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Quasi-Automorphisms of Lie Algebras

An invertible linear transformation ϕ on a Lie algebra L is called a quasi-automorphism of it if there exists an invertible linear transformation on L such that 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.