Affine sl(2) - 편집 역사

2020년 12월 28일 (월) 12:24에 Pythagoras0님의 편집

2020-12-28T12:24:03Z

Pythagoras0: /* construction from semisimple Lie algebra */

2020-11-14T19:21:50Z

construction from semisimple Lie algebra

2020년 11월 14일 (토) 19:21에 Pythagoras0님의 편집

2020-11-14T19:21:02Z

차이 보기

2020년 11월 14일 (토) 01:37에 imported>Pythagoras0님의 편집

2020-11-14T01:37:11Z

imported>Pythagoras0: /* articles */

2015-09-22T08:38:47Z

articles

imported>Pythagoras0: /* related items */

2015-07-27T10:59:18Z

imported>Pythagoras0: /* related items */

2015-07-27T10:58:57Z

imported>Pythagoras0: /* characters of irreducible representations */

2015-05-28T08:32:32Z

characters of irreducible representations

imported>Pythagoras0: /* basic representation */

2014-11-23T07:53:19Z

basic representation

imported>Pythagoras0: /* characters of irreducible representations */

2014-10-29T03:30:07Z

characters of irreducible representations

@@ 2번째 줄: / 2번째 줄: @@
 * affine sl(2) <math>A^{(1)}_1</math>
 * {{수학노트|url=Sl(2)의_유한차원_표현론}}
-−
 ==construction from semisimple Lie algebra==
@@ 8번째 줄: / 8번째 줄: @@
 * this is borrowed from [[affine Kac-Moody algebra]]
 * Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math>\langle \cdot,\cdot \rangle</math>
-* say <math>\mathfrak{g}=A_1</math>,  <math>\Phi=\{\alpha,-\alpha\}</math>
+* say <math>\mathfrak{g}=A_1</math>,  <math>\Phi=\{\alpha,-\alpha\}</math>
 *  Cartan matrix<math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math>
 *  Find the highest root  <math>\alpha</math>
-*  Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math> which is <math>\alpha_0=-\alpha</math>, but we regard this as an independent one now.
+*  Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math> which is <math>\alpha_0=-\alpha</math>, but we regard this as an independent one now.
 *  Construct a new Cartan matrix<math>A' = \begin{pmatrix} 2 & -2  \\ -2 & 2  \end{pmatrix}</math>
-*  Note that this matrix has rank 1 since <math>(1,1)</math> belongs to the null space
+*  Note that this matrix has rank 1 since <math>(1,1)</math> belongs to the null space
-*  construct a Lie algebra from the new Cartan matrix <math>A'</math>
+*  construct a Lie algebra from the new Cartan matrix <math>A'</math>
-*  Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix
+*  Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix
 :<math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0  \end{pmatrix}</math>
@@ 27번째 줄: / 27번째 줄: @@
 *  Weyl vector
-−
-−
 ==root systems==
@@ 36번째 줄: / 36번째 줄: @@
 *  real roots
 ** <math>\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}</math>
 *  imaginary roots
 ** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math>
 ** <math>\delta=\alpha_0+\alpha_1</math>
@@ 44번째 줄: / 44번째 줄: @@
 :<math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup  (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math>
-−
-−
 ==fixing a Cartan subalgebra and its dual==
@@ 76번째 줄: / 76번째 줄: @@
 </math>
 *  Weyl vector : <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math>
-−
 ==killing form==
@@ 83번째 줄: / 83번째 줄: @@
-−
 ==explicit construction==
@@ 139번째 줄: / 139번째 줄: @@
-−
 ==characters of irreducible representations==
 * [[Weyl-Kac character formula]]
@@ 188번째 줄: / 188번째 줄: @@
 from <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup  (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math>
 * we obtain {{수학노트|url=자코비_삼중곱(Jacobi_triple_product)}}
-−
 ===basic representation===
@@ 225번째 줄: / 225번째 줄: @@
 * [[vertex algebras]]
 * [[Quantum affine sl(2)]]
-−
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit
-−
 ==books==
-* Gannon 190p, 193p, 196p,371p
+* Gannon 190p, 193p, 196p,371p
-−
 ==articles==

@@ 6번째 줄: / 6번째 줄: @@
 ==construction from semisimple Lie algebra==
-* this is borrowed from [[affine Kac-Moody algebra]] entry
+* this is borrowed from [[affine Kac-Moody algebra]]
 * Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math>\langle \cdot,\cdot \rangle</math>
 * say <math>\mathfrak{g}=A_1</math>,  <math>\Phi=\{\alpha,-\alpha\}</math>
@@ 17번째 줄: / 17번째 줄: @@
 *  Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix
 :<math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0  \end{pmatrix}</math>
 ==basic quantities==

← 이전 판		2020년 11월 14일 (토) 01:37 판
247번째 줄:		247번째 줄:

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

@@ 241번째 줄: / 241번째 줄: @@
 ==articles==
 * Bakalov, Bojko, and Daniel Fleisher. “Bosonizations of $\widehat{\mathfrak{sl}}_2$ and Integrable Hierarchies.” arXiv:1407.5335 [math], July 20, 2014. http://arxiv.org/abs/1407.5335.
 * Dong, Jilan, and Naihuan Jing. 2014. “Realizations of Affine Lie Algebra A_^(1) at Negative Levels.” arXiv:1405.0339 [hep-Th], May. doi:10.1007/978-3-642-55361-5_36. http://arxiv.org/abs/1405.0339.

@@ 226번째 줄: / 226번째 줄: @@
 * [[sl(2) - orthogonal polynomials and Lie theory]]
 * [[vertex algebras]]
-* [[Quantum affine sl(2)]
+* [[Quantum affine sl(2)]]

@@ 157번째 줄: / 157번째 줄: @@
 * in general
 $$
 s_{\alpha_1}(m\omega_0+n\omega_1)=(m+2n)\omega_0-n\omega_1
 $$

@@ 208번째 줄: / 208번째 줄: @@
 $$
 where $z=e^{-\alpha_1}, q = e^{−\delta}$.
 ==related items==

@@ 154번째 줄: / 154번째 줄: @@
 * note that this is linear
 * $\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1$
 * $t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta$
 * $t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta$
-* so $w\in W$ can be written as $(n\alpha_1,\pm 1)$
+* $w\in W$ can be written as $(n\alpha_1,\pm 1)$