Cartan datum

Cartan datum <math>(A,P^{\vee},P,\Pi^{\vee},\Pi)</math>

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

fundamental weights

<math>\{\Lambda_{i}\in\mathfrak{h}^{*}|i\in I, \Lambda_{i}(h_j)=\delta_{ij},\Lambda_{i}(d_j)=0\}</math>

<math>Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}</math> : root lattice

Weyl group <math>W=\langle r_{i}|i\in I\rangle</math>

세르 관계식

• l : 리대수 <math>\mathfrak{g}</math>의 rank
• <math>(a_{ij})</math> : 카르탄 행렬
• 생성원 <math>e_i,h_i,f_i , (i=1,2,\cdots, l)</math>
• 세르 관계식
  • <math>\left[h_i,h_j\right]=0</math>
  • <math>\left[e_i,f_j\right]=\delta _{i,j}h_i</math>
  • <math>\left[h_i,e_j\right]=a_{i,j}e_j</math>
  • <math>\left[h_i,f_j\right]=-a_{i,j}f_j</math>
  • <math>\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0</math> (<math>i\neq j</math>)
  • <math>\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0</math> (<math>i\neq j</math>)