules (see [4]) can be viewed as particular cases of comodules over a coring. A fortiori, the Doi-Koppinen Hopf modules ( [13] and [20]) are special cases, and consequently Hopf modules, relative Hopf modules, graded modules, modules graded by G-sets and Yetter-Drinfel'd modules (see [13] and [9]), and we can apply the results we present below to all these types of modules. Let A be a ring, C an A-coring, and look at the forgetful functor…”

Section: Comodules Over Coringsmentioning

confidence: 82%

Frobenius Functors of the Second Kind

Caenepeel¹,

Groot²,

Militaru

2002

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

Abstract. A pair of adjoint functors (F, G) is called a Frobenius pair of the second type if G is a left adjoint of βF α for some category equivalences α and β. Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We recover the result that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind (cf.[27], [17]), and prove that the integral spaces of the Hopf algebra and its dual are isomorphic.

show abstract

Section: Comodules Over Coringsmentioning

confidence: 82%

Frobenius Functors of the Second Kind

Caenepeel¹,

Groot²,

Militaru

2002

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…ε H (a i h (2) )(c i · h (1) ) = and λ C * = C. Thus C is a k-Frobenius H-module coalgebra. Conversely, suppose that C is a k-Frobenius H-module coalgebra.…”

Section: Yetter-drinfel D Modules and Frobenius Type Properties 4331mentioning

confidence: 99%

“…Unless specified otherwise, all modules, algebras, coalgebras and Hopf algebras are over k, and unadorned ⊗ and Hom are ⊗ k and Hom k . In the sequel, H will be a Hopf algebra with invertible antipode S. For a coalgebra (C, ∆ C , ε C ) and a left C-comodule (M, ρ M ) we will use Sweedler's -notation ∆(c) = c (1) ⊗ c (2) and ρ M (m) = m <−1> ⊗ m <0> , where c ∈ C, m ∈ M.…”

Section: Preliminariesmentioning

confidence: 99%

“…Then the antipode S H is bijective (see [17]), and the Drinfel d double D(H) = H H * cop is defined as follows: D(H) = H ⊗ H * as a k-module. The multiplication, comultiplication, counit and antipode are respectively given by the formulas (1) f (2) ) ⊗ (h (2) f (1) ); • ε D(H) = ε H ⊗ ε H * cop ; • S D(H) (h f) = f (2) (S H h (1) ) (S(h (2) ) (S H * f (1) ), for h, h ∈ H and f, f ∈ H * . Radford [18] showed that, over a field k, D(H) is unimodular with integral λ Λ with nonzero λ ∈ r H and Λ ∈ l H * .…”

Section: Corollary 38 ([16 Theorem 214])mentioning

confidence: 99%

“…Several module structures appear as special cases; let us mention Sweedler's Hopf modules [20], Takeuchi's relative Hopf modules [21], graded modules, and modules graded by a G-set. It was proved recently [2] that Yetter-Drinfel d modules are also a special case of Doi-Hopf modules.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties

Caenepeel¹,

Militaru

Zhu

1997

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We study the following question: when is the right adjoint of the forgetful functor from the category of (H, A, C)-Doi-Hopf modules to the category of A-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that C ⊗ A and the smash product A#C * are isomorphic as (A, A#C * )-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case A = H, and this leads to the notion of k-Frobenius H-module coalgebra. In the special case of Yetter-Drinfel d modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if H is finite dimensional and unimodular.

show abstract