Automorphisms and derivations of algebras of infinite matrices

Derivation algebras and automorphism groups of a class of deformative super $W$-algebras $W^s_λ(2,2)$

In this paper, the derivation algebras and automorphism groups of a class of deformative super $W$-algebras $W_λ(2,2)$ are determined.

The automorphism groups and derivation algebras of two-dimensional algebras

The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.

Automorphisms, derivations and gradings of the split quartic Cayley algebra

The split quartic Cayley algebra is a structurable algebra which has been used to give constructions of Lie algebras of type D4. Here, we calculate its group of automorphisms, its algebra of derivations and its gradings.

The derivation algebra and automorphism group of the (generalized) twisted N=2 superconformal algebra

In this paper, we determine the derivation algebra and automorphism group of the twisted N=2 superconformal algebra. Then we generalize the relative results to the generalized twisted N=2 superconformal algebra in the final section.

Derivations and automorphisms of locally matrix algebras

We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description we show that for a countable--dimensional locally matrix algebra $A$ over a field $\mathbb{F}$ the dimension of the Lie algebra of outer derivations of $A$ and the order of the group of outer automorphisms of $A$ are both equal to $|\mathbb{F}|^{\aleph_0},$ where $|\mathbb{F}|$ is the cardinality of the field $\mathbb{F}.$

单 $ω$-Lie 代数的双导子、局部与 2-局部导子及自同构

Biderivations, local and 2-local derivation and automorphism of simple $ω$-Lie algebras

本文研究了复数域上有限维单 $ω$-Lie 代数的导子与自同构问题，证明了局部与2-局部导子及自同构的性质，并刻画了双导子与 $\frac{1}{2}$-导子等结构。

Derivations and automorphism groups of the original deformative Schrödinger-Virasoro algebras

In this paper, we determine the derivation algebra and the automorphism group of the original deformative ${\rm Schr\ddot{o}dinger}$-{\rm Virasoro} algebras which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module ${\rm I^g}(a,b)$.

关联环：自同构与导子

Incidence Rings: Automorphisms and Derivations

本文研究预序集$X$与交换环$T$上代数$R$构成的关联代数$I(X,R)$的自同构、同构及导子性质。

Automorphisms, derivations and central extensions of Lie algebras arising from quantum tori

For a Lie algebra $\mathfrak g$ related to a quantum torus, we compute its automorphisms, derivations and universal central extension. This Lie algebra $\mathfrak g$ is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.

Okubo algebras: automorphisms, derivations and idempotents

A survey of some properties of Okubo algebras is presented. Emphasis is put on automorphisms and derivations of these algebras, especially in characteristic three, where the situation is more involved and interesting. In this case, the Okubo algebra is closely related to a nodal noncommutative Jordan algebra. Dedicated to Helmut Strade on the occasion of his 70th birthday.

广义Loop Heisenberg-Virasoro代数的导子、2-局部导子、双导子与自同构

Derivations, 2-local derivations, biderivations and automorphisms of generalized Loop Heisenberg-Virasoro algebras

本文引入广义Loop Heisenberg-Virasoro代数，确定其导子、证明2-局部导子为导子、双导子为内双导子并给出应用，最后给出自同构群。

Derivations and automorphisms of a Lie algebra of Block type

Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the automorphism group of $\BB$ are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of $\BB$ is trivial.

结合代数的导子与自同构群的计算方法

Computational Approaches to Derivations and Automorphism Groups of Associative Algebras

本文利用低维结合代数的分类结果和符号计算工具，系统描述了二维至四维复数结合代数的导子与自同构群，揭示了其结构特征与对称性。

自由幂零李代数的导子与自同构及其商代数

Derivations and automorphisms of free nilpotent Lie algebras and their quotiens

本文通过研究自由幂零李代数的导子代数和自同构群，给出了同类型同幂零指数的一般幂零李代数的导子与自同构的自然刻画，并用例子加以说明。

局部导出与某些结合代数局部自同构的关系

The relationship between local derivations and local automorphisms of some associative algebras

本文研究了五维自然分级 nilpotent 结合代数的局部导出与局部自同构，揭示了 π₂ 和 π₃ 代数的矩阵一般形式，并证明这些代数的局部自同构（导出）不等于自同构（导出），同时建立了局部自同构与局部导出之间的指数关系，展示了局部导出集构成李代数。

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.

Derivations and automorphism groups of completed Witt Lie algebra

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a byproduct, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad-locally finite element.

Leibniz 2-代数的自同构与导子的Wells正合序列

Wells exact sequence for automorphisms and derivations of Leibniz 2-algebras

本文研究Leibniz 2-代数中自同构对与导子对的可诱导性，建立Wells正合序列的类比，并讨论Leibniz代数上交叉模的特殊情形。

Automorphisms and derivations of the insertion-elimination algebra and related graded Lie algebras

This paper addresses several structural aspects of the insertion-elimination algebra, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras, the automorphism group, the derivation group, and a generating set for the insertion-elimination Lie algebra. Many parts of the results are stated for a more general class of Lie algebras and reproduce results for the generalized Virasoro algebras.

关联余代数：自同构与导子

Incidence Coalgebras: Automorphisms and Derivations

本文利用关联余代数 Co(X,F) 的对偶代数同构于关联代数 I(X,F) 这一性质，刻画了域上偏序集关联余代数的自同构与导子。

Kac-Moody代数中Borel子代数及其零根的自同构和导子

Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras

本文确定了特征为0的域上的Kac-Moody代数（和逆成分李代数）中Borel子代数的导子及其导子零根，并确定了可对称化的Kac-Moody代数中Borel子代数的自同构.这些结果解决了R. V. Moody在大约30年前提出的一个结论，它推广了B. Kostant和A. Fialowski在仿射类型的情况下使用李代数上同调。

Witt型代数$\mathcal{W}(a,-1)$上的转置Poisson结构：导子、自同构和Rota-Baxter算子

Transposed Poisson structure on the Witt-type algebra $\mathcal{W}(a,-1)$: Derivations, Automorphisms, and Rota--Baxter operators

分类了Witt型代数W(a,-1)上的导子、自同构和Rota-Baxter算子，证明了非平凡δ-导子仅存在于δ=1和1/2，并建立了刚性结果。

Automorphisms and derivations of affine commutative and PI-algebras

We prove analogs of A.~Selberg's result for finitely generated subgroups of $\text{Aut}(A)$ and of Engel's theorem for subalgebras of $\text{Der}(A)$ for a finitely generated associative commutative algebra $A$ over an associative commutative ring. We prove also an analog of the theorem of W.~Burnside and I.~Schur about locally finiteness of torsion subgroups of $\text{Aut}(A)$.

Automorphisms and derivations of finite-dimensional algebras

Let $A$ be a finite-dimensional algebra over a field $F$ with char$(F)\ne 2$. We show that a linear map $D:A\to A$ satisfying $xD(x)x\in [A,A]$ for every $x\in A$ is the sum of an inner derivation and a linear map whose image lies in the radical of $A$. Assuming additionally that $A$ is semisimple and char$(F)\ne 3$, we show that a linear map $T:A\to A$ satisfies $T(x)^3- x^3 \in [A,A]$ for every $x\in A$ if and only if there exist a Jordan automorphism $J$ of $A$ lying in the multiplication algebra of $A$ and a central element $α$ satisfying $α^3=1$ such that $T(x)=αJ(x)$ for all $x\in A$. Th…

Local and 2-local derivations and automorphisms on simple Leibniz algebras

The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further we consider automorphisms of simple Leibniz algebras. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show that nilp…

The groups of automorphisms of the Lie algebras of unitriangular polynomial derivations

The groups of automorphisms of the Lie algebras of unitriangular polynomial derivations are found.

Semisimple Leibniz algebras and their derivations and automorphisms

The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms.

Derivations, Automorphisms, and Representations of Complex $ω$-Lie Algebras

Let $(\mathfrak{g},ω)$ be a finite-dimensional non-Lie complex $ω$-Lie algebra. We study the derivation algebra $Der(\mathfrak{g})$ and the automorphism group $Aut(\mathfrak{g})$ of $(\mathfrak{g},ω)$. We introduce the notions of $ω$-derivations and $ω$-automorphisms of $(\mathfrak{g},ω)$ which naturally preserve the bilinear form $ω$. We show that the set $Der_ω(\mathfrak{g})$ of all $ω$-derivations is a Lie subalgebra of $Der(\mathfrak{g})$ and the set $Aut_ω(\mathfrak{g})$ of all $ω$-automorphisms is a subgroup of $Aut(\mathfrak{g})$. For any 3-dimensional and 4-dimensional nontrivial $ω$-L…

关于只有平凡导子（自同构）的有限维代数与单代数

On finite dimensional algebras with only trivial derivations(automorphisms) and simple algebras

本文研究任意域上只有平凡导子或平凡自同构的 n 维代数及单代数，证明其集合非空且在代数闭域下于 Zariski 拓扑中稠密，并给出归纳构造方法及二维情形的完整同构分类。

广义Ramond N=2超共形代数的导子代数代数与自同构群

The Derivation algebra and automorphism group of the generalized Ramond N=2 superconformal algebra

给出了广义Ramond N=2超共形代数的定义，讨论了广义Ramond N=2超共形代数的导子代数和自同构群

Every monomorphism of the Lie algebra of unitriangular polynomial derivations is an automorphism

We prove that every monomorphism of the Lie algebra $\ggu_n$ of unitriangular derivations of the polynomial algebra $P_n=K[x_1,..., x_n]$ is an automorphism.

$σ$ -素环上的中心化自同构与Jordan左导子

Centralizing automorphisms and Jordan left derivations on $σ$-prime rings

设$R$是2-挠自由$σ$ -素环.本文证明了：如果$U\not\subset Z(R)$是$R$和$R$中的$a, b$的$σ$ -Lie理想，使得$aUb=σ(a)Ub=0,$，则要么是$a=0$，要么是$b=0.$.最后，我们描述了可加映射$d: R \longrightarrow R$使得$d(u^2) = 2ud(u)$，其中$u\in U,$是$R.$的非零$σ$ -平方闭李理想

具有非负元素的可逆矩阵半群的自同构

Automorphisms of the semigroup of invertible matrices with nonnegative elements

本文证明了线性序结合环上某些特殊子群上的非负元可逆矩阵半群的每一个自同构都与半群的一个内自同构、环的保序自同构和一个中心同素的合成一致。

Algebra endomorphisms and Derivations of Some Localized Down-Up Algebras

We study algebra endomorphisms and derivations of some localized down-up algebras $\A$. First, we determine all the algebra endomorphisms of $\A$ under some conditions on $r$ and $s$. We show that each algebra endomorphism of $\A$ is an algebra automorphism if $r^{m}s^{n}=1$ implies $m=n=0$. When $r=s^{-1}=q$ is not a root of unity, we give a criterion for an algebra endomorphism of $\A$ to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for $\A$. We also show that each surjective algebra endomorphism of the down-up algebra $A(r+s, -rs)$ is a…

The group of automorphisms of the Lie algebra of derivations of a polynomial algebra

We prove that the group of automorphisms of the Lie algebra $\Der_K (P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,..., x_n]$ over a field of characteristic zero is canonically isomorphic to the the group of automorphisms of the polynomial algebra $P_n$.

李代数$W$的导子、中心扩张和自同构群

The derivations, central extensions and automorphism group of the Lie algebra $W$

本文研究了文[8]和张东最近关于简单顶点算子代数分类的工作[22]中所出现的无限维李代数W的导子、中心扩张和自同构。

二元自由Poisson代数的自同构与导子

Automorphisms and derivations of free Poisson algebras in two variables

设P是特征为零的域上的二元自由Poisson代数。我们证明了P的自同构是驯服的，P的局部幂零导子是可三角化的。

division algebras的可换导子或自同构的可除子代数和特征值群的有限生成

Finite generation of division subalgebras and of the group of eigenvalues for commuting derivations or automorphisms of division algebras

设$D$是一个可除代数，使得$D\t D^o$是一个Noether代数，则$D$的任何可除子代数都是一个{ \em生成}可除代数.设$\D $是除代数$D$的可交换导子或自同构的有限集合，则公共特征值（即{ \em权}）群$\Ev (\D)$是{ \em权生成阿贝尔}群. $D$的典型例子是特征为零的域$K$上的光滑不可约仿射簇$X$上的微分算子环$\CD (X)$的商除代数${\rm Frac} (\CD (X))$，以及有限维李代数$\Gg $的通用包络代数$U(\Gg)$的商除代数${\rm Frac} (U (\Gg))$。证明了微分算子代数$\CD (X)$与它的反代数$\CD (X)^o$同构。

The group of automorphisms of the Lie algebra of derivations of a field of rational functions

We prove that the group of automorphisms of the Lie algebra $\Der_K (Q_n)$ of derivations of the field of rational functions $Q_n=K(x_1,..., x_n)$ over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the $K$-algebra $Q_n$.

超Virasoro代数上的局部导子与局部自同构

Local derivations and local automorphisms on the super Virasoro algebras

本文研究超Virasoro代数上的局部导子、局部自同构与2-局部自同构，证明局部导子必为导子，且局部或2-局部自同构必为自同构。

具有非平凡斜导子的Koszul Artin-Schelter正则代数的分次双重Ore扩张的中山自同构

Nakayama automorphisms of graded double Ore extensions of Koszul Artin-Schelter regular algebras with nontrivial skew derivations

本文研究Koszul Artin-Schelter正则代数的分次双重Ore扩张，通过构造极小自由预解证明其保持Koszul性质，并引入$ς$-散度同调不变量精确刻画中山自同构，进而构造扭曲超势实现导子商代数同构。

Description of 2-local derivations and automorphisms on finite dimensional Jordan algebras

In the present paper we introduce and investigate the notion of 2-local linear map on vector spaces. A sufficient condition is obtained for linearity of a 2-local linear map on finite dimensional vector spaces. Based on this result we prove that every 2-local derivation on a finite dimensional formally real Jordan algebra is a derivation. Also we show that every 2-local 1-automorphism (i.e. implemented by single symmetries) of mentioned Jordan algebras is an automorphism.

任意域上所有二维代数的自同构与导子的描述

A description of automorphisms and derivations of all two-dimensional algebras over an arbitrary field

本文对任意域上所有二维代数的自同构与导子进行了同构分类描述，彻底解决了相关的两个问题。

Tame and wild automorphisms of differential polynomial algebras of rank 2

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x,y$ with $m$ commuting derivations $δ_1, \ldots, δ_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.

交换偏序环上含非负元素的可逆矩阵半群的自同构

Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings

设R是一个序环，G_n（R）是$GL_n(R)$的一个子半群，由所有非负元素的矩阵组成. A.V. Mikhalev和M.A. Shatalova描述了G_n（R）的所有自同构，其中R是线性序斜体，n>1. E.I. Bunina和A. V. Mikhalev发现了G_n（R）的所有自同构，如果R是任意线性序结合环，n>2，且1/2。本文刻划了G_n（R）的自同构，如果R是交换偏序环，包含Q，n>2。

Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras

Let $A$ be a Koszul Artin-Schelter regular algebra, $σ$ a graded automorphism of $A$ and $δ$ a degree-one $σ$-derivation of $A$. We introduce an invariant for $δ$ called the $σ$-divergence of $δ$. We describe the Nakayama automorphism of the graded Ore extension $B=A[z;σ,δ]$ explicitly using the $σ$-divergence of $δ$, and construct a twisted superpotential $\hatω$ for $B$ so that it is a derivation quotient algebra defined by $\hatω$. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.

一个泛左对称包络代数的自同构与导子

Automorphisms and derivations of a universal left-symmetric enveloping algebra

本文研究了零乘法代数的泛左对称包络代数 $U_n$，基于局部幂零导子分析，证明了当 $n\geq 2$ 时其自同构群与多项式代数 $L_n$ 同构，并给出了 $n=1$ 时的完整描述。

赋予形式无限和的代数上的自同构与导子

Automorphisms and derivations on algebras endowed with formal infinite sums

本文在广义幂级数代数上利用形式指数泰勒展开建立自同构与导子的对应关系，并引入无限族可和性概念以定义指数映射，证明该对应适用于一大类代数。

Ternary mappings of triangular algebras

We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.

全正性矩阵的半群自同构

Semigroup automorphisms of total positivity

本文研究了全正和全非正矩阵半群的乘法自同构，证明其自同构相同且尊重乘法生成元。

无限矩阵环的导子

Derivations of rings of infinite matrices

本文刻画了一般系数环上无限矩阵的若干重要结合环与李环的导子结构。

Skew derivations on generalized Weyl algebras

A wide class of skew derivations on degree-one generalized Weyl algebras $R(a,φ)$ over a ring $R$ is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of $R$. It is determined which of the constructed derivations are $Q$-skew derivations. The compatibility of these skew derivations with the natural ${\mathbb Z}$-grading of $R(a,φ)$ is studied. Additional classes of skew derivations are constructed for generalized Weyl algebras given by an automorphism $φ$ of a finite order. Conditions that the central element $a$ that forms part of the structure of…

Inner Automorphisms of Lie Algebras Related with Generic 2 x 2 Matrices

Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion of F_m with respect to the formal power series topology. Our results are more precise for m=2 when F_2 is isomorphic to the Lie algebra L generated by two generic traceless 2 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain si…

量子环面与量子空间的斜导子

Skew derivations of quantum tori and quantum spaces

本文研究量子环面与量子仿射空间的σ-导子结构，给出其分解形式，推广了已有结论，并将结果应用于迭代Ore扩张，提出构造量子环面的算法，还探讨了斜导子与正规化序列的联系。