Let $L$ be a Lie conformal superalgebra and $A$ be an associative commutative algebra with unity. We define the current Lie conformal superalgebra by the tensor product $L \otimes A.$ We prove every conformal super-biderivation $φ_{λ}$ on $L$ is of the form of the centroid $Cent(L)$. Moreover, we show that every Lie conformal super-biderivation on $L\otimes A$ also has the same performance as $L$. We also prove that every Lie conformal linear super-commuting map $\varPsi_{λ}$ on $L \otimes A$ belongs to $Cent(L \otimes A)$, if the same holds for $L$ as well.
In this paper, we characterize the super-biderivations of Cartan type Lie superalgebras over the complex field $\mathbb{C}$. Furthermore, we prove that all super-biderivations of Cartan type simple Lie superalgebras are inner super-biderivations.
In this paper we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.
On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations, linear commuting maps and α-biderivations, and compute them for some typical Hom-Lie algebras and superalgebras, including q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebra.
Suppose the ground field is algebraically closed and of characteristic different from 2. In this paper, we described the intrinsic connections among linear super-commuting maps, super-biderivations and centroids for Lie superalgebras satisfying certain assumptions. This is a generalization of the results of Bre$\check{s}$ar and Zhao on Lie algebras.
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.
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.
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.
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…
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.
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…
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}_{\…
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.
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…
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…
