Cohomology for partial actions of Hopf algebras

Abstract

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra H over a commutative algebra A, we prove that there exists a new Hopf algebra A, over a commutative ring E(A), upon which H still acts partially and which gives rise to the same cohomologies as the original algebra A. We also study the partially cleft extensions of commutative algebras by partial actions of cocommutative Hopf algebras and prove that these partially cleft extensions can be viewed as a cleft extensions by Hopf algebroids.

Recent developments around partial actions

“…The algebraà 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 Gmodules in [124] mean unital partial actions of a group G on a commutative monoid A.…”

“…As it was mentioned above, partial Gmodules 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 submonoidÃ, which does not change the cohomology.…”

“…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). A significant new feature in [47] is the fact that the partial crossed productÃ# ω H is a Hopf algebroid over the base algebra E(A) = h•1 A | h ∈ H [47,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]).…”

Partial cohomology of groups and extensions of semilattices of abelian groups

“…. Therefore, using also (1) and (13) we may write Proof. We need to show that (δ 3 β)(g, h, k, l)a = a for all a ∈ C(D g D gh D ghk D ghkl ), that is θ g (θ g −1 (a)β(h, k, l))β(g, hk, l)β(g, h, k) = aβ(gh, k, l)β(g, h, kl).…”