"Symmetrizable generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (다른 사용자 한 명의 중간 판 4개는 보이지 않습니다) | |||
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* [[Generalized Cartan matrix]] | * [[Generalized Cartan matrix]] | ||
* symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | ||
| + | ;def | ||
| + | A [[generalized Cartan matrix]] <math>A</math> is symmetrisable if there exists a non-singular diagonal matrix <math>D</math> and a symmetric matrix <math>B</math> such that <math>A=DB</math>. | ||
| + | |||
| + | ==memo== | ||
| + | * from https://www.sharelatex.com/project/55caaef83e9789d92821b3e8 | ||
| + | * Let <math>\mathfrak g</math> be a simple Lie algebra of rank <math>\ell</math> | ||
| + | * <math>C</math> Cartan matrix | ||
| + | * Let <math>\langle \cdot,\cdot \rangle</math> be the invariant inner product on <math>\mathfrak g</math>, normalized as in \cite{Kac}, so that the square of length of the maximal root equals <math>2</math> with respect to the induced inner product on the dual space to the Cartan subalgebra <math>\mathfrak h</math> of <math>\mathfrak g</math> | ||
| + | * Let <math>r^\vee</math> be the maximal number of edges connecting two vertices of the Dynkin diagram of <math>\mathfrak g</math>. Thus, <math>r^\vee=1</math> for simply-laced <math>\mathfrak g</math>, <math>r^\vee=2</math> for <math>B_\ell, C_\ell, F_4, G_2</math>, and <math>r^\vee=3</math> for <math>D_4</math>. | ||
| + | * From now on we will use the inner product | ||
| + | :<math> | ||
| + | (\cdot,\cdot) = r^\vee \langle \cdot,\cdot \rangle | ||
| + | </math> | ||
| + | on <math>\mathfrak h^*</math> | ||
| + | * <math>D=\operatorname{diag}(d_1,\cdots, d_\ell)</math> such that <math>B:=D C</math> is symmetric | ||
| + | * Let <math>B = (B_{ij})_{1\leq i,j\leq \ell}</math> be the symmetric matrix | ||
| + | :<math> | ||
| + | B = D C, | ||
| + | </math> | ||
| + | i.e., | ||
| + | :<math> | ||
| + | B_{ij} = (\alpha_i,\alpha_j) = r^\vee \langle \alpha_i,\alpha_j \rangle. | ||
| + | </math> | ||
==example== | ==example== | ||
| − | * Cartan matrix of | + | * Cartan matrix of <math>G_2</math> |
| − | + | :<math> | |
A=\left( | A=\left( | ||
\begin{array}{cc} | \begin{array}{cc} | ||
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\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
| − | * take | + | * take <math>D</math> as follows : |
| − | + | :<math> | |
D=\left( | D=\left( | ||
\begin{array}{cc} | \begin{array}{cc} | ||
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\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
| − | * Then | + | * Then <math>DA=A^{t}D</math> is a symmetric matrix |
| − | + | :<math> | |
\left( | \left( | ||
\begin{array}{cc} | \begin{array}{cc} | ||
| 31번째 줄: | 54번째 줄: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
| 41번째 줄: | 64번째 줄: | ||
==computational resource== | ==computational resource== | ||
* https://docs.google.com/file/d/0B8XXo8Tve1cxdlBHdXA5THp3SFE/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxdlBHdXA5THp3SFE/edit | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 05:34 기준 최신판
introduction
- Generalized Cartan matrix
- symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms
- def
A generalized Cartan matrix \(A\) is symmetrisable if there exists a non-singular diagonal matrix \(D\) and a symmetric matrix \(B\) such that \(A=DB\).
memo
- from https://www.sharelatex.com/project/55caaef83e9789d92821b3e8
- Let \(\mathfrak g\) be a simple Lie algebra of rank \(\ell\)
- \(C\) Cartan matrix
- Let \(\langle \cdot,\cdot \rangle\) be the invariant inner product on \(\mathfrak g\), normalized as in \cite{Kac}, so that the square of length of the maximal root equals \(2\) with respect to the induced inner product on the dual space to the Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\)
- Let \(r^\vee\) be the maximal number of edges connecting two vertices of the Dynkin diagram of \(\mathfrak g\). Thus, \(r^\vee=1\) for simply-laced \(\mathfrak g\), \(r^\vee=2\) for \(B_\ell, C_\ell, F_4, G_2\), and \(r^\vee=3\) for \(D_4\).
- From now on we will use the inner product
\[ (\cdot,\cdot) = r^\vee \langle \cdot,\cdot \rangle \] on \(\mathfrak h^*\)
- \(D=\operatorname{diag}(d_1,\cdots, d_\ell)\) such that \(B:=D C\) is symmetric
- Let \(B = (B_{ij})_{1\leq i,j\leq \ell}\) be the symmetric matrix
\[ B = D C, \] i.e., \[ B_{ij} = (\alpha_i,\alpha_j) = r^\vee \langle \alpha_i,\alpha_j \rangle. \]
example
- Cartan matrix of \(G_2\)
\[ A=\left( \begin{array}{cc} 2 & -1 \\ -3 & 2 \\ \end{array} \right) \]
- take \(D\) as follows :
\[ D=\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \\ \end{array} \right) \]
- Then \(DA=A^{t}D\) is a symmetric matrix
\[ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) \]