Root system of affine Kac-Moody algebra

introduction

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

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