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## Karakalpak Scientific Journal

#### Article Title

LOCAL AND 2-LOCAL DERIVATIONS ON OCTONION ALGEBRAS

#### 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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#### References

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