"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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| 5번째 줄: | 5번째 줄: | ||
==relation to quantum mechanics== | ==relation to quantum mechanics== | ||
| − | * the position operators and momentum operators satisfy the relation | + | * 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== | ==differential operators== | ||
| − | * commutation relation | + | * commutation relation<math>x</math>, <math>p=\frac{d}{dx}</math><math>[x,p]=1</math> |
| 41번째 줄: | 41번째 줄: | ||
==infinite dimensional Heisenberg algebra== | ==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 | + | * 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 bracket | + | * Give a bracket <math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</math> |
| − | * add a derivation <math>d</math | + | * 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 | + | * 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 |
| 57번째 줄: | 57번째 줄: | ||
==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> |
| 74번째 줄: | 74번째 줄: | ||
==Heisenberg VOA== | ==Heisenberg VOA== | ||
| − | * [[VOA associated to Heisenberg algebra]] | + | * [[VOA associated to Heisenberg algebra]] |
| 82번째 줄: | 82번째 줄: | ||
==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]] |
2020년 11월 13일 (금) 09:10 판
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