Globalization of Twisted Partial Hopf Actions

Abstract

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

Dilations of partial representations of Hopf algebras

“…For example, in [7] the authors obtained a version of Blattner-Montgomery theorem for the case of partial actions, generalizing a result in [20] for the group case. Globalization theorems were also obtained in other contexts such as partial actions of Hopf algebras on k-linear categories [5] and twisted partial actions of Hopf algebras [8].…”

Cohomology for partial actions of Hopf algebras

“…for any g, h ∈ G and l, m ∈ L [6]. This translation invariance is a useful tool for searching solutions of partial 2-cocycles in specific cases.…”

“…This condition gives us 64 equations which are, in fact redundant, that is, every 2-cochain in this case is a 2-cocycle Finally, from ω * ω = e 2 , taking x = ω(p e , p e ) and y = ω(p e , p e ), we obtain the equation [6] 16xy − 3(x + y) + 1 2 = 0.…”

“…(3) The condition (7) refers to every linear combination of monomials involving the units of the cochain groups C • par (H, A) which vanish in the algebra A. Of course, some of these relations are in fact present among elements of the form (6), but there are other vanishing linear combinations in A involving partial actions of elements of H upon the unit 1 A which needed to be ruled out in order to remember the structure of A.…”

“…Moreover, ∆ l is a morphism of algebras. Take any a#h, b#l ∈ A# ω H, then (2) )ω(h (4) , l (2) )#h (6) l (4) On the other hand, (2) )ω(h (5) , l (3) )#h (6) l (4) .…”