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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.

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