Cartan datum

• \(A=(a_{ij})_{i,j\in I}\) GCM
• \(P^{\vee}=(\bigoplus_{i\in I}\mathbb{Z}h_{i})\bigoplus(\bigoplus_{j=1}^{\operatorname{corank}(A)}\mathbb{Z}d_{j})\) : dual weight lattice
• \(\mathfrak{h}=\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}\)
• \(P=\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}\) : weight lattice
• \(\Pi^{\vee}=\{h_{i}|i\in I\}\) : simple coroots
• \(\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in \alpha_{i}(h_j}=a_{ji}\}\) : simple coroots