Bulletin of the
Korean Mathematical Society BKMS

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 $\mathcal{T}_n(R)$.

Keywords: Upper triangular matrix Lie algebras, strong commutativity preserving maps, extremal inner automorphisms, idempotent scalar multiplications

MSC numbers: 15A04, 15A27, 15A86

Supported by: This work is supported by the National Natural Science Foundation of China (Grant No. 11871014) and the Natural Science Foundation of Fujian Province (Grant No. 2020J01162).