Abstract: Let g be a complex simple Lie algebra of rank I, b the standard Borel subalgebra. An invertible map on b is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In this article, by using some results of Chevalley groups, the theory of root systems and root space decomposition, the author gives an explicit description on such maps of b.

Key words: simple Lie algebra, abelian ideal, non-linear map