arXiv subjects

Rene Baltazar

Publications and source records attributed to Rene Baltazar.

8 recordsLinked to original sources

arXiv2026

Simple derivations and isotropy on Danielewski-type algebras

Let $\K$ be an algebraically closed field of characteristic zero. We study isotropy groups of simple derivations on Danielewski-type algebras \[ A_{c,q}=\K[x,y,z]/(c(x)z-q(x,y)). \] More precisely, motivated by the recent result of Mendes and Pan for simple derivations of $\K[x,y]$ (\cite[Theorem 1]{MP}), we ask whether every simple derivation $D$ of $A_{c,q}$ has trivial isotropy group. We prove that this property holds generically: for each fixed pair of degrees $\deg(c)\geq2$ and $\deg_y(q)\geq2$, there exists a nonempty Zariski open subset of the parameter space of reduced pairs $(c,q)$ such that every simple derivation of $A_{c,q}$ has trivial isotropy group. On the other hand, we construct a special Danielewski-type algebra admitting a simple derivation with nontrivial isotropy. Thus, the Mendes-Pan phenomenon holds generically for Danielewski-type algebras, but fails in full generality. Over $\K=\mathbb C$, we also discuss the associated foliations and formulate questions about their polynomial and birational symmetries.

math.RA↗

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↗

arXiv2025

The isotropy group of a derivation on a Danielewski-type algebra

Given an algebraically closed field $k$ of characteristic zero, we consider in this paper $k$-algebras of the form $$A_{c,q}=k[x,y,z]/\big(c(x)z-q(x,y)\big),$$ where $c(x)\in k[x]$ is a polynomial of degree at least two and $q(x,y)\in k[x,y]$ is a quasi-monic polynomial of degree at least two with respect to $y$. We give a complete description of the $k$-automorphism group of $A_{c,q}$ as an abstract group. Moreover, for every non-locally nilpotent $k$-derivation $\delta$ of $A_{c,q}$ we prove that the isotropy group of $\delta$ is a linear algebraic group of dimension at most three.

math.RA↗

arXiv2025

On Isotropy Groups of Quantum Plane

This paper investigates the isotropy groups of derivations on the Quantum Plane $\Bbbk_q[x, y]$, defined by the relation $yx = qxy$, where $q \in \Bbbk^*$, with $q^2\neq 1$. The main goal is to determine the automorphisms of the Quantum Plane that commutes with a fixed derivation $\delta$. We describe conditions under which the isotropy group $\text{Aut}_\delta(A)$ is trivial, finite, or infinite, depending on the structure of $\delta$ and whether $q$ is a root of unity: additionally, we present the structure of the group in the finite case. A key tool is the analysis of polynomial equations of the form $\mu_1^a \mu_2^b = 1$, arising from monomials in the inner part of $\delta$. We also make explicit which finite subgroups of $Aut(\Bbbk_q[x, y])$ are isotropy groups of some derivation: either $q$ root of unity or not. Techniques from algebraic geometry, such as intersection multiplicity, are also employed in the classification of the finite case.

math.RA↗

arXiv2024

On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

We study isotropy groups of $\sigma$-derivations of the quantum Weyl algebra and of ordinary derivations of the Jordanian plane. For the quantum Weyl algebra $A_q^1(\Bbbk)$, with $q$ not a root of unity, we use Brzezinski's classification to decompose every $\sigma$-derivation into inner and non-inner stable components. This yields an intersection formula for the isotropy group of an arbitrary $\sigma$-derivation and leads to explicit arithmetic descriptions. For the Jordanian plane $\Lambda_2(\Bbbk)$, we give a necessary and sufficient condition for an automorphism to belong to the isotropy group of an inner derivation. We compute the isotropy groups of monomial inner derivations and of locally nilpotent derivations. These examples show that isotropy groups in the Jordanian plane may contain large triangular subgroups, unlike the quantum Weyl algebra. In this way, isotropy groups provide a natural invariant that reflects the structural difference between the Jordanian plane and the quantum Weyl algebra.

math.RA↗

arXiv2020

On Isotropy Group of Danielewski Surfaces

In the present work we consider differential rings of the form $(\mathcal B,D)$ where $\mathcal B$ is a Danielewski surface and $D$ is a locally nilpotent derivation on $\mathcal B$. Influenced by several recent works, we describe the isotropy group of a locally nilpotent derivation, $D$, on Danielewski surfaces, in the cases $xy = φ(z)$, $x^ny=φ(Z)$, and $f(x)x = φ(z)$.

math.AG↗

arXiv2013

Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings

The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that both $\partial_x \in \mathfrak{D}$ and the ring is $\mathfrak{D}$-simple, then there is $d \in \mathfrak{D}$ such that $k[x,y]$ is $\{\partial_x,d\}$-simple. As applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.

math.RA↗

arXiv2013

On solutions for derivations of a Noetherian k-algebra and local simplicity

We introduce a general notion of solution for a Noetherian differential $k$-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be unique, with special emphasis in the cases of k-algebras of finite type and formal series rings over k. Using that notion we generalize a criterion for simplicity due to Brumatti-Lequain-Levcovitz and give a geometric characterization of that; as an application we give a new proof of a classification theorem for local simplicity due to Hart and obtain a general result for simplicity of formal series rings over k

math.AC↗