Related papers: On Shamsuddin derivations and the isotropy groups

Let $R=K[X_1,\dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ \text{Aut}(R)_d :=\{\rho \in \text{Aut}_K(R)|\; \rho d…

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…

We consider the subgroup Aut(D) consisting of automorphisms of K[x,y] commuting with a derivation D, where K is an algebraically closed field of characteristic 0. We prove that if D is simple (i.e. D does not stabilize non-trivial ideals),…

We introduce the tame isotropy group of a derivation of a polynomial ring. We study this group for certain triangular derivations up to three variables, for simple derivations in two variables, and for simple Shamsuddin derivations in any…

Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\geq 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\ldots,x_i]$. Let $d_n$ be an extension of $d_i$ to a…

In the paper, we first study the subgroup of $ K$-automorphisms of $K[x_1,\allowbreak \ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$. We show that the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which…

In the paper, we prove that the derivation $D=y\partial_x+(a_2(x)y^2+a_1(x)y+a_0(x))\partial_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^*$, $(2)$ $\deg a_1(x)\geq1$ or…

Let $k$ be a field of characteristic zero. Let $m$ and $\alpha$ be positive integers. For $n\geq 2$, let $R_n=k[x_1,x_2,\dots,x_n]$ with the $k$-derivation $d_n$ given by…

Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by…

We prove the main result that a groupoid of order n is an idempotent k-translatable quasigroup if and only if its multiplication is given by x.y = (ax+by)(mod n), where a+b = 1(mod n), a+bk = 0(mod n) and (k,n)= 1. We describe the structure…

In the paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu-Zhao spaces. In addition, we also show that the images of some…

In this paper, we study the isotropy group of Lotka-Volterra derivations of $K[x_{1},\cdots,x_{n}]$, i.e., a derivation $d$ of the form $d(x_{i})=x_{i}(x_{i-1}-C_{i}x_{i+1})$. If $n=3$ or $n \geq 5$, we have shown that the isotropy group of…

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…

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each…

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a…

In a previous paper, the author and his collaborators studied the phenomenon of isotropy in the context of single-sorted equational theories, and showed that the isotropy group of the category of models of any such theory encodes a notion…

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…

Let $\delta$ be a derivation in a $K$-algebra $R$ and let $Aut_{\delta}(R)$ be the isotropy group with respect to the natural conjugation action of $Aut(R)$ of $K$-automorphisms on the set $Der(R)$ of $K$-derivations: that is, the subgroup…

$ $Let $k$ be a field of characteristic zero. If $c_1, c_2\in k\setminus \{0\}, s,t\geq 1$ and $u\geq 0$, then it is shown that the $k$-derivations $\partial_x + x^u(c_1x^ty^s+c_2)\partial_y$ and $\partial_x +…

We show that the standard set of elementary generators of an elementary isotropic reductive group over a connected finitely generated ring is a Kazhdan subset. This generalizes the corresponding result of M. Ershov, A. Jaikin-Zapirain, and…