Quasi-automorphisms of lie algebras

摘要

An invertible linear transformation φ on a Lie algebra L is called a quasiautomorphism of it if there exists an invertible linear transformation φ? on L such that [φ(x), φ(y)] = φ?([x, y]) for ? x, y ∈ L. The group of all quasi-automorphisms of L is denoted by Q Aut(L). Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this article that, if l = 1, then Q Aut(p) = GL(p); otherwise, Q Aut(p) = Aut(p) × F*I_p, where F*I_p denotes the group of all non-zero scalar multiplication maps on p.