Partial actions: what they are and why we care

Batista, Eliezer

doi:10.36045/bbms/1489888814

Cited by 24 publications

(20 citation statements)

References 47 publications

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“…In this section, we recall a few results from [9] about partial representations of Hopf algebras. For more details and the relation with the group case, we refer to [9] or the review [10] and the references therein.…”

Section: Global Modules and Partial Modulesmentioning

confidence: 99%

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Dilations of partial representations of Hopf algebras

Alves¹,

Batista

Vercruysse

2019

J. London Math. Soc.

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We introduce the notion of a dilation for a partial representation (that is, a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (that is, a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial H-modules, a category of (global) H-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon H, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory.

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Section: Global Modules and Partial Modulesmentioning

confidence: 99%

“…The theory of partial actions soon became important in the theory of dynamical systems [19] and, after the seminal paper [14], attracted the attention of pure algebraists as well. For an overview of the historical development of the theory of partial actions, see, for example, [10] and [13].…”

Section: Introductionmentioning

confidence: 99%

Dilations of partial representations of Hopf algebras

Alves¹,

Batista

Vercruysse

2019

J. London Math. Soc.

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“…Take a#h ∈ A# ω H, then (1) )ω −1 (S(h (2) ), h (4) )#S(h (1) ))(a (2) #h (5) ) = (S(h (5) ) · S A (a (1) ))ω −1 (S(h (4) ), h (6) )(S(h (3) ) · a (2) )ω(S(h (2) ), h (7) )#S(h (1) )h (8) = (S(h (4) ) · S A (a (1) ))(S(h (3) ) · a (2) )ω −1 (S(h (2) ), h (5) )ω(S(h (1) ), h (6) )#S(h (7) )h (8) = (S(h (3) ) · S A (a (1) ))(S(h (2) ) · a (2) )(S(h (1) ) · (h (4) · 1 A ))#1 H = (S(h (3) ) · S A (a (1) ))(S(h (2) ) · a (2) )(S(h (1) )h (4) · 1 A )#1 H = (S(h (2) ) · S A (a (1) ))(S(h (1) ) · a (2) )#1 H = (S(h) · (S A (a (1) )a (2) ))#1 H = s r (S(h) · ǫ A (a)) = s • ǫ r (a#h).…”

Section: It Remains To Check Thatunclassified

“…and µ(Id ⊗ E(A),◮◭ S) • ∆ r (a#h) = (a (1) #h (1) )S(a (2) #h (2) ) =(a (1) #h (1) )(S(h (4) ) · S A (a (2) )ω −1 (S(h (3) ), h (5) )#S(h (2) )) =(a (1) (h (1) · (S(h (7) ) · S A (a (2) )ω −1 (S(h (6) ), h (8) )))ω(h (2) , S(h (5) ))#h (3) S(h (4) ) =(a (1) (h (1) · (S(h (6) ) · S A (a (2) )))(h (2) · ω −1 (S(h (5) ), h (7) ))ω(h (3) , S(h (4) ))#1 H =(a (1) (h (1) · (S(h (2) ) · S A (a (2) )))(h (3) · ω −1 (S(h (6) ), h (7) ))ω(h (4) , S(h (5) ))#1 H =(a (1) (h (1) S(h (2) ) · S A (a (2) )))(h (3) · ω −1 (S(h (6) ), h (7) ))ω(h (4) , S(h (5) ))#1 H =a (1) S A (a (2) )(h (1) · ω −1 (S(h (4) ), h (5) ))ω(h (2) , S(h (3) ))#1 H ( * ) =ǫ A (a)ω −1 (h (1) S(h (8) ), h (9) )ω(h (2) , S(h (7) )h (10) )ω −1 (h (3) , S(h (6) ))ω(h (4) , S(h (5) ))#1 H =ǫ A (a)ω −1 (h (1) S(h (6) ), h (7) )ω(h (2) , S(h (5) )h (8) )(h (3) · (S(h (4) ) · 1 A ))#1 H =ǫ A (a)ω −1 (h (1) S(h (5) ), h (6) )ω(h (2) , S(h (4) )h (7) )(h (3) · 1 A )#1 H =ǫ A (a)ω −1 (h (1) S(h …”

Section: It Remains To Check Thatunclassified

“…This work allowed the generalization of classical results in Hopf algebra theory as well several ideas developed for partial group actions, as globalization theorem [2], Morita equivalence between the partial smash product and the invariant subalgebra [3], duality for partial actions [4], partial representations [7], etc. For a more detailed account on recent developments of partial actions of groups and Hopf algebras, see [8] and references therein. Of particular interest for the present article are the notions of twisted partial actions of Hopf algebras, partial crossed products and partially cleft extensions of algebras by Hopf algebras [5].…”

Section: Introductionmentioning

confidence: 99%

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Cohomology for partial actions of Hopf algebras

Batista¹,

Mortari²,

Teixeira³

2019

Journal of Algebra

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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.

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