Generalized Cartan matrix - 편집 역사

2021년 2월 17일 (수) 09:33에 Pythagoras0님의 편집

2021-02-17T09:33:23Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T02:40:13Z

메타데이터: 새 문단

2020년 11월 16일 (월) 12:31에 Pythagoras0님의 편집

2020-11-16T12:31:11Z

2020년 11월 13일 (금) 15:50에 imported>Pythagoras0님의 편집

2020-11-13T15:50:34Z

imported>Pythagoras0: /* classification of generalized Cartan matrix */

2019-07-11T04:56:56Z

classification of generalized Cartan matrix

imported>Pythagoras0: /* generalized Cartan matrix */

2016-12-19T07:41:25Z

generalized Cartan matrix

imported>Pythagoras0: /* related items */

2015-04-02T06:58:44Z

imported>Pythagoras0: /* classification of generalized Cartan matrix */

2015-04-02T04:22:45Z

classification of generalized Cartan matrix

imported>Pythagoras0: /* computational resource */

2015-04-02T04:10:59Z

computational resource

2015년 4월 2일 (목) 04:03에 imported>Pythagoras0님의 편집

2015-04-02T04:03:31Z

@@ 64번째 줄: / 64번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q2004951 Q2004951]

← 이전 판		2020년 12월 28일 (월) 02:40 판
63번째 줄:		63번째 줄:
	[[분류:Lie theory]]		[[분류:Lie theory]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q2004951 Q2004951]

@@ 8번째 줄: / 8번째 줄: @@
 ** For non-diagonal entries, <math>a_{ij} \in {0,-1,-2,-3}</math>
 ** If <math>a_{ij} = -2\text{ or }-3</math> then <math>a_{ji} = 0</math>
-** $a_{ij} = 0$ if and only if $a_{ji} = 0$
+** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
 ==generalized Cartan matrix==
-* A generalized Cartan matrix is a square matrix $A = (a_{ij})$ with integer entries such that
+* A generalized Cartan matrix is a square matrix <math>A = (a_{ij})</math> with integer entries such that
-** For diagonal entries, $a_{ii} = 2$.
+** For diagonal entries, <math>a_{ii} = 2</math>.
-** For non-diagonal entries, $a_{ij} \leq 0 $.
+** For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
-** $a_{ij} = 0$ if and only if $a_{ji} = 0$
+** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
-* an $n\times n$ matrix $A=(a_{ij})$  is called a generalised Cartan matrix if it satisfies the conditions
+* an <math>n\times n</math> matrix <math>A=(a_{ij})</math>  is called a generalised Cartan matrix if it satisfies the conditions
-# $a_{ii}=2$ for $i=1,\cdots,n$
+# <math>a_{ii}=2</math> for <math>i=1,\cdots,n</math>
-# $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
+# <math>a_{ij}\in \mathbb{Z}</math> and <math>a_{ij}\leq 0</math> if <math>i\neq j</math>
-# $a_{ij}=0$ impies $a_{ji}=0$
+# <math>a_{ij}=0</math> impies <math>a_{ji}=0</math>
 ==classification of generalized Cartan matrix==
 * A GCM is called indecomposable if it is not equivalent to a diagonal sum of two smaller GCMs.
 * A GCM A has finite type if
-** $\text{det }A\neq 0$
+** <math>\text{det }A\neq 0</math>
-** there exists $u>0$ with $Au>0$
+** there exists <math>u>0</math> with <math>Au>0</math>
-** $Au\geq 0$ implies $u>0$ or $u=0$
+** <math>Au\geq 0</math> implies <math>u>0</math> or <math>u=0</math>
 * A GCM A has affine type if
-** $\text{corank }A=1$
+** <math>\text{corank }A=1</math>
-** there exists $u>0$ such that $Au=0$
+** there exists <math>u>0</math> such that <math>Au=0</math>
-** $Au\geq 0$ implies $Au=0$
+** <math>Au\geq 0</math> implies <math>Au=0</math>
 * A GCM A has indefinite type if
-** there exists $u>0$ with $Au<0$
+** there exists <math>u>0</math> with <math>Au<0</math>
-** $Au\geq 0$ and $u\geq 0$ implies $u>0$ or $u=0$
+** <math>Au\geq 0</math> and <math>u\geq 0</math> implies <math>u>0</math> or <math>u=0</math>
 ====main result====
-* Let $A$ be an indecomposable GCM. Then exactly one of the following three possibilities holds:
+* Let <math>A</math> be an indecomposable GCM. Then exactly one of the following three possibilities holds:
-** $A$ has finite type
+** <math>A</math> has finite type
-** $A$ has affine type
+** <math>A</math> has affine type
-** $A$ has indefinite type
+** <math>A</math> has indefinite type
-* Moreover the type of $A^t$ is the same as the type of $A$.
+* Moreover the type of <math>A^t</math> is the same as the type of <math>A</math>.
 ;cor
-Let $A$ be an indecomposable GCM. Then
+Let <math>A</math> be an indecomposable GCM. Then
-# A GCM A has finite type if and only if there exists $u>0$ with $Au>0$
+# A GCM A has finite type if and only if there exists <math>u>0</math> with <math>Au>0</math>
-# A GCM A has affine type if and only if there exists $u>0$ with $Au=0$
+# A GCM A has affine type if and only if there exists <math>u>0</math> with <math>Au=0</math>
-# A GCM A has indefinite type if and only if there exists $u>0$ with $Au<0$
+# A GCM A has indefinite type if and only if there exists <math>u>0</math> with <math>Au<0</math>
 * R.Carter's 'Lie algebras of finite and affine type' 337~344p

← 이전 판		2020년 11월 13일 (금) 15:50 판
62번째 줄:		62번째 줄:

	[[분류:Lie theory]]		[[분류:Lie theory]]
		+	[[분류:migrate]]

@@ 28번째 줄: / 28번째 줄: @@
 ** $Au\geq 0$ implies $u>0$ or $u=0$
 * A GCM A has affine type if
-** $\text{rank }A=1$
+** $\text{corank }A=1$
 ** there exists $u>0$ such that $Au=0$
 ** $Au\geq 0$ implies $Au=0$

@@ 57번째 줄: / 57번째 줄: @@
 * [[Skew-symmetrizable matrix]]
 * [[Killing form and invariant symmetric bilinear form]]
+* [[Symmetrizable generalized Cartan matrix]]
 ==computational resource==

@@ 54번째 줄: / 54번째 줄: @@
 ==computational resource==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxWGFXeWtoYTVyTlk/edit
 [[분류:Lie theory]]

← 이전 판		2015년 4월 2일 (목) 04:03 판
54번째 줄:		54번째 줄:

	==computational resource==		==computational resource==
−	* https://docs.google.com/file/d/0B8XXo8Tve1cxdlBHdXA5THp3SFE/edit


	[[분류:Lie theory]]		[[분류:Lie theory]]