"Affine sl(2)"의 두 판 사이의 차이

2010년 3월 4일 (목) 12:22 판

Gannon 190p, 196p,371p

Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
say \(\mathfrak{g}=A_2\), \(\Phi=\{\alpha_1,\alpha_2\}\)
Cartan matrix
\(\mathbf{A} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}\)
Find the highest root \(\sum a_l\alpha_l\)
- \(\alpha_1+\alpha_2\)
Add another simple root \(\alpha_0\) to the root system \(\Phi\)
- \(\alpha_0=-\alpha_1-\alpha_2\)
Construct a new Cartan matrix
\(A' = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}\)
Note that this matrix has rank 2 since \((1,1,1)\) belongs to the null space

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-Gannon 190p, 196p
+Gannon 190p, 196p,371p
+<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">construction</h5>
+* Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math><\cdot,\cdot></math>
+* say <math>\mathfrak{g}=A_2</math>,  <math>\Phi=\{\alpha_1,\alpha_2\}</math>
+*  Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 & -1 \\ -1 & 2  \end{pmatrix}</math><br>
+*  Find the highest root <math>\sum a_l\alpha_l</math><br>
+** <math>\alpha_1+\alpha_2</math><br>
+*  Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math><br>
+** <math>\alpha_0=-\alpha_1-\alpha_2</math><br>
+*  Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2  & -1 \\ -1 & -1 & 2 \end{pmatrix}</math><br>
+*  Note that this matrix has rank 2 since <math>(1,1,1)</math> belongs to the null space<br>