Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1007-1016, 2011
Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1007-1016, 2011

Triple automorphisms of simple Lie algebras

Dengyin Wang, Xiaoxiang Yu

Dengyin Wang, Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, P. R. China, e-mail: wdengyin@126.com; Xiaoxiang Yu, School of Mathematical Sciences, Xuzhou Normal University, Xuzhou 221116, P. R. China

Abstract: An invertible linear map $\varphi$ on a Lie algebra $L$ is called a triple automorphism of it if $\varphi([x,[y,z]])=[\varphi(x),[ \varphi(y),\varphi(z)]]$ for $\forall x, y, z\in L$. Let $\frak{g}$ 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 $\varphi$ on $\mathfrak{p}$ is a triple automorphism if and only if either $\varphi$ itself is an automorphism of $\mathfrak{p}$ or it is the composition of an automorphism of $\mathfrak{p}$ and an extremal map of order $2$.

Keywords: simple Lie algebras, parabolic subalgebras, triple automorphisms of Lie algebras

Classification (MSC 2010): 17B20, 17B30, 17B40


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