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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 29, Issue 3


Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams

Meng-Kiat Chuah / Mingjing Zhang
Published Online: 2016-07-30 | DOI: https://doi.org/10.1515/forum-2016-0023


Let 𝔤 be a simple Lie algebra. Let Aut(𝔤) be the group of all automorphisms on 𝔤, and let Int(𝔤) be its identity component. The outer automorphism group of 𝔤 is defined as Aut(𝔤)/Int(𝔤). If 𝔤 is complex and has Dynkin diagram D, then Aut(𝔤)/Int(𝔤) is isomorphic to Aut(D). We provide an analogous result for the real case. For 𝔤 real, we let 𝔤 be represented by a painted diagram P. Depending on whether the Cartan involution of 𝔤 belongs to Int(𝔤), we show that Aut(𝔤)/Int(𝔤) is isomorphic to Aut(P) or Aut(P)×2. This result extends to the outer automorphism groups of all real semisimple Lie algebras.

Keywords: Outer automorphism group; simple Lie algebra; Dynkin diagram; painted diagram

MSC 2010: 17B20; 17B40


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About the article

Received: 2016-01-27

Revised: 2016-04-28

Published Online: 2016-07-30

Published in Print: 2017-05-01

This work is partially supported by a research grant from the Ministry of Science and Technology of Taiwan.

Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 555–562, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0023.

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