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Leonardo Duarte Silva

Publications and source records attributed to Leonardo Duarte Silva.

4 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.

math.RT↗
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.

math.RA↗
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.

math.RA↗