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| + | ==introduction== | ||
| + | * [http://www.math.cornell.edu/People/Faculty/Heisen.pdf AUTOMORPHISMS OF THE DISCRETE HEISENBERG GROUP] | ||
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==relation to quantum mechanics== | ==relation to quantum mechanics== | ||
| − | * | + | * the position operators and momentum operators satisfy the relation<math>[X,P] = X P - P X = i \hbar</math> |
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==relation to Weyl algebra== | ==relation to Weyl algebra== | ||
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* a quotient of the universal enveloping algebra of the Heisenberg algebra | * a quotient of the universal enveloping algebra of the Heisenberg algebra | ||
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==finite dimensional Heisenberg algebra== | ==finite dimensional Heisenberg algebra== | ||
| − | * one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra< | + | * 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, z] = 0</math> | ||
| + | * <math>[q_j, z] = 0</math> | ||
| + | * Gannon 180p | ||
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==differential operators== | ==differential operators== | ||
| − | * commutation relation | + | * commutation relation<math>x</math>, <math>p=\frac{d}{dx}</math><math>[x,p]=1</math> |
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==infinite dimensional Heisenberg algebra== | ==infinite dimensional Heisenberg algebra== | ||
| − | * start with a | + | * start with a Lattice <math>\langle\cdot,\cdot\rangle</math> |
| − | * make a vector space from it | + | * make a vector space from it |
| − | * Construct a Loop algbera | + | * 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 | + | * Give a bracket <math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math> |
| − | * add a | + | * add a derivation <math>d</math><math>d(\alpha(n))=n\alpha(n)</math><math>d(c)=0</math> |
| − | * define a Lie bracket | + | * 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 |
| − | * commutator subalgebra | + | * commutator subalgebra |
| − | * The automorphisms of the Heisenberg group (fixing its center) form the symplectic group | + | * The automorphisms of the Heisenberg group (fixing its center) form the symplectic group |
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==highest weight module== | ==highest weight module== | ||
| − | * <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math | + | * <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math> |
| − | * <math>c.v_{h}=v_{h}</math | + | * <math>c.v_{h}=v_{h}</math> |
| − | * for <math>m>0</math>, <math>\alpha(m)v_{h}=0</math | + | * for <math>m>0</math>, <math>\alpha(m)v_{h}=0</math> |
| − | * <math>\alpha(0)v_{h}=hv_{h}</math | + | * <math>\alpha(0)v_{h}=hv_{h}</math> |
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==Stone-Von Neumann theorem== | ==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). |
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==Heisenberg VOA== | ==Heisenberg VOA== | ||
| − | * [[VOA associated to Heisenberg algebra]] | + | * [[VOA associated to Heisenberg algebra]] |
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==related items== | ==related items== | ||
| − | * [[half-integral weight modular forms|half-integral modular forms]] | + | * [[half-integral weight modular forms|half-integral modular forms]] |
| − | * [[Kac-Moody algebras]] | + | * [[Kac-Moody algebras]] |
| − | * [[central extension of groups and Lie algebras|central extension of semisimple lie algebra]] | + | * [[central extension of groups and Lie algebras|central extension of semisimple lie algebra]] |
| − | * [[Weyl algebra]] | + | * [[Weyl algebra]] |
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==books== | ==books== | ||
| + | * Michael Eugene Taylor [http://books.google.de/books?id=8y11t3mhLO8C Noncommutative Harmonic Analysis] | ||
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==encyclopedia== | ==encyclopedia== | ||
| 107번째 줄: | 102번째 줄: | ||
* http://en.wikipedia.org/wiki/Weyl_algebra | * http://en.wikipedia.org/wiki/Weyl_algebra | ||
* [http://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem] | * [http://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem] | ||
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==blogs== | ==blogs== | ||
* [http://www.math.columbia.edu/~woit/wordpress/?p=362 George Mackey 1916-2006] | * [http://www.math.columbia.edu/~woit/wordpress/?p=362 George Mackey 1916-2006] | ||
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==expositions== | ==expositions== | ||
| + | * Müller, Detlef. “Analysis of Invariant PDO’s on the Heisenberg Group.” arXiv:1408.2634 [math], August 12, 2014. http://arxiv.org/abs/1408.2634. | ||
| + | * 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] | ||
* [http://www.ms.unimelb.edu.au/documents/thesis/thesis-Matt-Collins Representations of Heisenberg Groups] | * [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/ | + | * 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] | * [http://www.math.umd.edu/~jmr/StoneVNart.pdf A Selective History of the Stone-von Neumann Theorem] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:Lie theory]] | [[분류:Lie theory]] | ||
| + | [[분류:theta]] | ||
| + | [[분류:migrate]] | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q1601337 Q1601337] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'heisenberg'}, {'LEMMA': 'group'}] | ||
2021년 2월 17일 (수) 03:13 기준 최신판
introduction
relation to quantum mechanics
- the position operators and momentum operators satisfy the relation\([X,P] = X P - P X = i \hbar\)
relation to Weyl algebra
- a quotient of the universal enveloping algebra of the Heisenberg algebra
finite dimensional Heisenberg algebra
- one dimensional central extension of abelian Lie algebra
- \([p_i, q_j] = \delta_{ij}z\)
- \([p_i, z] = 0\)
- \([q_j, z] = 0\)
- Gannon 180p
differential operators
- commutation relation\(x\), \(p=\frac{d}{dx}\)\([x,p]=1\)
infinite dimensional Heisenberg algebra
- start with a Lattice \(\langle\cdot,\cdot\rangle\)
- make a vector space from it
- Construct a Loop algbera\(\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)\(\alpha(m)=\alpha\otimes t^m\)
- Give a bracket \([\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c\)
- add a derivation \(d\)\(d(\alpha(n))=n\alpha(n)\)\(d(c)=0\)
- define a Lie bracket\([d,x]=d(x)\)
- 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
- \(\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c\)
- \(c.v_{h}=v_{h}\)
- for \(m>0\), \(\alpha(m)v_{h}=0\)
- \(\alpha(0)v_{h}=hv_{h}\)
Stone-Von Neumann theorem
- The 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
- half-integral modular forms
- Kac-Moody algebras
- central extension of semisimple lie algebra
- Weyl algebra
books
- Michael Eugene Taylor Noncommutative Harmonic Analysis
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Heisenberg_algebra
- http://en.wikipedia.org/wiki/Weyl_algebra
- http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem
blogs
expositions
- Müller, Detlef. “Analysis of Invariant PDO’s on the Heisenberg Group.” arXiv:1408.2634 [math], August 12, 2014. http://arxiv.org/abs/1408.2634.
- Kisil Lecture 18 The Heisenberg Group
- On the role of the Heisenberg group in harmonic analysis
- Representations of Heisenberg Groups
- Stephen Semmes, An Introduction to Heisenberg Groups in Analysis and Geometry, June/July 2003 Volume 50 Issue 6 , Notices of AMS
- A Selective History of the Stone-von Neumann Theorem
메타데이터
위키데이터
- ID : Q1601337
Spacy 패턴 목록
- [{'LOWER': 'heisenberg'}, {'LEMMA': 'group'}]