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http://dx.doi.org/10.4134/BKMS.b200770

STRONG COMMUTATIVITY PRESERVING MAPS OF UPPER TRIANGULAR MATRIX LIE ALGEBRAS OVER A COMMUTATIVE RING  

Chen, Zhengxin (School of Mathematics and Statistic Fujian Normal University)
Zhao, Yu'e (School of Mathematics and Statistic Qingdao University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.4, 2021 , pp. 973-981 More about this Journal
Abstract
Let R be a commutative ring with identity 1, n ≥ 3, and let 𝒯n(R) be the linear Lie algebra of all upper triangular n × n matrices over R. A linear map 𝜑 on 𝒯n(R) is called to be strong commutativity preserving if [𝜑(x), 𝜑(y)] = [x, y] for any x, y ∈ 𝒯n(R). We show that an invertible linear map 𝜑 preserves strong commutativity on 𝒯n(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 𝒯n(R).
Keywords
Upper triangular matrix Lie algebras; strong commutativity preserving maps; extremal inner automorphisms; idempotent scalar multiplications;
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