"Quantized universal enveloping algebra"의 두 판 사이의 차이

2012년 7월 27일 (금) 16:16 판

Cartan datum \((A,P^{\vee},P,\Pi^{\vee},\Pi)\)
\(A=(a_{ij})_{i,j\in I}\) symmetrizable GCM
- \(D=\operatorname{diag}(s_i\in\mathbb{Z}_{\geq 0})_{i \in I}\) diagonal matrix s.t. DA is symmetric
\(P^{\vee}=(\bigoplus_{i\in I}\mathbb{Z}h_{i})\bigoplus(\bigoplus_{j=1}^{\operatorname{corank}(A)}\mathbb{Z}d_{j})\) : 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

@@ 14번째 줄: / 14번째 줄: @@
 * Cartan datum <math>(A,P^{\vee},P,\Pi^{\vee},\Pi)</math>
-* <math>A=(a_{ij})_{i,j\in I}</math> symmetrizable GCM
+* <math>A=(a_{ij})_{i,j\in I}</math> symmetrizable GCM<br>
+** <math>D=\operatorname{diag}(s_i\in\mathbb{Z}_{\geq 0})_{i \in I}</math> diagonal matrix s.t. DA is symmetric
 * <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> : Cartan subalgebra
@@ 25번째 줄: / 26번째 줄: @@
-* <math>(\cdot|\cdot)</math> symmetric bilinear form on <math>\mathfrak{g}</math>
+* <math>(\cdot|\cdot)</math> symmetric bilinear form on <math>\mathfrak{g}^{*}</math>
+* <math>s_{i}=\frac{(\alpha_{i}|\alpha_{i})}{2}\in \mathbb{Z}_{>0}</math>
+* <math>q_i=q^{s_{i}}</math>
+* [[#]]