Globalization for Geometric Partial Comodules

Abstract: (based on a joint work [2] with Joost Vercruysse)

The study of partial symmetries (partial actions and coactions, partial representations and corepresentations, partial comodule algebras) is a relatively recent field in continuous expansion and, therein, one of the relevant questions is the existence and uniqueness of a so-called globalization (or enveloping action). For instance, in the framework of partial actions of groups any global action of a group $G$ on a set induces a partial action of the group on any subset by restriction. The idea behind the concept of globalization of a given partial action is to find a (universal) $G$-set such that the initial partial action can be realized as the restriction of this global one.

We propose here a categorical approach to partial symmetries and the globalization question, explaining several of the existing results and, at the same time, providing a procedure to construct globalizations in concrete contexts of interest. Our approach relies on the notion of geometric partial comodules, recently introduced by Hu and Vercruysse [1] in order to describe partial actions of algebraic groups from a Hopf-algebraic point of view.

Unlike classical partial actions, which exist only for (topological) groups and Hopf algebras, geometric partial comodules can be defined over any coalgebra in a monoidal category with pullbacks and they allow to describe phenomena that are out of the reach of the theory of partial (co)actions, even in the Hopf algebra framework. At the same time, geometric partial comodules allow to approach in a unified way partial actions of groups on sets, partial coactions of Hopf algebras on algebras and partial (co)actions of Hopf algebras on vector spaces. Thus, the question of studying the existence (and uniqueness) of globalization for geometric partial comodules naturally arises as a unifying way to address the issue.

References:

[1] J. Hu, J.Vercruysse - Geometrically partial actions. Trans. Amer. Math. Soc. 373 (2020), no. 6, 4085-4143.

[2] P. Saracco, J. Vercruysse - Globalization for geometric partial comodules. Part I: general theory. Preprint (2021).

category theoryquantum algebra

Audience: researchers in the topic

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Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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