Strong commutativity preserving maps of upper triangular matrix Lie algebras over a commutative ring
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초록
Let $R$ be a commutative ring with identity $1$, $n\geq 3$, and let $\mathcal{T}_n(R)$ be the linear Lie algebra of all upper triangular $n\times n$ matrices over $R$. A linear map $\varphi$ on $\mathcal{T}_n(R)$ is called to be strong commutativity preserving if $[\varphi(x),\varphi(y)]=[x,y]$ for any $x,y\in \mathcal{T}_n(R)$. We show that an invertible linear map $\varphi$ preserves strong commutativity on $\mathcal{T}_n(R)$ if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on $\mathcal{T}_n(R)$.
Abstract
Let $R$ be a commutative ring with identity $1$, $n\geq 3$, and let $\mathcal{T}_n(R)$ be the linear Lie algebra of all upper triangular $n\times n$ matrices over $R$. A linear map $\varphi$ on $\mathcal{T}_n(R)$ is called to be strong commutativity preserving if $[\varphi(x),\varphi(y)]=[x,y]$ for any $x,y\in \mathcal{T}_n(R)$. We show that an invertible linear map $\varphi$ preserves strong commutativity on $\mathcal{T}_n(R)$ if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on $\mathcal{T}_n(R)$.
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