Heisenberg group and Heisenberg algebra - 편집 역사

2021년 2월 17일 (수) 10:13에 Pythagoras0님의 편집

2021-02-17T10:13:28Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T12:56:15Z

메타데이터: 새 문단

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2020-12-28T12:04:08Z

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2020년 11월 13일 (금) 06:25에 imported>Pythagoras0님의 편집

2020-11-13T06:25:09Z

imported>Pythagoras0: /* finite dimensional Heisenberg algebra */

2015-05-26T06:20:49Z

finite dimensional Heisenberg algebra

2015년 4월 25일 (토) 11:31에 imported>Pythagoras0님의 편집

2015-04-25T11:31:08Z

2014년 8월 31일 (일) 04:58에 imported>Pythagoras0님의 편집

2014-08-31T04:58:11Z

imported>Pythagoras0: /* books */

2013-02-25T20:32:05Z

books

2013년 2월 25일 (월) 20:31에 imported>Pythagoras0님의 편집

2013-02-25T20:31:07Z

@@ 122번째 줄: / 122번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q1601337 Q1601337]

← 이전 판		2020년 12월 28일 (월) 12:56 판
121번째 줄:		121번째 줄:
	[[분류:theta]]		[[분류:theta]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1601337 Q1601337]

@@ 5번째 줄: / 5번째 줄: @@
 ==relation to quantum mechanics==
-*   the position operators and momentum operators satisfy the relation<math>[X,P] = X P - P X = i \hbar</math>
+*   the position operators and momentum operators satisfy the relation<math>[X,P] = X P - P X = i \hbar</math>
-−
-−
 ==relation to Weyl algebra==
@@ 15번째 줄: / 15번째 줄: @@
 * a quotient of the universal enveloping algebra of the Heisenberg algebra
-−
-−
 ==finite dimensional Heisenberg algebra==
@@ 27번째 줄: / 27번째 줄: @@
 *  Gannon 180p
-−
-−
 ==differential operators==
-*  commutation relation<math>x</math>, <math>p=\frac{d}{dx}</math><math>[x,p]=1</math>
+*  commutation relation<math>x</math>, <math>p=\frac{d}{dx}</math><math>[x,p]=1</math>
-−
-−
 ==infinite dimensional Heisenberg algebra==
-*  start with a Lattice <math>\langle\cdot,\cdot\rangle</math>
+*  start with a Lattice <math>\langle\cdot,\cdot\rangle</math>
 *  make a vector space from it
 *  Construct a Loop algbera<math>\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><math>\alpha(m)=\alpha\otimes t^m</math>
-*  Give a bracket <math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math>
+*  Give a bracket <math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math>
-*  add a derivation <math>d</math><math>d(\alpha(n))=n\alpha(n)</math><math>d(c)=0</math>
+*  add a derivation <math>d</math><math>d(\alpha(n))=n\alpha(n)</math><math>d(c)=0</math>
 *  define a Lie bracket<math>[d,x]=d(x)</math>
-*  In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra
+*  In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra
 *  commutator subalgebra
 *  The automorphisms of the Heisenberg group (fixing its center) form the symplectic group
-−
-−
 ==highest weight module==
@@ 62번째 줄: / 62번째 줄: @@
 * <math>\alpha(0)v_{h}=hv_{h}</math>
-−
 ==Stone-Von Neumann theorem==
-* The [[Heisenberg group and Heisenberg algebra|Heisenberg group]] has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem).
+* The [[Heisenberg group and Heisenberg algebra|Heisenberg group]] has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem).
-−
-−
 ==Heisenberg VOA==
@@ 76번째 줄: / 76번째 줄: @@
 * [[VOA associated to Heisenberg algebra]]
-−
-−
 ==related items==
@@ 87번째 줄: / 87번째 줄: @@
 * [[Weyl algebra]]
-−
-−
 ==books==
 * Michael Eugene Taylor [http://books.google.de/books?id=8y11t3mhLO8C Noncommutative Harmonic Analysis]
-−
 ==encyclopedia==
@@ 113번째 줄: / 113번째 줄: @@
 * [http://books.google.de/books?hl=en&lr=&id=P2Xe0lFFNO8C&oi=fnd&pg=PA333&dq=related:oJgsodjWPLsJ:scholar.google.com/&ots=SHNcihuGA0&sig=3qtjM71nZBTzBoSfq_6xLNe2FA0#v=onepage&q&f=false On the role of the Heisenberg group in harmonic analysis]
 * [http://www.ms.unimelb.edu.au/documents/thesis/thesis-Matt-Collins Representations of Heisenberg Groups]
-* Stephen Semmes, [http://www.ams.org/notices/200306/fea-semmes.pdf An Introduction to Heisenberg Groups in Analysis and Geometry], June/July  2003  Volume 50  Issue 6 , Notices of AMS
+* Stephen Semmes, [http://www.ams.org/notices/200306/fea-semmes.pdf An Introduction to Heisenberg Groups in Analysis and Geometry], June/July  2003  Volume 50  Issue 6 , Notices of AMS
 * [http://www.math.umd.edu/~jmr/StoneVNart.pdf A Selective History of the Stone-von Neumann Theorem]

@@ 5번째 줄: / 5번째 줄: @@
 ==relation to quantum mechanics==
-*   the position operators and momentum operators satisfy the relation<br><math>[X,P] = X P - P X = i \hbar</math><br>
+*   the position operators and momentum operators satisfy the relation<math>[X,P] = X P - P X = i \hbar</math>
@@ 33번째 줄: / 33번째 줄: @@
 ==differential operators==
-*  commutation relation<br><math>x</math>, <math>p=\frac{d}{dx}</math><br><math>[x,p]=1</math><br>
+*  commutation relation<math>x</math>, <math>p=\frac{d}{dx}</math><math>[x,p]=1</math>
@@ 41번째 줄: / 41번째 줄: @@
 ==infinite dimensional Heisenberg algebra==
-*  start with a Lattice <math>\langle\cdot,\cdot\rangle</math><br>
+*  start with a Lattice <math>\langle\cdot,\cdot\rangle</math>
-*  make a vector space from it<br>
+*  make a vector space from it
-*  Construct a Loop algbera<br><math>\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><br><math>\alpha(m)=\alpha\otimes t^m</math><br>
+*  Construct a Loop algbera<math>\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><math>\alpha(m)=\alpha\otimes t^m</math>
-*  Give a bracket <br><math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math><br>
+*  Give a bracket <math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math>
-*  add a derivation <math>d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br>
+*  add a derivation <math>d</math><math>d(\alpha(n))=n\alpha(n)</math><math>d(c)=0</math>
-*  define a Lie bracket<br><math>[d,x]=d(x)</math><br>
+*  define a Lie bracket<math>[d,x]=d(x)</math>
-*  In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra<br>
+*  In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra
-*  commutator subalgebra<br>
+*  commutator subalgebra
-*  The automorphisms of the Heisenberg group (fixing its center) form the symplectic group<br>
+*  The automorphisms of the Heisenberg group (fixing its center) form the symplectic group
@@ 57번째 줄: / 57번째 줄: @@
 ==highest weight module==
-* <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math><br>
+* <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math>
-* <math>c.v_{h}=v_{h}</math><br>
+* <math>c.v_{h}=v_{h}</math>
-*  for <math>m>0</math>, <math>\alpha(m)v_{h}=0</math><br>
+*  for <math>m>0</math>, <math>\alpha(m)v_{h}=0</math>
-* <math>\alpha(0)v_{h}=hv_{h}</math><br>
+* <math>\alpha(0)v_{h}=hv_{h}</math>
@@ 74번째 줄: / 74번째 줄: @@
 ==Heisenberg VOA==
-* [[VOA associated to Heisenberg algebra]]<br>
+* [[VOA associated to Heisenberg algebra]]
@@ 82번째 줄: / 82번째 줄: @@
 ==related items==
-* [[half-integral weight modular forms|half-integral modular forms]]<br>
+* [[half-integral weight modular forms|half-integral modular forms]]
-* [[Kac-Moody algebras]]<br>
+* [[Kac-Moody algebras]]
-* [[central extension of groups and Lie algebras|central extension of semisimple lie algebra]]<br>
+* [[central extension of groups and Lie algebras|central extension of semisimple lie algebra]]
-* [[Weyl algebra]]<br>
+* [[Weyl algebra]]

← 이전 판		2020년 11월 13일 (금) 06:25 판
120번째 줄:		120번째 줄:
	[[분류:Lie theory]]		[[분류:Lie theory]]
	[[분류:theta]]		[[분류:theta]]
		+	[[분류:migrate]]

@@ 21번째 줄: / 21번째 줄: @@
 ==finite dimensional Heisenberg algebra==
-*  one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra<br>
+*  one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra
 * <math>[p_i, q_j] = \delta_{ij}z</math>
-* <math>[p_i, q_j] = \delta_{ij}z</math><br>
+* <math>[p_i, z] = 0</math>
-* <math>[p_i, z] = 0</math><br>
+* <math>[q_j, z] = 0</math>
-* <math>[q_j, z] = 0</math><br>
+*  Gannon 180p

← 이전 판		2015년 4월 25일 (토) 11:31 판
120번째 줄:		120번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:Lie theory]]		[[분류:Lie theory]]
		+	[[분류:theta]]

@@ 110번째 줄: / 110번째 줄: @@
 ==expositions==
 * Kisil [http://www1.maths.leeds.ac.uk/~kisilv/courses/epal021.html Lecture 18  The Heisenberg Group]
 * [http://books.google.de/books?hl=en&lr=&id=P2Xe0lFFNO8C&oi=fnd&pg=PA333&dq=related:oJgsodjWPLsJ:scholar.google.com/&ots=SHNcihuGA0&sig=3qtjM71nZBTzBoSfq_6xLNe2FA0#v=onepage&q&f=false On the role of the Heisenberg group in harmonic analysis]

← 이전 판		2013년 2월 25일 (월) 20:32 판
93번째 줄:		93번째 줄:

	==books==		==books==
−		+	* Michael Eugene Taylor [http://books.google.de/books?id=8y11t3mhLO8C Noncommutative Harmonic Analysis]