Nonlinear Maps Satisfying Derivability on the Parabolic Subalgebras of the Full Matrix Algebras

Abstract:

      Let ${\mathbb{F}}$ be a field of characteristic $0$, $M_n({\mathbb{F}})$ the full matrix algebra over ${\mathbb{F}}$, ${\bf t}$ the subalgebra of $M_n({\mathbb{F}})$ consisting of all upper triangular matrices. Any subalgebra of $M_n({\mathbb{F}})$ containing ${\bf t}$ is called a parabolic subalgebra of $M_n({\mathbb{F}})$. Let ${\bf P}$ be a parabolic subalgebra of $M_n({\mathbb{F}})$. A map $\varphi$ on ${\bf P}$ is said to satisfy derivability if $\varphi (x\cdot y)=\varphi (x)\cdot y x\cdot \varphi(y)$ for all $x,y\in {\bf P}$, where $\varphi$ is not necessarily linear. Note that a map satisfying derivability on ${\bf P}$ is not necessarily a derivation on ${\bf P}$. In this paper, we prove that a map $\varphi$ on ${\bf P}$ satisfies derivability if and only if $\varphi$ is a sum of an inner derivation and an additive quasi-derivation on ${\bf P}$. In particular, any derivation of parabolic subalgebras of $M_n({\mathbb{F}})$ is an inner derivation.