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Invertible linear maps on simple Lie algebras preserving commutativity

Authors: Dengyin Wang and Zhengxin Chen
Journal: Proc. Amer. Math. Soc. 139 (2011), 3881-3893
MSC (2010): Primary 17B20, 17B40
Published electronically: April 7, 2011
MathSciNet review: 2823034
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Abstract: Let $\mathfrak {g}$ be a finite-dimensional simple Lie algebra of rank $l$ over an algebraically closed field of characteristic zero. An invertible linear map $\varphi$ on $\mathfrak {g}$ is called preserving commutativity in both directions if, for any $x, y\in \mathfrak {g}$, $[x,y]=0$ $\Leftrightarrow$ $[\varphi (x),\varphi (y)]=0$. The group of all such maps on $\mathfrak {g}$ is denoted by $Pzp (\mathfrak {g})$. It is shown in this paper that, if $l=1$, then $Pzp(\mathfrak {g})=GL(\mathfrak {g})$; otherwise, $Pzp(\mathfrak {g})=Aut (\mathfrak {g})\times F^*I_{\mathfrak {g}}$, where $F^*I_{\mathfrak {g}}$ denotes the group of all non-zero scalar multiplication maps on $\mathfrak {g}$.

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Additional Information

Dengyin Wang
Affiliation: Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, People’s Republic of China

Zhengxin Chen
Affiliation: School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, People’s Republic of China

Keywords: Simple Lie algebras, maps preserving commutativity, automorphisms of Lie algebras
Received by editor(s): March 7, 2010
Received by editor(s) in revised form: October 1, 2010
Published electronically: April 7, 2011
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.