Invertible linear maps on Borel subalgebras preserving zero Lie products

摘要

Let g be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, b a Borel subalgebra of g. An invertible linear map φ on b is said preserving zero Lie products in both directions if for x,y ? b, [x, y] = 0 if and only if [φ(x), φ(y)] = 0. In this paper, it is shown that an invertible linear map φ 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.