Super-Biderivations and Linear Super-Commuting Maps on the Lie Superalgebras

特殊李超代数 S(m, n; t) 的斜对称超双导子

Skew-symmetric super-biderivations of the special Lie superalgebra S(m, n; t)

本文研究了特征 p > 2 域上特殊李超代数 S(m, n; t) 的斜对称超双导子，利用交换子代数 TS 及其权空间分解，证明了所有斜对称超双导子均为内导子。

Super-biderivations of the contact Lie superalgebra $K(m,n;\underline{t})$

Let $K$ denote the contact Lie superalgebra $K(m,n;\underline{t})$ over a field of characteristic $p>3$, which has a finite $\mathbb{Z}$-graded structure. Let $T_K$ be the canonical torus of $K$, which is an abelian subalgebra of $K_{0}$ and operates on $K_{-1}$ by semisimple endomorphisms. Utilizing the weight space decomposition of $K$ with respect to $T_K$, %we show the action of the skew-symmetric super-biderivation on the elements of $T$ and the contact of $K$. %Moreover, we prove that each skew-symmetric super-biderivation of $K$ is inner.

特殊奇哈密顿超代数 SHO(n, n; t) 的斜对称超双导子

Skew-symmetric super-biderivations of the special odd Hamiltonian superalgebra SHO(n, n; t)

本文研究特征 p > 2 域上特殊奇哈密顿超代数 SHO(n, n; t) 的斜对称超双导子，利用交换子代数 TSHO 及其权空间分解，证明了 SHO 的所有斜对称超双导子均为内导子。

Linear commuting maps and skew-symmertric biderivations of the deformative Schrodinger-Virasoro Lie algebras

In this paper, we investigate the skew-symmertric biderivations of the deformative Schrodinger-Virasoro Lie algebras which contain the twisted and original deformative Schrodinger-Virasoro Lie algebras. As an application, we give the explicit form of each linear commuting map on the deformative Schrodinger-Virasoro Lie algebras. In particular, we obtain that there exist non-inner biderivations and non-standard linear commuting maps for the certain deformative Schrodinger-Virasoro Lie algebras.

Biderivations, commuting mappings and (2-)local derivations of $\mathbb{N}$-graded Lie algebras of maximal class

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with 1-dimensional homogeneous components: $\mathfrak{m}_0$, $L_1$ and $\mathfrak{m}_2$. In this paper, we studied their biderivations by considering the embedded mapping to derivation algebras. Then we determined commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras were characterized as the given gradings.

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。

$σ$-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…

Super-biderivations and post-Lie superalgebras on some Lie superalgebras

In this paper, all symmetric super-biderivations of some Lie superalgebras are determined. As an application, commutative post-Lie superalgebra structures on these Lie superalgebras are also obtained.

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}_{\…

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.

广义四元数代数上的局部超导子与超双导子

Local superderivation and super-biderivation on generalized quaternion algebra

本文以含单位元交换环上的广义四元数代数为研究对象，将其视作李超代数，探究其上的超双导子与局部超导子，证明了该代数上任意局部超导子均为超导子。

李超代数的超双导子的一种新定义

A New Definition of Superbiderivations for Lie Superalgebras

本文从结构和几何视角研究李超代数超双导子，提出要求括号为超双导子的新定义，并在完全李超代数中引入线性超交换映射，最后展示了其在海森堡超代数与形变理论中的两项应用。

两类与Heisenberg-Virasoro李共形超代数相关的李共形超代数

Two classes of Lie conformal superalgebras related to the Heisenberg--Virasoro Lie conformal algebra

本文构造了两类与Heisenberg-Virasoro李共形超代数相关的李共形超代数$\mathcal{HVS}(α)$和$\mathcal{HVS}(β,γ,τ)$，并完全确定了它们的共形导子、共形双导子、自同构群及秩$(1+1)$的共形模。

Tensor product representations for orthosymplectic Lie superalgebras

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the action of the orthosymplectic Lie superalgebra $\spo(V)$ and the orthosymplectic Lie color algebra $\spo(V,β)$. We use the Brauer algebra action to compute maximal vectors in $V^{\otimes k}$ and to decompose $V^{\otimes k}$ into a direct sum of submodules $T^λ$. We compute the characters of the modules $T^λ$, give a combinatorial description of these characters in terms o…

交换的仿射超杨代数边收缩子代数：源自边收缩

Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions

本文研究了仿射超杨代数边收缩的交换性，并应用该结果建立了与W超代数相关的一系列同态关系。

On associated variety for Lie superalgebras

We define the associated variety $ X_{M} $ of a module $ M $ over a finite-dimensional superalgebra $ {\mathfrak g} $, and show how to extract information about $ M $ from these geometric data. $ X_{M} $ is a subvariety of the cone $ X $ of self-commuting odd elements. For finite-dimensional $ M $, $ X_{M} $ is invariant under the action of the underlying Lie group $ G_{0} $. For simple superalgebra with invariant symmetric form, $ X $ has finitely many $ G_{0} $-orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible f…