"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
| − | * [www.math.cornell.edu/People/Faculty/Heisen.pdf AUTOMORPHISMS OF THE DISCRETE HEISENBERG GROUP] | + | * [http://www.math.cornell.edu/People/Faculty/Heisen.pdf AUTOMORPHISMS OF THE DISCRETE HEISENBERG GROUP] |
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==books== | ==books== | ||
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* 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== | ||
| + | * [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/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] | * [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]] | ||
2013년 2월 25일 (월) 08:28 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
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
- 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