Symmetrizable generalized Cartan matrix - 편집 역사

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imported>Pythagoras0: /* memo */

2015-10-14T21:55:04Z

memo

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imported>Pythagoras0: 새 문서: ==introduction * Generalized Cartan matrix * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms ==example== * Cartan ma...

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새 문서: ==introduction * Generalized Cartan matrix * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms ==example== * Cartan ma...

새 문서

==introduction
* [[Generalized Cartan matrix]]
* symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms

==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
$$
\left(
\begin{array}{cc}
6 & -3 \\
-3 & 2 \\
\end{array}
\right)
$$

==related items==
* [[Killing form and invariant symmetric bilinear form]]

@@ 3번째 줄: / 3번째 줄: @@
 * 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 $A=DB$.
+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 $\mathfrak g$ be a simple Lie algebra of rank $\ell$
+* Let <math>\mathfrak g</math> be a simple Lie algebra of rank <math>\ell</math>
-* $C$ Cartan matrix
+* <math>C</math> 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 <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 $r^\vee$ 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 $D_4$.
+* 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 $\mathfrak h^*$
+on <math>\mathfrak h^*</math>
-* $D=\operatorname{diag}(d_1,\cdots, d_\ell)$ such that $B:=D C$ is symmetric
+* <math>D=\operatorname{diag}(d_1,\cdots, d_\ell)</math> such that <math>B:=D C</math> is symmetric
-* Let $B = (B_{ij})_{1\leq i,j\leq \ell}$ be the symmetric matrix
+* 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==
-* Cartan matrix of $G_2$
+* Cartan matrix of <math>G_2</math>
-$$
+:<math>
 A=\left(
 \begin{array}{cc}
@@ 36번째 줄: / 36번째 줄: @@
 \end{array}
 \right)
-$$
+</math>
-* take $D$ as follows :
+* take <math>D</math> as follows :
-$$
+:<math>
 D=\left(
 \begin{array}{cc}
@@ 45번째 줄: / 45번째 줄: @@
 \end{array}
 \right)
-$$
+</math>
-* Then $DA=A^{t}D$ is a symmetric matrix
+* Then <math>DA=A^{t}D</math> is a symmetric matrix
-$$
+:<math>
 \left(
 \begin{array}{cc}
@@ 54번째 줄: / 54번째 줄: @@
 \end{array}
 \right)
-$$
+</math>

← 이전 판		2020년 11월 14일 (토) 04:16 판
64번째 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]]

@@ 8번째 줄: / 8번째 줄: @@
 ==memo==
 * from https://www.sharelatex.com/project/55caaef83e9789d92821b3e8
 * $C$ Cartan matrix
-* Let $\langle \cdot,\cdot \rangle$ be the invariant inner product
+* 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 $\mathfrak g$
-on $\g$, normalized as in \cite{Kac}, so that the square of length of
+* 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 $D_4$.
-the maximal root equals $2$ with respect to the induced inner product
+* From now on we will use the inner product
 $$
 (\cdot,\cdot) = r^\vee \langle \cdot,\cdot \rangle
 $$
-on $\h^*$
+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
@@ 30번째 줄: / 26번째 줄: @@
 B_{ij} = (\alpha_i,\alpha_j) = r^\vee \langle \alpha_i,\alpha_j \rangle.
 $$
 ==example==

← 이전 판		2015년 10월 14일 (수) 21:52 판
4번째 줄:		4번째 줄:
	;def		;def
	A [[generalized Cartan matrix]] $A$ is symmetrisable if there exists a non-singular diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$.		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
		+	* $C$ Cartan matrix
		+	* Let $\langle \cdot,\cdot \rangle$ be the invariant inner product
		+	on $\g$, 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 $\h$ of $\g$
		+	* Let $r^\vee$ be the maximal number of edges connecting two vertices of
		+	the Dynkin diagram of $\g$. Thus, $r^\vee=1$ for simply-laced $\g$,
		+	$r^\vee=2$ for $B_\el, C_\el, F_4, G_2$, 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 $\h^*$
		+	* $D=\operatorname{diag}(d_1,\cdots, d_\ell)$ 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.,
		+	$$
		+	B_{ij} = (\alpha_i,\alpha_j) = r^\vee \langle \alpha_i,\alpha_j \rangle.
		+	$$

← 이전 판		2015년 4월 2일 (목) 07:06 판
2번째 줄:		2번째 줄:
	* [[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]] $A$ is symmetrisable if there exists a non-singular diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$.

← 이전 판		2015년 4월 2일 (목) 04:03 판
36번째 줄:		36번째 줄:
	==related items==		==related items==
	* [[Killing form and invariant symmetric bilinear form]]		* [[Killing form and invariant symmetric bilinear form]]
		+
		+
		+
		+	==computational resource==
		+	* https://docs.google.com/file/d/0B8XXo8Tve1cxdlBHdXA5THp3SFE/edit

@@ 1번째 줄: / 1번째 줄: @@
-==introduction
+==introduction==
 * [[Generalized Cartan matrix]]
 * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms