Maps on upper triangular matrices preserving Lie products

Abstract

Abstract Let be an arbitrary field with characteristic zero, let Tn be the Lie algebra of all n × n upper triangular matrices over with the Lie product [A,B] = A B − B A, and let a bijective map φ : Tn → Tn satisfy φ ([A,B]) = [φ (A) , φ (B)], A,B ∊ Tn . Then there exist an invertible matrix T ∊ Tn , a function satisfying ϕ (C) =0 for every strictly upper triangular matrix C ∊ Tn , and an automorphism f of the field , such that for all [aij ]∊ Tn , or for all [aij ] ∊ Tn , where . Keywords: Non-linear preserver problemLie productUpper triangular matricesKeywords: 2000 Mathematics Subject Classifications:15A0417B40 Acknowledgement The author was supported in part by a grant from the Ministry of Higher Education, Science and Technology of Slovenia.