arXiv subjects

R. Baltazar

Publications and source records attributed to R. Baltazar.

4 recordsLinked to original sources

arXiv2026

On Locally Finite Derivations in Ore Extensions

Let $\Bbbk$ be an algebraically closed field of characteristic zero. We classify the locally finite derivations of arbitrary Ore extensions of $\Bbbk[x]$, thus extending van den Essen's \cite{V92} classification for the polynomial algebra $\Bbbk[x,y]$ to this noncommutative setting. More precisely, we consider the three families arising in the classification of Ore extensions of $\Bbbk[x]$: the quantum plane, the first quantum Weyl algebra, and the differential Ore extensions \[ A_h=\Bbbk[x][t;h(x)\partial_x]. \] For both the quantum plane and the first quantum Weyl algebra, we determine the locally finite derivations and explain how the resulting classifications are related to the work of Su\'arez-Alvarez and Vivas \cite{SuarezVivas} on generalized Weyl algebras. For the algebras $A_h$, with $h$ nonconstant, we obtain a complete classification in both the square-free and non-square-free cases. As a consequence, we show that $\LFD(A_h)$ is a solvable and weakly locally finite Lie subalgebra of $\Der(A_h)$, although it is not locally finite as a set of derivations.

math.RA↗

arXiv2026

Isotropy Groups of $\sigma$-Derivations on the Quantum Plane

Let k be an algebraically closed field of characteristic zero and let k_q[x,y] be the quantum plane. We study sigma-derivations of k_q[x,y] and their isotropy groups under the conjugation action of automorphisms. For q\neq\pm1, we use Jordan's recent classification of skew derivations for toric automorphisms, which generalizes the description of Almulhem and Brzezi\'nski for the quantum plane. Using this classification, we determine the isotropy groups of arbitrary sigma-derivations. These groups are described by character equations on the torus k^2, reducing the problem to arithmetic conditions. We recover the ordinary derivation case when sigma=id and exhibit new phenomena for nontrivial sigma-derivations, including cases where q is a root of unity. We also analyze the singular case q=-1. In this setting, we classify the sigma-derivations and describe the corresponding isotropy groups. In particular, for sigma=id, we obtain an explicit description of the isotropy groups of ordinary derivations of k_q[x,y], completing the singular case left open in previous work \cite{SBVA}.

math.RA↗

arXiv2019

On the Automorphism Group of a Polynomial Differential Ring in Two Variables

We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for deciding whether that group is an algebraic group. In most cases, from that study we deduce a primary classification of this type of differential ring up to conjugation with a polynomial automorphism.

math.AC↗

arXiv2014

On simple Shamsuddin derivations in two variables

We study the subgroup of $k$-automorphisms of $k[x,y]$ which commute with a simple derivation $D$ of $k[x,y].$ We prove, for example, that this subgroup is trivial when $D$ is a Shamsuddin simple derivation. In the general case of simple derivations, we obtain properties for the elements of this subgroup.

math.AC↗