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Journal of Lie Theory 20 (2010), No. 4, 709--737
Copyright Heldermann Verlag 2010

The Component Group of the Automorphism Group of a Simple Lie Algebra and the Splitting of the Corresponding Short Exact Sequence

Hasan Gündogan
Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany


\def\g{{\frak g}} \def\C{{\Bbb C}} \def\K{{\Bbb K}} \def\R{{\Bbb R}} \def\Aut{\mathop{\rm Aut}\nolimits} \def\sdir#1{\hbox{$\mathrel\times{\hskip -4.6pt {\vrule height 4.7 pt depth .5 pt}}\hskip 2pt_{#1}$}} Let $\g$ be a simple Lie algebra of finite dimension over $\K \in \left\{\R,\C\right\}$ and $\Aut(\g)$ the finite-dimensional Lie group of its automorphisms. We will calculate the component group $\pi_0(\Aut(\g)) = \Aut(\g)/\Aut(\g)_0$ and the number of its conjugacy classes, and we will show that the corresponding short exact sequence $$ {\bf1}\to\Aut(\g)_0\to\Aut(\g)\to\pi_0(\Aut(\g))\to{\bf1} $$ is split or, equivalently, there is an isomorphism $\Aut(\g)\cong \Aut(\g)_0 \sdir{}\pi_0(\Aut(\g))$. Indeed, since $\Aut(\g)_0$ is open in $\Aut(\g)$, the quotient group $\pi_0(\Aut(\g))$ is discrete. Hence a section $\pi_0(\Aut(\g))\to\Aut(\g)$ is automatically continuous, giving rise to an isomorphism of Lie groups $\Aut(\g)\cong\Aut(\g)_0 \sdir{}\pi_0(\Aut(\g))$.

Keywords: Automorphism group, simple, semisimple, Lie algebras splitting, semidirect product.

MSC: 17B20, 22E15

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