Maps on upper triangular matrices preserving Lie products

Abstract

Let be an arbitrary field with characteristic zero, let T_n 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 : T_n → T_n satisfy φ ([A,B]) = [φ (A) , φ (B)], A,B ∊ T_n . Then there exist an invertible matrix T ∊ T_n , a function satisfying ϕ (C) =0 for every strictly upper triangular matrix C ∊ T_n , and an automorphism f of the field , such that for all [a_ij ]∊ T_n , or for all [a_ij ] ∊ T_n , where .

Keywords:

• Non-linear preserver problem
• Lie product
• Upper triangular matrices

Keywords:

• 2000 Mathematics Subject Classifications:15A04
• 17B40

Acknowledgement

The author was supported in part by a grant from the Ministry of Higher Education, Science and Technology of Slovenia.