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

**Full text** available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.

Subscribers of Springer need to access the articles on their site, which is http://www.springeronline.com/10587.