Invertible linear maps on Borel subalgebras preserving zero Lie products

Let g be a simple Lie algebra of rank 1 over an algebraically closed field of characteristic zero, b a Borel subalgebra of g. An invertible linear map phi on b is said preserving zero Lie products in both directions if for x, y is an element of b, [x, y] = 0 if and only if [phi(x), phi(y)] = 0. In this paper, it is shown that an invertible linear map phi on b preserving zero Lie products in both directions if and only if it is a composition of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism, which extends the main result in [8] from a linear solvable Lie algebra to far more general cases. (C) 2014 Elsevier Inc. All rights reserved.