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Karakalpak Scientific Journal

Abstract

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

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Last Page

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References

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