Nonlinear maps preserving solvability
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.
scientific article; zbMATH DE number 2136739