Lie automorphisms of incidence algebras

Incidence Monoids: Automorphisms and Complexity

The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.

Proper Lie automorphisms of incidence algebras

Let $X$ be a finite connected poset and $K$ a field. We study the question, when all Lie automorphisms of the incidence algebra $I(X,K)$ are proper. Without any restriction on the length of $X$ we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of $X$. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns and ordinal sums of two antichains we give a complete answer.

Lie-type Derivations of Finitary Incidence Algebras

Let $P$ be an arbitrary partially ordered set, $R$ a commutative ring with identity and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. Under some natural assumption on $R$, we prove that each Lie-type derivation of $FI(P,R)$ is proper.

Lie Derivations of Incidence Algebras

Let $X$ be a locally finite preordered set, $\mathcal R$ a commutative ring with identity and $I(X,\mathcal R)$ the incidence algebra of $X$ over $\mathcal R$. In this note we prove that each Lie derivation of $I(X,\mathcal R)$ is proper, provided that $\mathcal R$ is $2$-torsion free.

Classification of involutions on finitary incidence algebras of non-connected posets

Let $FI(X,K)$ be the finitary incidence algebra of a non-connected partially ordered set $X$ over a field $K$ of characteristic different from $2$. For the case where every multiplicative automorphism of $FI(X,K)$ is inner, we present necessary and sufficient conditions for two involutions on $FI(X,K)$ to be equivalent.

Regular Hom-Lie structures on incidence algebras

We fully characterize regular Hom-Lie structures on the incidence algebra $I(X,K)$ of a finite connected poset $X$ over a field $K$. We prove that such a structure is the sum of a central-valued linear map annihilating the Jacobson radical of $I(X,K)$ with the composition of certain inner and multiplicative automorphisms of $I(X,K)$.

有限入射代数上的第二类对合

Involutions of the second kind on finitary incidence algebras

本文研究有限入射代数上的第二类对合，给出了其存在的充要条件、特征描述，以及在特定条件下两个此类对合等价的判定条件。

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $φ:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $φ(a_1)φ(a_2)=b$. In this paper we study bijective linear maps $φ:I(X,F)\to I(X,F)$ preserving products equal to primitive idempotents of $I(X,F)$, where $I(X,F)$ is the incidence algebra of a finite connected poset $X$ over a field $F$. We fully characterize the situation, when such a map $φ$ exists, and whenever it does, $φ$ is either an automorphism of $I(X,F)$ or the negative of an automorphi…

关于幂零李代数的交换自同构

On Commuting automorphisms of nilpotent Lie algebras

本文研究余类≤3的幂零李代数的交换自同构，揭示其构成自同构群子群的条件。

可解李代数中无不动点的自同构

Fixed-point-free automorphisms of solvable Lie algebras

研究可解李代数中无不动点自同构的存在性，证明此类李代数必为强单模的，且为复近近阿贝尔李代数存在无不动点自同构的充要条件，复filiform李代数中无不动点自同构等价于非特征 nilpotent。

乘法李代数扩张的自同构

Automorphisms of Multiplicative Lie algebra Extensions

本文研究乘法李代数扩张中自同构的可诱导性问题，证明其障碍由第二个上同调群刻画，并建立联系自同构群与第二个上同调群的Wells型正合序列。

The automorphism group functor of the N=4 Lie conformal superalgebra

In this paper, we study the structure and representability of the automorphism group functor of the N=4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.

Derivations and automorphisms of twisted deformative Schrödinger-Virasoro Lie algebras

In this paper the derivation algebra and automorphism group of the twisted deformative Schrödinger-Virasoro Lie algebras are determined.

Gradings on Incidence Algebras and their Graded Polynomial Identities

Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of characteristic zero, and A, B are both elementary G-graded incidence algebras satisfying the same G-graded polynomial identities, and the automorphisms group of P acts transitively on the maximal chains of P , we show that A and B are graded isomorphic.

Inner and Outer Automorphisms of Free Metabelian Nilpotent Lie algebras

We describe the groups of inner and outer automorphisms of the free metabelian nilpotent Lie algebra of finite rank over a field of characteristic 0. To obtain this result we first describe the groups of inner and continuous outer automorphisms of the completion with respect to the formal power series topology of the free metabelian Lie algebra finite rank.

Local automorphisms of finite dimensional simple Lie algebras

Let ${\mathfrak g}$ be a finite dimensional simple Lie algebra over an algebraically closed field $K$ of characteristic $0$. A linear map $φ:{\mathfrak g}\to {\mathfrak g}$ is called a local automorphism if for every $x$ in ${\mathfrak g}$ there is an automorphism $φ_x$ of ${\mathfrak g}$ such that $φ(x)=φ_x(x)$. We prove that a linear map $φ:{\mathfrak g}\to {\mathfrak g}$ is local automorphism if and only if it is an automorphism or an anti-automorphism.

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.

Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras

Let L be the free m-generated metabelian nilpotent of class c Lie algebra over a field of characteristic 0. An automorphism f of L is called normal if f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal subgroup N(L) of Aut(L) containing the group of inner automorphisms. We describe the group of normal automorphisms of L and the quotient group of Aut(L) modulo N(L).

On the automorphisms of a free Lie algebra of rank 3 over an integral domain

We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.

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.

Extensions and automorphisms of Lie algebras

Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $\Aut(A) \times \Aut(B)$ to be induced by an automorphism in $\Aut(L)$ lies in the Lie algebra cohomology $\Ha^2(B;A)$. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in $\Aut\big(L_{n,2}^{(1)}\big) \times \Aut\big(L_{n,2}^{ab}\big)$ to be induced by an au…

幂零李代数的局部导子与自同构

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。

Lie algebras with Frobenius dihedral groups of automorphisms

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ of order 2, such that the fixed-point subalgebra of $F$ is trivial and the fixed-point subalgebra of $H$ is metabelian. Then the derived length of $L$ is bounded by a constant.

可数无限秩自由幂零李代数的驯顺自同构

Tame Automorphisms of Free Nilpotent Lie Algebras of Countably Infinite Rank

本文证明特征零域上可数无限秩自由幂零李代数L_{∞,c}的每个自同构都由自由李代数L_∞的自同构诱导。

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。

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.

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.

Automatic classification of automorphisms of lower-dimensional Lie algebras

We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-commutative Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are delivered as the MATHEMATICA package cwsAutoClass. In order to facilitate the application of this package to symmetry Lie algebras of differential equations, we also provide a package, cwsLieSymTools, for manipulating finite-dimensional Lie algebras of vector fields. In particular, this package allows the computations of Lie brackets, structure constants, and…

Nilpotency of Lie type algebras with metacyclic Frobenius groups of automorphisms

Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is nilpotent of class c. If the ground field K contains a primitive n-th root of unity, then L is nilpotent and the nilpotency class of L is bounded in terms of |H| and c. The result extends the known theorem of Khukhro, Makarenko and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.

非阿贝尔扩展和后李代数的自同构

Non-abelian extensions and automorphisms of post-Lie algebras

本文研究了后李代数的交叉模与cat$^1$-后李代数的等价性，构建了非阿贝尔上同调以分类扩展，并通过Wells精确序列解决自同构诱导性问题。

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。

Integrality over fixed rings of automorphisms in a Lie nilpotent setting

Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].

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…

Lie type algebras with an automorphism of finite order

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $α, β\in \Bbb F$ such that $α\neq 0$ and $[[a,b],c]=α[a,[b,c]]+β[[a,c],b]$. Examples of Lie type algebras are associative algebras, Lie algebras, Leibniz algebras, etc. It is proved that if a Lie type algebra $L$ admits an automorphism of finite order $n$ with finite-dimensional fixed-point subalgebra of dimension $m$, then $L$ has a soluble ideal of finite codimension bounded in terms of $n$ and $m$ and of derived length bounded …

Lie-Yamaguti代数扩张的自同构与可诱导性问题

Automorphisms of extensions of Lie-Yamaguti algebras and Inducibility problem

本文研究Lie-Yamaguti代数扩张自同构的可诱导性问题，将其与(2,3)-上同调关联，构建Wells正合序列，应用于半直积与指数2的幂零Lie-Yamaguti代数，给出例子并设计求解可诱导配对的算法。

Lie Triple Derivations of Incidence Algebras

Let $\mathcal{R}$ be a $2$-torsion free commutative ring with unity, $X$ a locally finite pre-ordered set and $I(X,\mathcal{R})$ the incidence algebra of $X$ over $\mathcal{R}$. If $X$ consists of a finite number of connected components, we prove in this paper that every Lie triple derivation of $I(X,\mathcal{R})$ is proper.

3-李代数匹配对的上同调与自同构

Cohomology and automorphisms for a matched pair of 3-Lie algebras

本文研究3-李代数匹配对的表示、低维上同调、形变与扩张，并利用Wells正合序列刻画其自同构的可诱导性。

Nijenhuis Lie 共形代数的上同调、同伦、扩张与自同构

Cohomology, Homotopy, Extensions, and Automorphisms of Nijenhuis Lie Conformal Algebras

本文研究 Nijenhuis Lie 共形代数的上同调、同伦结构、非交换扩张及自同构，定义其上同调并建立与形变理论、交叉模及 Wells 型正合序列的联系。

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…

自由亚阿贝尔李代数某些自同构的野性的错误证明

Erroneous proofs of the wildness of some automorphisms of free metabelian Lie algebras

本文指出 Bahturin–Nabiyev 与 Özcurt–Ekici 关于自由亚阿贝尔李代数野性自同构的证明存在不可修正错误，强调秩 n≥4 情形的驯顺/野性问题仍未解决。

半单李代数的标准自同构及其关系

Standard automorphisms of semisimple Lie algebras and their relations

本文针对迹为零方阵单李代数中标准自同构的“民间常识”关系，给出一种初等验证方法，避免依赖 Millson 与 Toledano Laredo 证明中的非平凡拓扑论证。

$2$-local automorphisms on finite-dimensional Lie algebras

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an automorphism.

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.

Outer Automorphism Groups of Simple Lie Algebras and Symmetries of Painted Diagrams

For a real semisimple Lie algebra, we consider its automorphism group quotient by its identity component. This is known as the outer automorphism group. In this article, we compute the outer automorphism groups of all real semisimple Lie algebras.

Inner automorphisms of Lie algebras of symmetric polynomials

Let $L_{n}$ be the free Lie algebra, $F_{n}$ be the free metabelian Lie algebra, and $L_{n,c}$ be the free metabelian nilpotent of class $c$ Lie algebra of rank $n$ generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ symmetric in these Lie algebras if $p(x_1,\ldots,x_n)=p(x_{π(1)},\ldots,x_{π(n)})$ for each element $π$ of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincide with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determi…

The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras

The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras are found.

关联代数的斜导子

Skew derivations of incidence algebras

本文研究局部有限偏序集上关联代数的斜导子，证明其可分解，且斜导子空间模内斜导子子空间与第一上同调群同构。

Double automorphisms of graded Lie algebras

We introduce the concept of a double automorphism of an A-graded Lie algebra L. Roughly, this is an automorphism of L which also induces an automorphism of the group A. It is clear that the set of all double automorphisms of L forms a subgroup in Aut(L). In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. We also give an application of our results to groups admitting a Frobenius group of automorphisms.

The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence

The group of automorphisms is found for the Lie algebra of polynomial vector fields with constant divergence.

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence are found.

Automorphisms and derivations of algebras of infinite matrices

We describe automorphisms and derivations of several important associative and Lie algebras of infinite matrices over a field.

Lie algebras admitting a metacyclic Frobenius group of automorphisms

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the k…

Evolution algebras, automorphisms, and graphs

The affine group scheme of automorphisms of an evolution algebra that is equal to its square, is shown to lie in an exact sequence, such that the other terms depend solely on the directed graph associated to the algebra. As a consequence, the Lie algebra of derivations is shown to be trivial in characteristic 0 or 2, and to be abelian, with a precise description depending just on the graph, otherwise.

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.