Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more general case when the ring $A$ is a finitely generated domain over $K$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer $C_{{\rm Der}A}(D)$ is a "large" \ subalgebra in ${\rm Der}_{K} A$, namely $\rk_A C_{\Der A}(D) := \dim_R RC_{\Der A}(D)$ equals ${\rm tr}.°_{K}R,$ where $R$ is the field of fraction of the ring $A.
Let B be an algebra over a field k and let Der(B) be the set of k-derivations from B to B. We define what it means for a subset of Der(B) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of Der(B); if every element of L is a locally nilpotent derivation then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?
It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebra is the maximal Lie algebra which lies in the set of locally nilpotent derivations of the ring. Also it was conjectured that each other maximal Lie algebra which lies in the set of locally nilpotent derivations of the polynomial ring is conjugated to the triangular Lie algebra. In the present work we prove the first part of this conjecture and provide the counterexample to the second part. Also we show that the second part of the conjecture holds for the maximal Lie algebras among locally nilpoten…
In commutative algebra, if $δ$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $δ$, then $Δ=wδ$ is also a locally nilpotent derivation with the same kernel as $δ$. In this paper we prove that the locally nilpotent derivation $Δ$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $Δ$ is a free associative algebra and give an explicit set of its free generators.
Let $R$ be a commutative ring and $\mathcal A$ an $R$-algebra. An $R$-$\mathcal E$-derivation of $\mathcal A$ is an $R$-linear map of the form $\operatorname{I}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of $\mathcal A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In this paper we discuss some open problems on whether or not the image of a locally finite $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ is a Mathieu subspace [Z2, Z3] of $\mathcal A$, and whether or not a locally nilpotent $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ maps every …
In this exposition, we propose a notion of rank and rigidity of locally nilpotent derivations of affine fibrations. We show that the concept is analogous to the perception of rank and rigidity of locally nilpotent derivations of polynomial algebras. Our results characterize locally nilpotent derivations of $\mathbb{A}^3$-fibrations having slice by classifying the fixed point free locally nilpotent derivations in terms of their ranks.
Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra Der A of all K-derivations of A is an A-module in a natural way and if R is the quotient field of A, then RDer A is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDer A of rank k over R (i.e. such that dim_{R}RL=k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of …
In this article, we show that a fixed point free locally nilpotent derivation of an $\mathbb{A}^2$-fibration over a Noetherian ring containing $\mathbb{Q}$ has slice.
In this paper, we give a characterization of locally nilpotent derivations on $\mathbb{A}^2$-fibrations over Noetherian domains containing $\mathbb{Q}$ having kernel isomorphic to an $\mathbb{A}^1$-fibration.
We investigate the filtration corresponding to the degree function induced by a non-zero locally nilpotent derivations and its associated graded algebra. As an application we provide an efficient method to recover the Makar-Limanov invariants, isomorphism classes and automorphism groups of classically known algebras. We also present a new class of examples which can be fully described with this method.
Given a locally nilpotent derivation on an affine algebra $B$ over a field $k$ of characteristic zero, we consider a finitely generated $B$-module $M$ which admits a locally nilpotent module derivation $δ_M$ (see Definition 1.1 below). Let $A=\Ker δ$ and $M_0=\Ker δ_M$. We ask if $M_0$ is a finitely generated $A$-module. In general, there exist counterexamples which are closely related to the fourteenth problem of Hilbert. We also look for some sufficient conditions for finite generation.
Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie algebra is embedded into the Lie algebra $R\Der_{\mathbb K} A\subseteq \Der_{\mathbb K}R$, where $R={\rm Frac}(A)$ is the fraction field of $A.$ The rank $rk_{R}L$ of a subalgebra $L$ of $R\Der_{\mathbb K} A$ is defined as dimension $\dim_R RL.$ We prove that every locally nilpotent subalgebra $L$ of $R\Der_{\mathbb K} A$ with $rk_{R}L=n$ has a series of ideals…
In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three.
We describe homogeneous locally nilpotent derivations of the algebra of regular functions for a class of affine trinomial hypersurfaces. This class comprises all non-factorial trinomial hypersurfaces.
In this paper we give an algorithm to recognize triangulable locally nilpotent derivations in dimension three. In case the given derivation is triangulable, our method produces a coordinate system in which it exhibits a triangular form.
By using classical invariant theory approach a formulas for computation of the Poincare series of the kernel of linear locally nilpotent derivations is found.
We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.
Let A be a commutative associative integrally closed k-algebra without zero divisors effectively graded by a lattice. We obtain a criterion of local nilpotency of the sum of two homogeneous locally nilpotent derivations (LNDs) of fiber type on A in terms of their degrees. The same problem is solved for commutators of two homogeneous LNDs.
Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and $δ$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of $\mathcal A$ (i.e., $δ=\operatorname{Id}_A-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $\mathcal A$). Motivated by the Idempotent conjecture proposed in [Z4], we first show that for every idempotent $e$ lying in both the kernel ${\mathcal A}^δ$ and the image $\operatorname{Im}δ\!:=δ({\mathcal A})$ of $δ$, the principal ideal $(e)\subseteq \operatorname{Im} δ$ if $δ$ is a locally finite $K$-derivation or a locally nilpotent $K$-$\mathcal E$-derivation …
We prove Freudenburg's Freeness Conjecture: Let B be the polynomial ring in three variables over a field of characteristic zero, let D : B --> B be a nonzero locally nilpotent derivation, and let A = ker(D). Then B is a free A-module, and there exists a basis $(e_i)_{i \in \mathbb{N}}$ of B such that deg$_D(e_i) = i$ for all $i \in \mathbb{N}$.
We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y - φ(X,Z))$ constructed from the defining equation $f(X)Y = φ(X,Z)$ of a generalized Danielewski surface in $\mathbb K^3$ for a specific choice of polynomials $f$ and $φ$, with $\mathbb K$ an algebraically closed field of characteristic zero. As a consequence of this description we calculate the $ML$-invariant and the Derksen invariant of this ring. We also determine a set of generators for the group of $\mathbb K$-automorphisms of $\mathbb K[X,Y,Z]/(f(X)Y - φ(Z))$ also for a specific choice …
Let B be a commutative $\mathbb{Z}$-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that $B_{(f)}$ is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d > 0, we give sufficient conditions that guarantee that every derivation of $B^{(d)} = \oplus_i B_{di}$ can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaiden…
Let $K$ be a field of characteristic zero and $x$ a free variable. A $K$-$\mathcal E$-derivation of $K[x]$ is a $K$-linear map of the form $\operatorname{I}-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $K[x]$, where $\operatorname{I}$ denotes the identity map of $K[x]$. In this paper we study the image of an ideal of $K[x]$ under some $K$-derivations and $K$-$\mathcal E$-derivations of $K[x]$. We show that the LFED conjecture proposed in [Z4] holds for all $K$-$\mathcal E$-derivations and all locally finite $K$-derivations of $K[x]$. We also show that the LNED conjecture proposed in [Z4] holds …
Let k be an algebraically closed field of zero characteristic. The Lie algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts on the polynomial ring k[x, y] and also on the field of rational functions k(x, y). For a fixed non-constant rational function u from k(x,y) we consider the set A_{W_2}(u) of all derivations D from W_2 such that D(u)=0. We prove that A_{W_2}(u) is a free submodule of rank 1 of the k[x,y]-module W_2. A description of the maximal abelian subalgebras as well of the centralizers of elements in the Lie algebra A_{W_2}(u) has been obtained.
Let k be an algebraically closed field of characteristic zero, D a locally nilpotent derivation on the polynomial ring k[X_1, X_2,X_3,X_4] and A the kernel of D. A question of M. Miyanishi asks whether projective modules over A are necessarily free. Implicit is a subquestion: whether the Grothendieck group K_0(A) is trivial. In this paper we shall demonstrate an explicit k[X_1]-linear fixed point free locally nilpotent derivation D of k[X_1,X_2, X_3, X_4] whose kernel A has an isolated singularity and whose Grothendieck group K_0(A) is not finitely generated; in particular, there exists an inf…
Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is arbitrary big.
We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$, where $n\geq 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, …
Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric algebra of all derivations of $P_n$ \cite{Dzhuma99,UU2014-1}. Using the language of $\mathscr{L}_n$, for every derivation $D\in \mathscr{L}_n$ we define the associative algebra $A_D$, the Lie algebra $L_D$, and the left-symmetric algebra $\mathscr{L}_D$ related to the study of the Jacobian Conjecture. For every derivation $D\in \mathscr{L}_n$ there is a unique $n$-tuple $F=(f_1,f_2,\ldots,f_n)$ of elements of $P_n$ suc…
