全矩阵代数的抛物子代数上具有可导性的非线性映射

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

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

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.