Local and 2-local derivations on octonion algebras

Abstract

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 [Formula: see text]. This description implies that the space of all local derivations on [Formula: see text] when equipped with Lie bracket is isomorphic to the Lie algebra [Formula: see text] of all real skew-symmetric [Formula: see text]-matrices. We also consider [Formula: see text]-local derivations on an octonion algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero and prove that every [Formula: see text]-local derivation on [Formula: see text] 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 [Formula: see text] and Malcev algebra [Formula: see text] are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.