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Triple automorphisms of simple Lie algebras

Deng Yin Wang, Xiaoxiang Yu (2011)

Czechoslovak Mathematical Journal

An invertible linear map ϕ on a Lie algebra L is called a triple automorphism of it if ϕ ( [ x , [ y , z ] ] ) = [ ϕ ( x ) , [ ϕ ( y ) , ϕ ( z ) ] ] for x , y , z 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 paper that an invertible linear map ϕ on 𝔭 is a triple automorphism if and only if either ϕ itself is an automorphism of 𝔭 or it is the composition of an automorphism of 𝔭 and an extremal map of order 2 .

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