Modules over weak entwining structures

Partial (Co)Actions of Hopf Algebras and Partial Hopf-Galois Theory

“…It follows from (6) that the A-bimodule structure on R is induced by η. So an A-ring is a ring R together with a ring morphism η : A → R. Let R be a left unital A-bimodule, and consider the projection π : R → R = R1 A , and an A-bimodule map µ : R ⊗ A R → R satisfying (5). µ restricts to a map µ : R ⊗ A R → R, since µ(r1 A ⊗ A s1 A ) = µ(r1 A ⊗ A s)1 A ∈ R, for all r, s ∈ R. We will write µ(r ⊗ A s) = rs, as usual.…”

“…Right unital lax and weak A-rings are introduced in a similar way. Let R be a right unital A-bimodule and R = 1 A R. Consider an A-bimodule map µ : R ⊗ A R → R satisfying (5). µ restricts to µ : R ⊗ A R → R. η : A → R corestricts to the map π • η : A → R. (R, µ, η) is a right unital lax (resp.…”

“…The right A-module structure of the coring is induced by a kind of entwining map. In the weak Hopf algebra case, it is a weak entwining map, as introduced in [5]. The map in the partial group action case, however, does not satisfy the axioms of a weak entwining structure.…”

The Affineness Criterion for Weak Yetter-Drinfel’d Modules

“…Let H be a weak Hopf algebra over the field k. Recall from [3] that a left H-comodule algebra is an algebra A together with a multiplicative left…”

The Galois Theory of Matrix C-rings

“…Recall from [16] that a (right-right) weak entwining structure is a triple (A,C, ψ R ), where A is an algebra, C a coalgebra, and ψ R : C ⊗ A → A ⊗ C a k-linear map which, writing,…”