Heisenberg group and Heisenberg algebra

introduction

• AUTOMORPHISMS OF THE DISCRETE HEISENBERG GROUP

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

• VOA associated to Heisenberg algebra
  

 

 

related items

• half-integral modular forms
  
• Kac-Moody algebras
  
• central extension of semisimple lie algebra
  
• Weyl algebra
  

 

 

 

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

• George Mackey 1916-2006

expositions

• 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