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JOURNALS // Ural Mathematical Journal // Archive

Ural Math. J., 2022 Volume 8, Issue 2, Pages 46–58 (Mi umj171)

This article is cited in 2 papers

A characterization of derivations and automorphisms on some simple algebras

Farhodjon Arzikulova, Furkat Urinboyevb, Shahlo Ergashevac

a V.I. Romanovskiy Institute of Mathematics
b Namangan State University
c Kokand State Pedagogical Institute

Abstract: In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic $0$ without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra $\mathcal{D}$ and the seven-dimensional central simple commutative algebra $\mathcal{C}$. We prove that every local derivation of these algebras $\mathcal{D}$ and $\mathcal{C}$ is a derivation, and every $2$-local derivation of these algebras $\mathcal{D}$ and $\mathcal{C}$ is also a derivation. We also prove that every local automorphism of these algebras $\mathcal{D}$ and $\mathcal{C}$ is an automorphism, and every $2$-local automorphism of these algebras $\mathcal{D}$ and $\mathcal{C}$ is also an automorphism.

Keywords: simple algebra, derivation, local derivation, 2-local derivation, automorphism, local automorphism, 2-local automorphism, basis of identities.

Language: English

DOI: 10.15826/umj.2022.2.004



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