Commuting maps on certain incidence algebras

简单李超代数上的超双导子和线性超交换映射

Super-biderivations and linear super-commuting maps on simple Lie superalgebras

本文证明了特征不为2的代数闭域上有限维简单李超代数的斜超对称超双导子均为内导子，且其上的线性超交换映射均为标量乘法映射。

Heisenberg代数上的交换映射

Commuting maps on the Heisenberg algebra

研究Heisenberg代数上的线性交换映射，证明其不必具有标准形式。

幂交换与中心化映射在严格上三角矩阵环上

Power commuting and centralizing maps on the ring of strictly upper triangular matrices

本文研究了严格上三角矩阵环上满足特定交换和中心化条件的映射性质，揭示了2幂交换映射与中心化映射的结构特征。

$σ$-Biderivations and $σ$-commuting maps of triangular algebras

Let $\A$ be an algebra and $σ$ an automorphism of $\A$. A linear map $d$ of $\A$ is called a $σ$-derivation of $\A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$ is said to be a $σ$-biderivation of $\A$ if it is a $σ$-derivation in each component. An additive map $Θ$ of $\A$ is $σ$-commuting if it satisfies $Θ(x)x - σ(x)Θ(x) = 0$, for all $x \in \A$. In this paper, we introduce the notions of inner and extremal $σ$-biderivations and of proper $σ$-commuting maps. One of our main results states that, under certain assumptions, every $σ$-biderivatio…

Biderivations and commuting linear maps on Lie algebras

Let $L$ be a Lie algebra over a field of characteristic different from $2$. If $L$ is perfect and centerless, then every skew-symmetric biderivation $δ:L\times L\to L$ is of the form $δ(x,y)=γ([x,y])$ for all $x,y\in L$, where $γ\in{\rm Cent}(L)$, the centroid of $L$. Under a milder assumption that $[c,[L,L]]=\{0\}$ implies $c=0$, every commuting linear map from $L$ to $L$ lies in ${\rm Cent}(L)$. These two results are special cases of our main theorems which concern biderivations and commuting linear maps having their ranges in an $L$-module. We provide a variety of examples, some of them sho…

Biderivations and commuting linear maps on Hom-Lie algebras

The purpose of this paper is to determine skew-symmetric biderivations $\text{Bider}_{\text{s}}(L, V)$ and commuting linear maps $\text{Com}(L, V)$ on a Hom-Lie algebra $(L,α)$ having their ranges in an $(L,α)$-module $(V, ρ, β)$, which are both closely related to $\text{Cent} (L, V)$, the centroid of $(V, ρ, β)$. Specifically, under appropriate assumptions, every $δ\in\text{Bider}_{\text{s}}(L, V)$ is of the form $δ(x,y)=β^{-1}γ([x,y])$ for some $γ\in \text{Cent} (L, V)$, and $\text{Com}(L, V)$ coincides with $\text{Cent} (L, V)$. Besides, we give the algorithm for describing $\text{Bider}_{\…

Commuting maps on rank-$k$ matrices

Let $n\geq2$ be a natural number. Let $M_n(\mathbb{K})$ be the ring of all $n \times n$ matrices over a field $\mathbb{K}$. Fix natural number $k$ satisfying $1<k\leq n$. Under a mild technical assumption over $\mathbb{K}$ we will show that additive maps $G:M_n(\mathbb{K})\to M_n(\mathbb{K})$ such that $[G(x),x]=0$ for every rank-$k$ matrix $x\in M_n(\mathbb{K})$ are of form $λx + μ(x)$, where $λ\in Z$, $μ:M_n(\mathbb{K})\to Z$, and $Z$ stands for the center of $M_n(\mathbb{K})$. Furthermore, we shall see an example that there are additive maps such that $[G(x),x]=0$ for all rank-1 matrices th…

关于流李共形超代数上的共形双导子与共形交换映射

On the Conformal biderivations and conformal commuting maps on the current Lie Conformal superalgebras

本文研究流李共形超代数上的共形双导子与共形交换映射，证明其共形超双导子均可由中心化子刻画，并将该性质从李共形超代数 L 推广到张量积形式 L⊗A。

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…

有限关联代数的可逆性保持映射

Invertibility preservers of finitary incidence algebras

该研究针对任意偏序集$X$与任意域$K$，通过幂集代数映射与线性映射刻画了有限关联代数$FI(X,K)$的线性单位可逆性保持映射，丰富了关联代数的映射理论。

交换代数

The Commuting Algebra

该研究在路代数框架下探讨由平行路径差生成理想的商代数，证明其有限维且整体维数有限、与关联代数Morita等价，并推广至含一般双边理想的商路代数情形。

Commutativity preservers of incidence algebras

Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $φ:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $φ(D(X,K))=D(X,K)$. We prove that such a map $φ$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(θ,σ,c,κ)$ of simpler maps $θ$, $σ$, $c$ and a sequence $κ$ of elements of $K$.

含幂等元的环上的交换Jordan导子

Commuting Jordan derivations over a Ring with idempotents

该文研究含非平凡幂等元的素环上的交换Jordan导子，证明其为零映射，并将结论推广至交换Jordan高阶导子与交换广义Jordan导子，丰富了环上导子理论。

关于箭图的可达性范畴、持续性与交换代数

On reachability categories, persistence, and commuting algebras of quivers

本文通过箭图的可达性范畴，从范畴与拓扑角度研究其性质，证明交换代数与可达性偏序集的关联代数 Morita 等价，并引入持续性 Hochschild 同调。

Potent preservers of incidence algebras

Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $φ:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $φ$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $φ$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $φ$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain de…

Functional identities and zero Lie product determined Banach algebras

Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.

关联代数上双侧零积处的二次泛函恒等式

Functional identities of degree 2 at two-sided zero products on incidence algebras

该论文刻画关联代数上满足双侧零积处二次泛函恒等式的线性映射形式，证明其为标准形式的充要条件，涉及可比图2-连通性刻画与关联代数的双侧零积确定性质。

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 …

解析合成代数中的函数方程：解决猜想并考察其推论

Unraveling Functional Equations in Composition Algebra: Resolving Conjectures and Examining Implications

本文处理了 Ferreira 等人近期文章中遗留的猜想，并通过实例说明了回答这些猜想的结果所需条件的必要性。

Biderivations of finite dimensional complex simple Lie algebras

In this paper, we prove that a biderivation of a finite dimensional complex simple Lie algebra without the restriction of skewsymmetric is inner. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also get the forms of linear commuting maps on the finite dimensional complex simple Lie algebra or general linear Lie algebra.

Lax-Sato formulation of the Novikov-Veselov Hierarchy

We construct a hierarchy of pairwise commuting flows $d/dt_{i,n}$ indexed by $i \in \{1,2 \}$ and $n \in \mathbb{Z}_{\geq 0}$ on triples $(\mathcal{L}_1, \mathcal{L}_2, \mathcal{H})$ where $\partial_1$ and $\partial_2$ are two commuting derivations, $\partial_i \mathcal{L}_i$ is a self-adjoint pseudodifferential operator in $\partial_i$ and $\mathcal{H}$ is the formal Schrödinger operator $\mathcal{H}=\partial_1 \partial_2 +u$. $\mathcal{L}_1, \mathcal{L}_2$ and $\mathcal{H}$ are coupled by the relations $\mathcal{H} \mathcal{L}_i+\mathcal{L}_i^* \mathcal{H}=0$. We show that the flows $d/dt_{1…