arXiv subjects

William Hautekiet

Publications and source records attributed to William Hautekiet.

6 recordsLinked to original sources

arXiv2026

One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras

In this work, we complete the description of the one-dimensional partial actions of 8-dimensional Hopf algebras by computing the remaining cases: the Kac-Paljutkin algebra $\mathcal{A}$ and the unique non-semisimple non-pointed Hopf algebra $\mathcal{K}$. We prove that all these partial actions are symmetric, study their associated partial smash products and we determine all their partial coactions of dimension one. Beyond the 8-dimensional setting, we investigate which right coideal subalgebras of a Hopf algebra $H$ can be realized as partial smash products $\underline{ \Bbbk \# H}$ over the base field. In particular, we show that every right coideal subalgebra of a finite-dimensional cosemisimple Hopf algebra arises in this way.

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arXiv2025

Globalization and the biactegory of partial modules

We show that the category of partial modules over a Hopf algebra $H$ is a biactegory (a bimodule category) over the category of global $H$-modules. The corresponding enrichment of partial modules over global modules is described, and the close relation between the dilation of partial modules and Hom-objects arising from this enrichment is investigated. In particular, for finite-dimensional pointed Hopf algebras, the standard dilation of a partial module $M$ is isomorphic to the Hom-object from the monoidal unit to $M$.

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arXiv2024

Partial representations of connected and smash product Hopf algebras

We show that every partial representation of a connected Hopf algebra is global. Some interesting classes of partial representations of smash product Hopf algebras are studied, and a description of the partial "Hopf" algebra if the first tensorand is connected is given. If $H$ is cocommutative and has finitely many grouplikes, this allows to see $H_{par}$ as the weak Hopf algebra coming from a Hopf category.

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arXiv2023

Towards a classification of simple partial comodules of Hopf algebras

Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra $H$ using central idempotents in right coideal subalgebras and show that any $1$-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial $H$-comodules arise this way. For $H = kG$ for some finite group $G$, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If $H = kG^*,$ then our construction recovers the work of M. Dokuchaev and N. Zhukavets. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra $\mathcal{A}$.

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arXiv2022

A comonadicity theorem for partial comodules

We show that the category of partial comodules over a Hopf algebra $H$ is comonadic over ${\sf Vect}_k$ and provide an explicit construction of this comonad using topological vector spaces. The case when $H$ is finite dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.

math.RA↗

arXiv2020

Partial and global representations of finite groups

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is trivial), we develop further an effective theory that allows explicit computations. As a case study, we apply our theory to the symmetric group and its subgroup of permutations fixing 1: this provides a natural extension of the classical representation theory of the symmetric group.

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