Jump to ContentJump to Main Navigation
Show Summary Details
More options …

# 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

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
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

## Abstract

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

MSC 2010: 17B20; 17B40

## References

• [1]

Chuah M. K., Finite order automorphisms on real simple Lie algebras, Trans. Amer. Math. Soc. 364 (2012), 3715–3749. Google Scholar

• [2]

Gündoğan H., The component group of the automorphism group of a simple Lie algebra and the splitting of the corresponding short exact sequence, J. Lie Theory 20 (2010), 709–737. Google Scholar

• [3]

Heintze E. and Groß C., Finite order automorphisms and real forms of affine Kac–Moody algebras in the smooth and algebraic category, Mem. Amer. Math. Soc. 219 (2012), no. 1030. Google Scholar

• [4]

Helgason S., Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math. 34, American Mathematical Society, Providence, 2001. Google Scholar

• [5]

Knapp A., Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002. Google Scholar

• [6]

Murakami S., On the automorphisms of a real semi-simple Lie algebra, J. Math. Soc. Japan 4 (1952), 103–133. Google Scholar

• [7]

Onishchik A. L. and Vinberg E. B., Lie Groups and Algebraic Groups, Springer, Berlin, 1990. Google Scholar

## 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,

Export Citation

© 2017 by De Gruyter.

## Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Mauro Patraõ
Israel Journal of Mathematics, 2019
[2]
M. Aa. Solberg
Journal of High Energy Physics, 2018, Volume 2018, Number 5

## Comments (0)

Please log in or register to comment.
Log in