Root system of affine Kac-Moody algebra

introduction

\(\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n \neq 0\}\)
real roots
- \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
- multiplicity 1 [Carter Cor14 .16 and Prop16 .18]
- roots coming from the simple Lie algebra
imaginary roots
- \(\{n\delta|n\in\mathbb{Z},n \neq 0\}\)
- has norm zero i.e. \(\delta^2=0\)
- multiplicity is not always 1 but equal to the rank of the simple Lie algebra
simple roots
- \(\alpha_0,\alpha_1,\cdots,\alpha_r\)
positive roots

\[\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\]