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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} }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.