A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

Farhodjon Arzikulov, Furqatjon Urinboyev, Shahlo Ergasheva
2022 Ural Mathematical Journal  
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
more » ... mple 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.
doi:10.15826/umj.2022.2.004 fatcat:ifaw6tub2jevjbglikmfxo22uq