Cohomology for partial actions of Hopf algebras

Recent developments around partial actions

“…The algebra à possesses a partial action of H, so that, as in the group case, the isomorphism H n par (H, A) ∼ = H n par (H, Ã) of the corresponding cohomology groups holds. Furthermore, à enjoys a structure of a commutative and co-commutative Hopf algebra [47,Theorem 4.5]. In addition, by a result from [20] one naturally concludes that the partial crossed products A# ω H (with commutative A and co-commutative H) are in a bijective correspondence with the cohomology classes [ω] ∈ H 2 par (H, A).…”

“…Hopf algebroids appear also with respect to partial Hopf cohomology, which is the matter of the hot off the press preprint [47]. As it was mentioned above, partial G-modules in [124] mean unital partial actions of a group G on a commutative monoid A.…”

“…As it was mentioned above, partial G-modules in [124] mean unital partial actions of a group G on a commutative monoid A. Assuming that A is a (unital) algebra, it is possible to extend the notion of a partial cohomology group from [124] to the context of a partial action of a co-commutative Hopf algebra H on a commutative algebra A [47], giving a natural generalization of Sweedler's cohomology [278] to the partial action setting. In [124] the monoid A is replaced by an appropriate inverse submonoid Ã, which does not change the cohomology.…”

“…A significant new feature in [47] is the fact that the partial crossed product Ã# ω H is a Hopf algebroid over the base algebra Theorem 5.10]. The latter suggests that it should be possible to look at the partial Hopf cohomology from the point of view of the cohomology of Hopf algebroids (see [64]).…”

“…More precisely, given a co-commutative Hopf algebra H acting partially on a commutative and co-commutative Hopf algebra Ã, and a partial 2-cocycle ω ∈ Z 2 par (H, Ã), the partial smash product E( Ã)#H is a Hopf algebroid over E( Ã) by [48,Theorem 3.5]. Then [47,Theorem 6.6] states that the partial crossed product A# ω H is a right E( Ã)#H-module algebra whose subalgebra of the co-invariants is Ã, and the extension à ⊆ A# ω H is a cleft extension by the Hopf algebroid E( Ã)#H. Recalling from [20] that any partially cleft H-extension over the co-invariants can be seen as a crossed product extension, we obtain the desired fact. The article is enriched with examples and suggestions for further research.…”