Year: 2012
Author: Zhengxin Chen, Qiong Chen
Communications in Mathematical Research , Vol. 28 (2012), Iss. 1 : pp. 26–42
Abstract
Let $M_n$ be the algebra of all $n × n$ complex matrices and $gl(n, \mathbb{C})$ be the general linear Lie algebra, where $n ≥ 2$. An invertible linear map $ϕ: gl(n, \mathbb{C}) → gl(n, \mathbb{C})$ preserves solvability in both directions if both $ϕ$ and $ϕ^{−1}$ map every solvable Lie subalgebra of $gl(n, \mathbb{C})$ to some solvable Lie subalgebra. In this paper we classify the invertible linear maps preserving solvability on $gl(n, \mathbb{C})$ in both directions. As a sequence, such maps coincide with the invertible linear maps preserving commutativity on $M_n$ in both directions.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2012-CMR-19067
Communications in Mathematical Research , Vol. 28 (2012), Iss. 1 : pp. 26–42
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: general linear Lie algebra solvability automorphism of Lie algebra.