Chengkang Xu
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.
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We describe automorphisms and derivations of several important associative and Lie algebras of infinite matrices over a field.
In this article, we determine the derivation algebra and the automorphism group of the twisted Schrödinger-Virasoro algebra.
In this paper the derivation algebra and automorphism group of the twisted deformative Schrödinger-Virasoro Lie algebras are determined.
The present work is devoted to the extension of some general properties of automorphisms and derivations which are known for Lie algebras to finite dimensional complex Leibniz algebras. The analogues of the Jordan-Chevalley decomposition for derivations and the multiplicative decomposition for automorphisms of finite dimensional complex Leibniz algebras are obtained.
Local derivations and automorphisms of nilpotent Lie algebra
本文研究幂零李代数的局部导子与自同构，证明幂零指数为3和4时存在非导子的局部导子及非自同构的局部自同构，并给出存在性充分条件及一个局部导子空间与导子空间重合的非结合代数例子。
In the paper we give a complete classification of $2$-dimensional evolution algebras over algebraically closed fields, describe their groups of automorphisms and derivation algebras.
Algebraic derivations and radicals of automorphism groups of algebras
本文研究半素结合代数的代数导子与相关自同构群的根性质，证明其素根与弱可解根重合，并联系Martindale商环内导子描述。
In this paper, the derivation algebras and automorphism groups of a class of deformative super $W$-algebras $W_λ(2,2)$ are determined.
The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.
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.
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.
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 $\mathbb{F}.$
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。
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 ${\rm I^g}(a,b)$.
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。
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.
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。
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.
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。
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。
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。
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.
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.
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。
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.
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。
In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type.
An n-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.
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。
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 $\text{Aut}(A)$.
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…
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 are found.
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…
The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms.
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。
In this paper, we give the definition of the generalized Ramond N=2 superconformal algebras and discuss the derivation algebra and the automorphism group
Let $R$ be a 2-torsion free $σ$-prime ring. It is shown here that if $U\not\subset Z(R)$ is a $σ$-Lie ideal of $R$ and $a, b$ in $R$ such that $aUb=σ(a)Ub=0,$ then either $a=0$ or $b=0.$ This result is then applied to study the relationship between the structure of $R$ and certain automorphisms on $R$. To end this paper, we describe additive maps $d: R \longrightarrow R$ such that $d(u^2) = 2ud(u)$ where $u\in U,$ a nonzero $σ$-square closed Lie ideal of $R.$
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.
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…
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 $P_n$.
In this paper, we study the derivations, central extensions and the automorphisms of the infinite-dimensional Lie algebra W which appeared in [8] and Dong-Zhang's recent work [22] on the classification of some simple vertex operator algebras.
Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
Let $D$ be a division algebra such that $D\t D^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a {\em finitely generated} division algebra. Let $\D $ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\Ev (\D)$ of common eigenvalues (i.e. {\em weights}) is a {\em finitely generated abelian} group. Typical examples of $D$ are the quotient division algebra ${\rm Frac} (\CD (X))$ of the ring of differential operators $\CD (X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient …
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 $Q_n$.
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。
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。
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.
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。
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.
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.
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。
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。
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。
We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of g. The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster 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.
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…
Non-abelian cohomology of Nijenhuis Lie algebras and the inducibility of automorphisms and derivations
该研究引入Nijenhuis李代数的非阿贝尔上同调群，给出其非阿贝尔扩张同构类的参数化与阿贝尔扩张分类，并刻画了自同构、导子可诱导性的上同调阻碍。
We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coor…
An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.
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