Iterated Differential Polynomial Rings over Locally Nilpotent Rings

On Lie algebras consisting of locally nilpotent derivations

Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of $LND(A)$ is of great interest, and the question about properties of Lie algebras contained in $LND(A)$ is still open. An answer to it in the finite dimensional case is given. It is proved that any finite dimensional (over $K$) subalgebra of $Der_{K}(A)$ consisting of locally nilpotent derivations is nilpotent. In the case $A=K[x, y],$ it is also proved that any …

通过局部幂零导出物解决取消问题

Cancellation problem via locally nilpotent derivations

本文通过局部幂零导出物方法研究取消问题，探讨其在刚性环境中的有效性及非斜情况下的局限性。

On the proof of some theorem on locally nilpotent subgroups in division rings

In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.

关于带局部幂零导子的微分多项式环

On differential polynomial rings with locally nilpotent derivations

该研究针对带局部幂零导子的微分多项式环的根结构问题，证明了其Jacobson根可表示为底环某诣零理想对应的微分多项式环，解决了Agata Smoktunowicz提出的相关问题。

Locally nilpotent sets of derivations

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?

$\mathcal{D}$-局部幂零代数、其理想结构及单性判据

$\D$-locally nilpotent algebras, their ideal structure and simplicity criteria

本文引入并研究$\mathcal{D}$-局部幂零代数，刻画其理想结构并给出单性判据，将微分算子代数的已有结果推广到更广泛的代数类。

Combinatorics of words and semigroup algebras which are sums of locally nilpotent subalgebras

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Maximal Lie subalgebras of locally nilpotent derivations

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…

Modular group algebras whose group of unitary units is locally nilpotent

We characterize those modular group algebras FG whose group of unitary units is locally nilpotent under the classical involution of FG.

Poisson catenarity in Poisson nilpotent algebras

We prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length.

Locally Nilpotent Derivations of Free Algebra of Rank Two

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.

Group algebra whose unit group is locally nilpotent

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.

Locally quasi-nilpotent elementary operators

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $ϕ=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(ϕ)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(ϕ)=\frac{n(n-1)}{2} $, then there exists a representation of $ϕ$ as $ϕ=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of leng…

关于幂零李代数的极大可解扩张

On maximal solvable extensions of nilpotent Lie algebras

本文完整刻画了一类幂零李代数的极大可解扩张，证明其在d-局部可对角化条件下唯一并实现为半直积，解决Šnobl猜想，并推广至李超代数。

Hyperinvariant subspaces of locally nilpotent linear transformations

A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $ x \in V $ is annihilated by some power of $f$, and that $V$ is an infinite direct sum of $f$-cyclic subspaces. In this note we describe the lattice of hyperinvariant subspaces of $V$. We extend results of Fillmore, Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite dimensional spaces.

Injective hulls of simple modules over finite dimensional nilpotent complex Lie superalgebras

We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U(g)-modules are locally Artinian are precisely those whose even part g_0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.

Some Open Problems on Locally Finite or Locally Nilpotent Derivations and ${\mathcal E}$-Derivations

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 …

关于李代数$W_n(K)$的某些可解子代数与局部幂零子代数的极大性

On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$

本文研究特征零代数闭域上李代数$W_n(K)$的子代数结构，证明了三角子代数$u_n(K)$是极大局部幂零子代数，可解子代数$s_n(K)$是极大可解子代数，丰富了$W_n(K)$的子代数分类结果。

Locally nilpotent Lie algebras of derivations of integral domains

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…

Injective Hulls of Simple Modules Over Nilpotent Lie Color Algebras

Using cocycle twists for associative graded algebras, we characterize finite dimensional nilpotent Lie color algebras $L$ graded by arbitrary abelian groups whose enveloping algebras $U(L)$ have the property that the injective hulls of simple right $U(L)$-modules are locally Artinian. We also collect together results on gradings on Lie algebras arising from this characterization.

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S…

Differential polynomial rings over rings satisfying a polynomial identity

Let $R$ be a ring satisfying a polynomial identity and let $δ$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $δ(N)\subseteq N$ and the Jacobson radical of $R[x;δ]$ is equal to $N[x;δ]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x;δ]$ is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski.

Centralizers of linear and locally nilpotent derivations

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 gener…

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let $A$ be an algebra over a field $K$ of characteristic zero, let $\d_1, >..., \d_s\in \Der_K(A)$ be {\em commuting locally nilpotent} $K$-derivations such that $\d_i(x_j)=\d_{ij}$, the Kronecker delta, for some elements $x_1,..., x_s\in A$. A set of algebra generators for the algebra $A^\d:= \cap_{i=1}^s\ker (\d_i)$ is found {\em explicitly} and a set of {\em defining relations} for the algebra $A^\d$ is described. Similarly, given a set $\s_1, ..., \s_s\in \Aut_K(A)$ of {\em commuting} $K$-automorphisms of the algebra $A$ such that the maps $\s_i-{\rm id_A}$ are {\em locally nilpotent} and …

On Radicals of Ore Extensions and Related Questions

We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Niels…

关于多项式环导子的有限维齐次李代数

On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings

该研究针对非局部幂零的有限个齐次导子生成的李代数，推导得出其有限维判定准则，补充了以往仅针对局部幂零齐次导子相关结论的研究空白。

Ore扩张的导子的局部幂零性刻画

A Characterization of Local Nilpotence for Derivations of Ore Extensions

本文研究Pan关于局部幂零导子迷向群刻画的非交换推广，证明其对微分Ore扩张成立，对第一Weyl代数局部有限导子成立，并构造自由结合代数中反例。

Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal E$-derivations

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 …

Nilpotent Lie algebras of derivations with the center of small corank

Let $\mathbb K$ be a field of characteristic zero, $A$ an integral domain over $\mathbb K$ with the field of fractions $R = \text{Frac}(A),$ and $\text{Der}_{\mathbb{K}}A$ the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=R\text{Der}_{\mathbb{K}} A$ and $L$ a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $ W(A)$ of rank $n$ over $R$ with a natural basis over t…

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 $f(x)x = φ(z)$.

The LFED and LNED Conjectures for Laurent Polynomial Algebras

Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all locally finite or locally nilpotent $R$-derivations and $R$-$\mathcal E$-derivations of ${\mathcal A}_n$, where by an $R$-$\mathcal E$-derivation of ${\mathcal A}_n$ we mean an $R$-linear map of the form $\operatorname{Id}_{{\mathcal A}_n}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of ${\mathcal A}_n$. In particular, we show that ${\mathcal A}_n$ has no nonzero l…