Narrow Results

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

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.

This paper is devoted to the nilpotent finite-dimensional evolution algebras $E$ with $dim{E}^2=dimE-1$. We describe the Lie algebra of derivations of these algebras. Moreover, in terms of these Lie algebras, we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. Besides, all automorphisms and local automorphisms of these algebras are found.

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra ${𝕆}_{ℝ}$. This description implies that the space of all local derivations on ${𝕆}_{ℝ}$ when equipped with Lie bracket is isomorphic to the Lie algebra ${𝔰𝔬}_7(ℝ)$ of all real skew-symmetric $7\times 7$-matrices. We also consider $2$-local derivations on an octonion algebra ${𝕆}_{𝔽}$ over an algebraically closed field $𝔽$ of characteristic zero and prove that every $2$-local derivation on ${𝕆}_{𝔽}$ is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra ${𝕆}_{ℝ}$ and Malcev algebra $M_7(ℝ)$ are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.

Let R be a prime ring with a nontrivial idempotent. In this paper, we prove that if g is an additive map of R into itself such that xg(y)z = 0 for all x, y, z ∈ R with xy = yz = 0, then g is a generalized derivation. As an application of this result, we show that every local generalized derivation in a prime ring with a nontrivial idempotent is a generalized derivation.