Non-linear maps preserving solvability

Abstract

Let M n be the algebra of all n × n complex matrices and let L be the general linear Lie algebra gl ( n , C ) or the special linear Lie algebra sl ( n , C ) . A bijective (not necessarily linear) map ϕ : L → L preserves solvability in both directions if both ϕ and ϕ − 1 map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n ⩾ 3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [ a i j ] ↦ [ f ( a i j ) ] induced by a field automorphism f : C → C , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n = 2 . Using Lie's theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on M n preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on M n that preserve inclusion for lattices of invariant subspaces in both directions.