"Affine sl(2)"의 두 판 사이의 차이
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| 7번째 줄: | 7번째 줄: | ||
<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> | <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> | ||
| + | * <br> | ||
| + | * this is borrowed from [[affine Kac-Moody algebra]] entry | ||
* 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> | ||
| − | * say <math>\mathfrak{g}= | + | * say <math>\mathfrak{g}=A_1</math>, <math>\Phi=\{\alpha\}</math> |
| − | * Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 | + | * Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br> |
| − | * Find the highest root | + | * Find the highest root <br> |
| − | ** <math>\ | + | ** <math>\alpha</math><br> |
* Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math><br> | * Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math><br> | ||
| − | ** <math>\alpha_0=-\ | + | ** <math>\alpha_0=-\alpha</math><br> |
| − | * Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & - | + | * Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}</math><br> |
| − | * Note that this matrix has rank | + | * Note that this matrix has rank 1 since <math>(1,1)</math> belongs to the null space<br> |
| + | * construct a Lie algebra from the new Cartan matrix <math>A'</math><br> | ||
| + | * <br> Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix<br><math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}</math><br> | ||
| 23번째 줄: | 27번째 줄: | ||
<h5>level k highest weight representation</h5> | <h5>level k highest weight representation</h5> | ||
| − | * integrable highest weight<br><math>\lambda=\ | + | * integrable highest weight<br><math>\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1</math>, <math>\lambda_{i}\in\mathbb{N}</math><br> |
| − | * | + | * level<br><math>k=a_{0}^{\vee}\lambda_{0}+a_{1}^{\vee}\lambda_{1}</math><br> |
2010년 3월 5일 (금) 03:39 판
Gannon 190p, 193p, 196p,371p
construction
-
- this is borrowed from affine Kac-Moody algebra entry
- Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
- say \(\mathfrak{g}=A_1\), \(\Phi=\{\alpha\}\)
- Cartan matrix
\(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\) - Find the highest root
- \(\alpha\)
- \(\alpha\)
- Add another simple root \(\alpha_0\) to the root system \(\Phi\)
- \(\alpha_0=-\alpha\)
- \(\alpha_0=-\alpha\)
- Construct a new Cartan matrix
\(A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}\) - Note that this matrix has rank 1 since \((1,1)\) belongs to the null space
- construct a Lie algebra from the new Cartan matrix \(A'\)
-
Add a outer derivation\(d=-l_0\) to compensate the degeneracy of the Cartan matrix
\(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)
level k highest weight representation
- integrable highest weight
\(\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1\), \(\lambda_{i}\in\mathbb{N}\) - level
\(k=a_{0}^{\vee}\lambda_{0}+a_{1}^{\vee}\lambda_{1}\)