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 $\mathbb{F}.$
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.
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.
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.
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 ${\rm I^g}(a,b)$.
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.
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.
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.
The relationship between local derivations and local automorphisms of some associative algebras
V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；Andijan State University（安集延国立大学）；Namangan State University（纳曼干国立大学）
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.
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.
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。
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 present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms.
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…
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 $P_n$.
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 $Q_n$.
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.
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.
In the present paper we study local and 2-local derivations of locally finite split simple Lie algebras. Namely, we show that every local and 2-local derivation on such Lie algebra is a derivation.
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…
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 …
