Volume 28, Issue 1
Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability

Commun. Math. Res., 28 (2012), pp. 26-42.

Published online: 2021-05

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• 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.

• Keywords

general linear Lie algebra, solvability, automorphism of Lie algebra.

15A01, 17B40

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@Article{CMR-28-26, author = {Zhengxin and Chen and and 18439 and and Zhengxin Chen and Qiong and Chen and and 18440 and and Qiong Chen}, title = {Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {28}, number = {1}, pages = {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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19067.html} }
TY - JOUR T1 - Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability AU - Chen , Zhengxin AU - Chen , Qiong JO - Communications in Mathematical Research VL - 1 SP - 26 EP - 42 PY - 2021 DA - 2021/05 SN - 28 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19067.html KW - general linear Lie algebra, solvability, automorphism of Lie algebra. AB -

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.

Zhengxin Chen & Qiong Chen. (2021). Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability. Communications in Mathematical Research . 28 (1). 26-42. doi:
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