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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 ℝ . This description implies that the space of all local derivations on ℝ when equipped with Lie bracket is isomorphic to the Lie algebra 7 (ℝ) of all real skew-symmetric 7 × 7 -matrices. We also consider 2 -local derivations on an octonion algebra over an algebraically closed field of characteristic zero and prove that every 2 -local derivation on 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 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. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations on the so-called infinite dimensional generalized Witt algebras. Firstly, we prove that every 2-local derivation on the generalized Witt algebra W n (F) over the vector space F n is a derivation, where F is a field of characteristic zero. Further we consider generalized Witt algebras of the form W = W (G , I) over the field F , where I is an infinite index set and G = ⨁ i ∈ I Z = (a i) i ∈ I : a i = 0 e x c e p t f o r a f i n i t e n u m b e r o f i ∈ I , and prove that all 2-local derivations on W (G , I) are also derivations. Finally, we show that every 2-local derivation on B (Z n , I) , the Borel subalgebra of W n (F) , is a derivation. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of the structures of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations of infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are (global) derivations. We also give an example of an infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation. [ABSTRACT FROM AUTHOR]