Triple automorphisms of simple Lie algebras

Wang, Dengyin ; Yu, Xiaoxiang

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1007-1016.

Résumé

An invertible linear map $\phi$ on a Lie algebra $L$ is called a triple automorphism of it if $\phi ([x,[y,z]])=[\phi (x),[\phi (y),\phi (z)]]$ for $\mathrm{\forall }x,y,z\in L$. Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $l$ defined over an algebraically closed field $F$ of characteristic zero, $\mathfrak{p}$ an arbitrary parabolic subalgebra of $\mathfrak{g}$. It is shown in this paper that an invertible linear map $\phi$ on $\mathfrak{p}$ is a triple automorphism if and only if either $\phi$ itself is an automorphism of $\mathfrak{p}$ or it is the composition of an automorphism of $\mathfrak{p}$ and an extremal map of order $2$.

Détail

DOI : 10.1007/s10587-011-0043-9

Classification : 17B20, 17B30, 17B40
Mots-clés : simple Lie algebras; parabolic subalgebras; triple automorphisms of Lie algebras