"Cartan datum"의 두 판 사이의 차이
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| + | Cartan datum <math>(A,P^{\vee},P,\Pi^{\vee},\Pi)</math> | ||
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* <math>A=(a_{ij})_{i,j\in I}</math> GCM | * <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> : dual weight lattice | * <math>P^{\vee}=(\bigoplus_{i\in I}\mathbb{Z}h_{i})\bigoplus(\bigoplus_{j=1}^{\operatorname{corank}(A)}\mathbb{Z}d_{j})</math> : dual weight lattice | ||
| − | * <math>\mathfrak{h}=\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}</math> | + | * <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>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^{\vee}=\{h_{i}|i\in I\}</math> : simple coroots | ||
| − | * <math>\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in \alpha_{i}(h_j}=a_{ji}\}</math> : | + | * <math>\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)}=a_{ji}\}</math> : simple roots |
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| + | fundamental weights | ||
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| + | <math>\{\Lambda_{i}\in\mathfrak{h}^{*}|i\in I, \Lambda_{i}(h_j)}=\delta_{ij},\Lambda_{i}(d_j)=0\}</math> | ||
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| + | <math>Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}</math> : root lattice | ||
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| + | Weyl group <math>W=\langle r_{i}|i\in I\rangle</math> | ||
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| + | 캐츠-무디 대수 | ||
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| + | <h5>세르 관계식</h5> | ||
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| + | * 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<br> | ||
| + | ** <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>) | ||
2012년 7월 23일 (월) 08:24 판
Cartan datum \((A,P^{\vee},P,\Pi^{\vee},\Pi)\)
- \(A=(a_{ij})_{i,j\in I}\) 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
- \(\mathfrak{h}=\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}\) : Cartan subalgebra
- \(P=\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}\) : weight lattice
- \(\Pi^{\vee}=\{h_{i}|i\in I\}\) : simple coroots
- \(\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)}=a_{ji}\}\) : simple roots
fundamental weights
\(\{\Lambda_{i}\in\mathfrak{h}^{*}|i\in I, \Lambda_{i}(h_j)}=\delta_{ij},\Lambda_{i}(d_j)=0\}\)
\(Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}\) : root lattice
Weyl group \(W=\langle r_{i}|i\in I\rangle\)
캐츠-무디 대수
세르 관계식
- l : 리대수 \(\mathfrak{g}\)의 rank
- \((a_{ij})\) : 카르탄 행렬
- 생성원 \(e_i,h_i,f_i , (i=1,2,\cdots, l)\)
- 세르 관계식
- \(\left[h_i,h_j\right]=0\)
- \(\left[e_i,f_j\right]=\delta _{i,j}h_i\)
- \(\left[h_i,e_j\right]=a_{i,j}e_j\)
- \(\left[h_i,f_j\right]=-a_{i,j}f_j\)
- \(\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0\) (\(i\neq j\))
- \(\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0\) (\(i\neq j\))