Narrow Results

Let $L$ be a Lie algebra and let $\mathrm{\text{Der}}(L)$ be the set of all derivations of $L$. Then the derivation $\alpha$ of $L$ is called an ID-derivation if $\alpha (x)\in L^2$ for all $x\in L$. The set of all ID-derivations is denoted by $\mathrm{\text{ID}}(L)$. Let ${\mathrm{\text{Der}}}_z(L)$ be the set of all central derivations and let $C^{\ast }(L)$ and ${\mathrm{\text{ID}}}^{\ast }(L)$ be, respectively, the set of all center derivations and ID-derivations that map $Z(L)$ to zero. In this paper, we verify relations between $\mathrm{\text{ID}}(L)$ and ${\mathrm{\text{ID}}}^{\ast }(L)$ with ${\mathrm{\text{Der}}}_z(L)$ and $C^{\ast }(L)$. Also, we prove that if $L$ is a nilpotent Lie algebra of class $k$, then $\mathrm{\text{ID}}(L)$ is a nilpotent Lie algebra of class $k-1$.

In this paper, we study the derivations of group algebras of some important groups, namely, Dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($S{D}_{8n}$). First, we explicitly classify all inner derivations of a group algebra $𝔽G$ of a finite group $G$ over an arbitrary field $𝔽$. Then we classify all $𝔽$-derivations of the group algebras $𝔽D_{2n}$, $𝔽T_{4n}$ and $𝔽(S{D}_{8n})$ when $𝔽$ is a field of characteristic $0$ or an odd rational prime $p$ by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when $𝔽$ is an algebraic extension of a prime field.

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have $1$ dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning $2$-local derivations of such algebras is investigated and an example of solvable Leibniz algebra is given such that any $2$-local derivation on it is a derivation, but which admits local derivations which are not derivations.

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.

The paper is devoted to local derivations on the algebra of τ-measurable operators affiliated with a von Neumann algebra and a faithful normal semi-finite trace τ. We prove that every local derivation on which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra for the algebra to admit local derivations which are not derivations.

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical.

To give a Z₃-graded Cartan calculus on the extended quantum plane, the noncommutative differential calculus on the extended quantum plane is extended by introducing inner derivations and Lie derivatives.

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra $\mathcal{D}(\text{𝔤})$ and describe the relation between $\mathcal{D}(\text{𝔤})$ and the usual derivation Lie superalgebra Der(𝔤). We show that the set $\mathcal{D}(\text{𝔤})$ of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of $\mathcal{D}(\text{𝔤})$. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in $\mathcal{D}(\text{𝔤})$ is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

Let L be a Lie algebra, and Der(L) and IDer(L) be the set of all derivations and inner derivations of L, respectively. Also, let Der_c(L) denote the set of all derivations α ∈ Der(L) for which α(x) ∈ Imad_x for all x ∈ L. We give necessary and sufficient conditions under which Der_c(L) = Der_z(L), where Der_z(L) is the set of all derivations of L whose images lie in the center of L. Moreover, it is shown that any two isoclinic Lie algebras L₁ and L₂ satisfy Der_c(L₁) ≅ Der_c(L₂).