Nonlinear maps preserving solvability
type: Item
instance of: scholarly article
title: Nonlinear maps preserving solvability
zbMATH Open document ID: 1068.17007
author: H. Radjavi
author: Peter Šemrl
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: It is shown that a bijective nonlinear map \(\phi:{\mathcal L}\to{\mathcal L}\) preserves solvability in both directions if \(\phi\) maps every Lie subalgebra of a Lie algebra \({\mathcal L}\) into some solvable Lie subalgebra. Here \({\mathcal L}\) is the Lie algebra \(\text{gl}(n,\mathbb{C})\) or the Lie algebra \(\text{sl}(n,\mathbb C)\), which both are associated with the algebra \(M_n\) of \(n\times n\)-matrices \(A= [a_{ij}]\) defined in the complex space \(\mathbb C\). It is proved that if \(n\geq 3\) then each such a map is either a composition of a bijective lattice preserving map with a similarity transformation and a map \([a_{ij}]\mapsto[f(a_{ij})]\) induced by a field automorphism \(f: \mathbb C\to \mathbb C\), or a map of the same type composed with the transposition. The general form of such maps when \(n= 2\) is also given. With the help of Lie's theorem, the proof of the statement under consideration is reduced to characterizing bijective maps on \(M_n\) preserving triangularizability of matrix pairs in both directions. The bijective maps on \(M_n\) that preserve inclusion for lattices of invariant subspaces in both directions are also considered.
reviewed by: A. A. Bogush
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 2136739
zbMATH Keywords: bijective nonlinear map
zbMATH Keywords: solvability
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lie subalgebra
zbMATH Keywords: matrix algebra
zbMATH Keywords: bijective lattice
zbMATH Keywords: triangularizability
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.06.016
OpenAlex ID: W2094526033
cites work: Non-Linear Spectrum-Preserving Maps
cites work: Q3748420
cites work: Geometries of Matrices. I 1 Arithmetical Construction
cites work: Q5843659
cites work: Geometries of Matrices. II. Study of Involutions in the Geometry of Symmetric Matrices
cites work: Geometries of Matrices. III. Fundamental Theorems in the Geometries of Symmetric Matrices
cites work: Geometry of symmetric matrices over any field with characteristic other than two
cites work: Linear maps preserving the isomorphism class of lattices of invariant subspaces
cites work: Linear maps that preserve the commutant, double commutant or the lattice of invariant subspaces
cites work: Automorphisms of the Field of Complex Numbers
cites work: Linear preserver problems: A brief introduction and some special techniques
cites work: Adjacency preserving maps on matrices and operators
cites work: Simultaneous triangularization
cites work: A short proof of Hua's fundamental theorem of the geometry of Hermitian matrices.
cites work: Q5609479
cites work: On Hua's fundamental theorem of the geometry of rectangular matrices
cites work: Hua's fundamental theorem of the geometry of matrices.
cites work: Q5478096
cites work: Q4092109
DOI: 10.1016/J.JALGEBRA.2004.06.016
description: scientific article; zbMATH DE number 2136739