2-local superderivations on a superalgebra \(M_n(\mathbb{C})\).
Let \(A\) be an associative algebra over a field \(K\). A mapping \(\delta\colon A\to A\) is called a 2-local derivation (respectively, 2-local automorphism) if for every \(a,b\) in \(A\) there exists a derivation (respectively, automorphism) \(\delta_{a,b}\colon A\to A\) depending on \(a\) and \(b\) such that \(\delta(a)=\delta_{a,b}(a)\) and \(\delta(b)=\delta_{a,b}(b)\), and no linearity is assumed. This definition was introduced by \textit{P. Šemrl} [Proc. Am. Math. Soc. 125, No. 9, 2677-2680 (1997; Zbl 0887.47030)]. In the paper under review, the authors give the definition of 2-local superderivation (respectively, 2-local superautomorphism). Let \(A\) be an associative superalgebra. A mapping \(\delta\colon A\to A\) is called a 2-local superderivation of degree \(i\), \(0\leq i\leq 1\), (respectively, 2-local superautomorphism), if for every \(a,b\) in \(A\) there exists a superderivation of degree \(i\) (respectively, superautomorphism) \(\delta_{a,b}\colon A\to A\) depending on \(a\) and \(b\) such that \(\delta(a)=\delta_{a,b}(a)\) and \(\delta(b)=\delta_{a,b}(b)\). The authors prove that every 2-local superderivation on the associative superalgebra \(M_n(\mathbb{C})\), the algebra of all \(n\times n\) complex matrices, is a superderivation.
scientific article; zbMATH DE number 5547507
scientific article