Dilations of partial representations of Hopf algebras

Abstract

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.

Geometrically partial actions

“…If moreover X is geometric, than we can also restrict and corestrict θ −1 X to obtain an inverse θ −1 Y of θ Y and Y is again geometric. The following corollary describes a phenomenon that was also observed in [2] for the case of partial representations. Proof.…”

“…(iii) The associated map α ′ : G → Par(X, X) preserves the unit. (2) The following assertions are equivalent (i) The partial action datum defines a global action of G on X;…”

“…However, in contrast to the classical case, the monoidal category in play is no longer the usual monoidal category of representations (or modules) of the Hopf algebra H, but rather the category of partial representations which coincides with the category of representations over a newly constructed Hopf algebroid H par . Lately, it was shown in [2] how partial representations can be globalized and the partial representations of Sweedler's 4-dimensional Hopf algebra were completely classified. A recent development in the theory of partial actions, is the approach of [15], where the initial theory of parial actions over C * -algebras is merged with the Hopf-algebra setting, in the study of partial actions of C * -quantum groups.…”

Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

“…Namely, in §3.2 we show that the partial modules over a Hopf algebra H, introduced in [8] in terms of partial representations of H, provide examples of geometric partial comodules in the opposite of the category of vector spaces. We also show that the so-called standard dilation of a partial representation as described in [9] (see [2] for the case of partial representations of groups) coincides with the globalization of the associated geometric partial comodule. After, in §3.3, we verify that the globalization for partial representations of finite groups, as introduced and applied in [19], can be recovered by our approach, too.…”

Partial actions: what they are and why we care

“…The notion of a globalization can be extended to partial representations. This is what we call a dilation of a partial representation and it leads to a functor from the category of partial H-modules to the category of H-modules with projections [12]. The relationship between the subcategory of H-modules inside the category of partial H-modules need to be better clarified.…”