"Affine sl(2)"의 두 판 사이의 차이
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29번째 줄: | 29번째 줄: | ||
* c_i=a_i^{\vee}=1<br> | * c_i=a_i^{\vee}=1<br> | ||
* a_{ij}<br> | * a_{ij}<br> | ||
+ | * coxeter number<br> | ||
+ | ** 2<br> | ||
* dual Coxeter number<br> | * dual Coxeter number<br> | ||
+ | ** 2<br> | ||
* Weyl vector<br> | * Weyl vector<br> | ||
43번째 줄: | 46번째 줄: | ||
* imaginary roots <br> | * imaginary roots <br> | ||
** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | ** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | ||
+ | ** <math>\delta=\alpha_0+\alpha_1</math><br> | ||
* simple roots<br> | * simple roots<br> | ||
** <br><math>\alpha_0,\alpha_1</math><br> | ** <br><math>\alpha_0,\alpha_1</math><br> | ||
− | * positive | + | * positive roots<br> |
− | * <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math><br> | + | ** <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math><br> |
− | + | ||
+ | |||
2010년 3월 22일 (월) 14:57 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
\(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)
basic quantities
- a_i=1
- c_i=a_i^{\vee}=1
- a_{ij}
- coxeter number
- 2
- 2
- dual Coxeter number
- 2
- 2
- Weyl vector
root systems
- \(\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}\)
- real roots
- \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
- \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
- imaginary roots
- \(\{n\delta|n\in\mathbb{Z},n\neq 0\}\)
- \(\delta=\alpha_0+\alpha_1\)
- simple roots
-
\(\alpha_0,\alpha_1\)
-
- positive roots
- \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)
- \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)
fixing a Cartan subalgebra and its dual
- basis of the Cartan subalgebra H
\(h_0=C-h_1\)
\(h_1\)
\(d=-l_0\) - dual basis for H
\(\omega_0,\omega_1,\delta\)
killing form
- invariant symmetric non-deg bilinear forms
\(<h_i,h_j>=A_{ij}\)
\(<h_0,d>=1\)
\(<h_1,d>=0\)
\(<d,d>=0\)
level k highest weight representation
- integrable highest weight
\(\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1\), \(\lambda_{i}\in\mathbb{N}\) - level
\(k=\lambda_{0}+\lambda_{1}\) - therefore \(\lambda_{0}\in\{0,1,\cdots,k\}\)