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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Invertible linear maps on simple Lie algebras preserving commutativity
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by Dengyin Wang and Zhengxin Chen PDF
Proc. Amer. Math. Soc. 139 (2011), 3881-3893 Request permission

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
  • Email: wdengyin@126.com
  • Zhengxin Chen
  • Affiliation: School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, People’s Republic of China
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3881-3893
  • MSC (2010): Primary 17B20, 17B40
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10834-7
  • MathSciNet review: 2823034