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| {{for|W*-algebra|Von Neumann algebra}}
| | Friends call her Felicidad and her husband doesn't like it at all. Arizona is her beginning place and she will never move. Interviewing is what she does but quickly she'll be on her personal. To perform croquet is some thing that I've carried out for many years.<br><br>Here is my page - [http://brazil.Amor-amore.com/nilindon http://brazil.Amor-amore.com/nilindon] |
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| In mathematics, a '''W-algebra''' is a structure in conformal field theory related to generalizations of the [[Virasoro algebra]]. They were introduced by {{harvtxt|Zamolodchikov|1985}}, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
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| There are at least three different but related notions called W-algebras: classical W-algebras, quantum W-algebras, and finite W-algebras.
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| ==Classical W-algebras==
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| Performing classical [[Vladimir Drinfeld|Drinfeld]]-Sokolov reduction on a Lie algebra provides the [[Poisson bracket]] on this algebra.
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| ==Quantum W-algebras==
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| {{harvtxt|Bouwknegt|1993}} defines a (quantum) W-algebra to be a [[meromorphic]] [[conformal field theory]] (roughly a [[vertex operator algebra]]) together a distinguished set of generators satisfying various properties.
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| They can be constructed from a Lie (super)algebra by quantum Drinfeld-Sokolov reduction. Another approach is to look for other conserved currents besides the [[Stress-energy tensor]] in a similar manner to how the [[Virasoro algebra]] can be read off from the expansion of the stress tensor.
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| ==Finite W-algebras==
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| {{harvtxt|Wang|2011}} compares several different definitions of finite W-algebras, which are certain associative algebras associated to nilpotent elements of semisimple Lie algebras.
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| The original definition, provided by Alexander Premet, starts with a pair <math>(\mathfrak{g}, e)</math> consisting of a reductive Lie algebra <math>\mathfrak{g}</math> over the complex numbers and a nilpotent element e.
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| By the [[Jacobson-Morozov theorem]], e is part of an sl<sub>2</sub> triple (e,h,f). The eigenspace decomposition of ad(h) induces a <math>\mathbb{Z}</math>-grading on g:
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| ::<math>
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| \mathfrak{g} = \bigoplus \mathfrak{g} (i).
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| </math>
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| Define a [[character (mathematics)|character]] <math> \chi </math> (ie. a [[homomorphism]] from g to the trivial 1-dimensional Lie algebra) by the rule <math> \chi(x) = \kappa(e,x) </math>, where <math> \kappa </math> denotes the [[Killing form]]. This induces a [[degenerate form| non-degenerate]] anti-symmetric [[bilinear form]] on the -1 graded piece by the rule:
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| ::<math>
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| \omega_\chi (x,y) = \chi ( [x,y] ).
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| </math>
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| After choosing any [[Lagrangian subspace]] <math>l</math>, we may define the following [[nilpotent Lie algebra|nilpotent]] subalgebra which acts on the universal enveloping algebra by the [[adjoint representation|adjoint action]].
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| ::<math>
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| \mathfrak{m} = l + \bigoplus_{i \leq -2} \mathfrak{g} (i).
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| </math>
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| The left [[ideal (ring theory)| ideal]] <math> I </math> of the [[enveloping algebra]] <math> U(\mathfrak{g}) </math> generated by <math> \{ x - \chi(x) : x \in \mathfrak{m} \} </math> is invariant under this action. It follows from a short calculation that the invariants in <math> U(\mathfrak{g})/I</math> under ad<math>(\mathfrak{m})</math> inherit the [[associative algebra]] structure from <math> U(\mathfrak{g}) </math>. The invariant subspace <math>(U(\mathfrak{g})/I)^{\text{ad}(\mathfrak{m})}</math> is called the finite W-algebra constructed from (g,e) and is usually denoted <math> U(\mathfrak{g},e)</math>.
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| ==References==
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| *{{Citation | last3=Schoutens | first2=Kareljan | last1=Bouwknegt | first1=Peter | title=W symmetry in conformal field theory | url=http://dx.doi.org/10.1016/0370-1573(93)90111-P | doi=10.1016/0370-1573(93)90111-P | id={{MR|1208246}} | year=1993 | journal=Physics Reports. A Review Section of Physics Letters | issn=0370-1573 | volume=223 | issue=4 | pages=183–276}}
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| *{{Citation | editor2-last=Schoutens | editor2-first=K. | editor1-last=Bouwknegt | editor1-first=P. | title=W-symmetry | url=http://books.google.com/books?id=wQ6G0sPOoH8C | publisher=World Scientific Publishing Co. Inc. | location=River Edge, NJ | series=Advanced Series in Mathematical Physics | isbn=9789810217624 | id={{MR|1338864}} | year=1995 | volume=22}}
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| *{{Citation | last2=Tjin | first2=Tjark | last1=de Boer | first1=Jan | title=Quantization and representation theory of finite W algebras | url=http://projecteuclid.org/getRecord?id=euclid.cmp/1104254359 | id={{MR|1255424}} | year=1993 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=158 | issue=3 | pages=485–516}}
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| *{{Citation | last1=Dickey | first1=L. A. | title=Lectures on classical W-algebras | url=http://dx.doi.org/10.1023/A:1017903416906 | doi=10.1023/A:1017903416906 | year=1997 | journal=Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications | issn=0167-8019 | volume=47 | issue=3 | pages=243–321}}
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| *{{Citation | last1=Gan | first1=Wee Liang | last2=Ginzburg | first2=Victor | title=Quantization of Slodowy slices | url=http://dx.doi.org/10.1155/S107379280210609X | doi=10.1155/S107379280210609X | id={{MR|1876934}} | year=2002 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=5 | pages=243–255}}
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| *{{Citation | last1=Losev | first1=Ivan | title=Quantized symplectic actions and W-algebras | url=http://arxiv.org/abs/0707.3108 | doi=10.1090/S0894-0347-09-00648-1 | id={{MR|2552248}} | year=2010 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=23 | issue=1 | pages=35–59}}
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| *{{Citation | last1=Pope | first1=C.N. | title=Lectures on W algebras and W gravity | url=http://arxiv.org/abs/hep-th/9112076 | series=Lectures given at the Trieste Summer School in High-Energy Physics, August 1991 | year=1991}}
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| *{{Citation | last1=Wang | first1=Weiqiang | editor3-last=Wang | editor3-first=Weiqiang | editor2-last=Savage | editor2-first=Alistair | editor1-last=Neher | editor1-first=Erhard | title=Geometric representation theory and extended affine Lie algebras | url=http://arxiv.org/abs/0912.0689 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Fields Inst. Communications | isbn=978-0-8218-5237-8 | id={{MR|2777648}} | year=2011 | volume=59 | chapter=Nilpotent orbits and finite W-algebras | pages=71–105}}
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| *{{Citation | last1=Watts | first1=Gerard M. T. | editor2-last=Palla | editor2-first=László | editor1-last=Horváth | editor1-first=Zalán | title=Conformal field theories and integrable models (Budapest, 1996) | url=http://dx.doi.org/10.1007/BFb0105278 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Phys. | isbn=978-3-540-63618-2 | doi=10.1007/BFb0105278 | id={{MR|1636798}} | year=1997 | volume=498 | chapter=W-algebras and their representations | pages=55–84}}
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| *{{Citation | last1=Zamolodchikov | first1=A. B. | title=Infinite extra symmetries in two-dimensional conformal quantum field theory | language=Russian | id={{MR|829902}} | year=1985 | journal=Akademiya Nauk SSSR. Teoreticheskaya i Matematicheskaya Fizika | issn=0564-6162 | volume=65 | issue=3 | pages=347–359}}
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| *{{Citation| last=Brown | first=Jonathan | title= Finite W-algebras of Classical Type| url=https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/10201/Brown_Jonathan_phd2009sp.pdf?sequence=1}}
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| [[Category:Conformal field theory]]
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| [[Category:Integrable systems]]
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| [[Category:Representation theory]]
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Friends call her Felicidad and her husband doesn't like it at all. Arizona is her beginning place and she will never move. Interviewing is what she does but quickly she'll be on her personal. To perform croquet is some thing that I've carried out for many years.
Here is my page - http://brazil.Amor-amore.com/nilindon