Strong commutativity preserving maps of upper triangular matrix Lie algebras over a commutative ring
Bull. Korean Math. Soc. 2021 Vol. 58, No. 4, 973-981
Published online June 28, 2021
Printed July 31, 2021
Zhengxin Chen, Yu'e Zhao
Fujian Normal University; Qingdao University
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 $\mathcal{T}_n(R)$.
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).
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