"Affine sl(2)"의 두 판 사이의 차이
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| 5번째 줄: | 5번째 줄: | ||
| − | <h5 style=" | + | <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; 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> | * Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math><\cdot,\cdot></math> | ||
| 16번째 줄: | 16번째 줄: | ||
* Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}</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> | * Note that this matrix has rank 2 since <math>(1,1,1)</math> belongs to the null space<br> | ||
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| + | <h5>level k highest weight representation</h5> | ||
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| + | * integrable highest weight<br><math>\lambda=\sum_{i=0}^{r}\lambda_{i}\omega_i</math>, <math>\lambda_{i}\in\mathbb{N}</math><br> | ||
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2010년 3월 5일 (금) 04:33 판
Gannon 190p, 193p, 196p,371p
construction
- 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 \(\sum a_l\alpha_l\)
- \(\alpha_1+\alpha_2\)
- \(\alpha_1+\alpha_2\)
- Add another simple root \(\alpha_0\) to the root system \(\Phi\)
- \(\alpha_0=-\alpha_1-\alpha_2\)
- \(\alpha_0=-\alpha_1-\alpha_2\)
- 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
level k highest weight representation
- integrable highest weight
\(\lambda=\sum_{i=0}^{r}\lambda_{i}\omega_i\), \(\lambda_{i}\in\mathbb{N}\)