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Grasiela Martini

Publications and source records attributed to Grasiela Martini.

12 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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arXiv2026

On the isotropy of differential Ore extensions

Let Ah = k[x][t; d] be the differential Ore extension. We study the action of the automorphism group of Ah on the derivations of Ah and explicitly describe, using Nowicki's decomposition of the derivations of Ah, the isotropy groups of this action. More precisely, we first obtain an explicit description of the automorphism group of Ah for deg(h) >= 1. Then we determine the isotropy groups of derivations of the form D = ad_w + Delta_s(x), which exhaust all derivations in the square-free case, that is, when gcd(h,h') = 1. In the singular case, where gcd(h,h') is not equal to 1 and special derivations of type EH appear, we show that the isotropy problem is governed by a suitable localization and by the element w* = w + psi^(-1)H, where psi = gcd(h,h'). This yields a general criterion for the isotropy of a derivation of the form D = ad_w + EH + Delta_s(x). Finally, we provide explicit examples illustrating the new phenomena that arise in this setting.

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arXiv2024

On partial actions of Hopf-Ore extensions

In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in particular, generalized Taft algebras and Radford algebras), under suitable conditions.

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arXiv2024

Weak Hopf algebras arising from weak matched pairs

This work extends the idea of matched pairs presented by Majid in \cite{Majid} and Takeuchi in \cite{Takeuchi} for the context of weak bialgebras and weak Hopf algebras. We introduce, also inspired by partial matched pairs \cite{matchedpair}, the notion of weak matched pairs and establish conditions for a subspace of the smash product be a weak bialgebra/Hopf algebra. Further, some new examples of (co)actions of weak bialgebras over algebras and some results about integral elements are presented.

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arXiv2022

Partial (co)actions of Taft and Nichols Hopf algebras on algebras

In this paper, we characterize suitable partial (co)actions of Taft and Nichols Hopf algebras on algebras, and moreover we get that such partial (co)actions are symmetric. For certain algebras, these partial (co)actions obtained are, indeed, all of them. This work generalizes the results obtained by the authors in \cite{taft_corpo_revista}.

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arXiv2022

Partial Actions of a Hopf algebra on its base field and the corresponding partial smash product algebra

We introduce the concept of a $λ$-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a $λ$-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.

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arXiv2020

Partial Smash Coproduct of Multiplier Hopf Algebras

In this work we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.

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arXiv2020

Partial Coactions of Weak Hopf Algebras on Coalgebras

It will be seen that if $H$ is a weak Hopf algebra in the definition of coaction of weak bialgebras on coalgebras \cite{Wang}, then a definition property is suppressed giving rise to the (global) coactions of weak Hopf algebras on coalgebras. The next step will be introduce the more general notion of partial coactions of weak Hopf algebras on coalgebras as well as a family of examples via a fixed element on the weak Hopf algebra, illustrating both definitions: global and partial. Moreover, it will also be presented how to obtain a partial comodule coalgebra from a global one via projections, giving another way to find examples of partial coactions of weak Hopf algebras on coalgebras. In addition, the weak smash coproduct \cite{Wang} will be studied and it will be seen under what conditions it is possible to generate a weak Hopf algebra structure from the coproduct and the counit defined on it. Finally, a dual relationship between the structures of partial action and partial coaction of a weak Hopf algebra on a coalgebra will be established.

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arXiv2019

Hopf algebras arising from partial (co)actions

In this paper, extending the idea presented by M. Takeuchi in [13], we introduce the notion of partial matched pair $(H,L)$ involving the concepts of partial action and partial coaction between two Hopf algebras $H$ and $L$. Furthermore, we present necessary conditions for the corresponding bismash product $L\# H$ to generate a new Hopf algebra and, as illustration, a family of examples is provided.

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arXiv2019

Multiplier Hopf Algebras: Globalization for partial actions

In partial action theory, a pertinent question is whenever given a partial (co)action of a Hopf algebra A on an algebra R, it is possible to construct an enveloping (co)action. The authors Alves and Batista, in [2],have shown that this is always possible if R has unit. We are interested in investigating the situation where both algebras A and R are nonunitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [11], which is called multiplier Hopf algebra. Therefore, we will consider partial (co)actions of multipliers Hopf algebras on algebras not necessarily unitary and we will present globalization theorems for these structures. Moreover, Dockuchaev, Del Rio and Simón, in [5], have shown when group partial actions on nonunitary algebras are globalizable. Based in [5], we will establish a bijection between group partial actions on an algebra R not necessarily unitary and partial actions of a multiplier Hopf algebra on the algebra R.

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arXiv2018

Partial Actions of Weak Hopf Algebras on Coalgebras

In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

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