"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이

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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">introduction</h5>
 
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">introduction</h5>
  
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* start with a Lattice <math><\cdot,\cdot></math><br>
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* <math>[p_i, q_j] = \delta_{ij}z</math><br>
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*  make a vector space from it<br>
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* <math>[p_i, z] = 0</math><br>
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*  Construct a Loop algbera<br><math>A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><br><math>\alpha(m)=\alpha\otimes t^m</math><br>
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* <math>[q_j, z] = 0</math><br>
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* Give a bracket <br><math>[\alpha(m),\beta(n)]=m\delta_{m,-n}<\alpha,\beta>c</math><br>
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*  add a derivation <math>d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br>
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* define a Lie bracket<br><math>[d,x]=d(x)</math><br>
 
  
 
 
 
 
  
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">relation to quantum mechanics</h5>
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">relation to quantum mechanics</h5>
  
 
*   the position operators and momentum operators<br>
 
*   the position operators and momentum operators<br>
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">infinite dimensional Heisenberg algebra</h5>
 +
 +
*  start with a Lattice <math><\cdot,\cdot></math><br>
 +
*  make a vector space from it<br>
 +
*  Construct a Loop algbera<br><math>A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><br><math>\alpha(m)=\alpha\otimes t^m</math><br>
 +
*  Give a bracket <br><math>[\alpha(m),\beta(n)]=m\delta_{m,-n}<\alpha,\beta>c</math><br>
 +
*  add a derivation <math>d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br>
 +
*  define a Lie bracket<br><math>[d,x]=d(x)</math><br>
 +
 +
<br>
  
 
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">fock space representation</h5>
 
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">fock space representation</h5>
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* [[Kac-Moody algebras]]<br>
 
* [[Kac-Moody algebras]]<br>
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* [[central extension of groups and Lie algebras|central extension of semisimple lie algebra]]<br>
  
 
 
 
 

2009년 8월 13일 (목) 01:50 판

introduction
  • \([p_i, q_j] = \delta_{ij}z\)
  • \([p_i, z] = 0\)
  • \([q_j, z] = 0\)

 

 

relation to quantum mechanics
  •  the position operators and momentum operators

 

 

infinite dimensional Heisenberg algebra
  • start with a Lattice \(<\cdot,\cdot>\)
  • make a vector space from it
  • Construct a Loop algbera
    \(A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
    \(\alpha(m)=\alpha\otimes t^m\)
  • Give a bracket 
    \([\alpha(m),\beta(n)]=m\delta_{m,-n}<\alpha,\beta>c\)
  • add a derivation \(d\)
    \(d(\alpha(n))=n\alpha(n)\)
    \(d(c)=0\)
  • define a Lie bracket
    \([d,x]=d(x)\)


fock space representation

 

 

Representation theory

 

related items

 

books

 

 

encyclopedia

 

blogs

 

articles

 

TeX 

 

 

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