"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이

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<h5>introduction</h5>
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==introduction==
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* replace cluster variables by modules of quantum groups
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* motivation comes from [[Positivity conjecture on cluster algebras]]
  
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* replace cluster variables by modules
 
  
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==main results==
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* Hernandez-Leclerc and Nakajima categorified [[Classification of cluster algebras of finite type|cluster algebras of finite type]] using [[Kirillov-Reshetikhin (KR) modules]] of [[Quantum affine algebra]]
  
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<h5>notions</h5>
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* quiver : oriented graph
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==monoidal categorification==
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* represetation of a quiver : collection of vector space and linear maps between them
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* <math>A</math> : cluster algebra
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* homomorphism of 2 quiver representations
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* <math>M</math> : monoidal categorify
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*  path algebra of a quiver<br>
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* <math>M</math> is a monoidal categorification of <math>A</math> if the Grothendieck ring <math>K_0(M)</math> of <math>M</math> is isomorphic to <math>A</math> which induces bijection between
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** given a quiver Q, a path p is a sequence of arrows with some conditions
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# cluster monomials of <math>A</math>
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** path algebra : set of all k-linear combinations of all paths (including e_i's)
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# the classes of real simple objects of <math>M</math> where <math>V</math> is ''real'' if <math>V\otimes V</math> is simple
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** p_1p_2 will correspond to a composition <math>p_2\circ p_1</math> of two maps (U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W)
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* cluster variables of <math>A</math> (including coefficients) corresponds to classes of real prime simple objects
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* quiver representation is in fact, a representaion of path algebra of a quiver
 
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* quiver has finite type of there are finitely many indecomposables
 
  
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===proposition===
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* Suppose that <math>A</math> has a monoidal categorification <math>M</math> and also that each object <math>B</math> in <math>M</math> has unique finite composition series, (i.e., find simple subobject <math>A_1</math>, then simple subobject of <math>A_2</math> of <math>B/A_1</math>, etc ... composition series if colleciton of all <math>A</math>'s)
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* Then
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# each cluster variable of a has positivie Laurent expansion with respect to any cluster
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# cluster monomials are linearly independent
  
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\thm (Gabriel)
 
  
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A connected quiver Q has finite type iff corresponding graph is Dynking diagram (A,D,E)
 
  
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==history==
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outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
 
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<h5>history</h5>
 
  
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  
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<h5>related items</h5>
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==related items==
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* [[Additive categorifications of cluster algebras]]
 
* [[categorification of quantum groups]]
 
* [[categorification of quantum groups]]
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* [[Quiver Hecke algebras]]
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* [[Coordinate ring of maximal unipotent subgroup]]
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* [http://eom.springer.de/ http://eom.springer.de]
 
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* http://www.proofwiki.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2011년 books and articles]]
 
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* http://library.nu/search?q=
 
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* http://library.nu/search?q=
 
  
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxbXM4aFBiRWotdk0/edit?usp=drivesdk
  
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<h5>expositions</h5>
 
  
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==expositions==
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* collection of notes http://mathserver.neu.edu/~sstella/seminars/cac-2011.shtml
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* Leclerc, 2012, Cluster algebras and Lie theory, http://www.emis.de/journals/SLC/wpapers/s69vortrag/leclerc.pdf
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* Leclerc, Bernard. 2011. “Quantum loop algebras, quiver varieties, and cluster algebras”. <em>1102.1076</em> (2월 5). http://arxiv.org/abs/1102.1076.
 
* Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. <em>0807.1960</em> (7월 12). http://arxiv.org/abs/0807.1960.
 
* Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. <em>0807.1960</em> (7월 12). http://arxiv.org/abs/0807.1960.
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* [http://www.math.jussieu.fr/%7Ekeller/publ/Reisensburg.pdf Cluster algebras and quiver representations], Keller, Bernhard, 2006
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* [http://www.mpim-bonn.mpg.de/node/365 Total positivity, cluster algebras and categorification]
  
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==articles==
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* Kang, Seok-Jin, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. ‘Monoidal Categorification of Cluster Algebras II’. arXiv:1502.06714 [math], 24 February 2015. http://arxiv.org/abs/1502.06714.
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* Kang, Seok-Jin, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. “Monoidal Categorification of Cluster Algebras.” arXiv:1412.8106 [math], December 27, 2014. http://arxiv.org/abs/1412.8106.
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* Ma, Hai-Tao, Yan-Min Yang, and Zhu-Jun Zheng. “Quantum Cluster Algebra Structure on the Finite Dimensional Representations of <math>U_q(\widehat{sl_{2}})</math>.” arXiv:1406.2452 [math-Ph], June 10, 2014. http://arxiv.org/abs/1406.2452.
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
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* Yang, Yan-Min, and Zhu-Jun Zheng. 2014. “Cluster Algebra Structure on the Finite Dimensional Representations of <math>U_q(\widehat{A_{3}})</math> for <math>l</math>=2.” arXiv:1403.5124 [math-Ph], March. http://arxiv.org/abs/1403.5124.
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* Hernandez, David, and Bernard Leclerc. ‘Monoidal Categorifications of Cluster Algebras of Type A and D’. arXiv:1207.3401 [math], 14 July 2012. http://arxiv.org/abs/1207.3401.
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*   <br>
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* Nakajima, Hiraku. 2011. “Quiver varieties and cluster algebras”. <em>Kyoto Journal of Mathematics</em> 51 (1): 71-126. doi:10.1215/0023608X-2010-021.
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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* http://math.stackexchange.com/search?q=
 
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* http://math.stackexchange.com/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5>links</h5>
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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[[분류:cluster algebra]]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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[[분류:math and physics]]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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[[분류:math]]
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* http://functions.wolfram.com/
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[[분류:quantum groups]]
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[[분류:migrate]]

2020년 11월 14일 (토) 01:54 기준 최신판

introduction


main results



monoidal categorification

  • \(A\) : cluster algebra
  • \(M\) : monoidal categorify
  • \(M\) is a monoidal categorification of \(A\) if the Grothendieck ring \(K_0(M)\) of \(M\) is isomorphic to \(A\) which induces bijection between
  1. cluster monomials of \(A\)
  2. the classes of real simple objects of \(M\) where \(V\) is real if \(V\otimes V\) is simple
  • cluster variables of \(A\) (including coefficients) corresponds to classes of real prime simple objects


proposition

  • Suppose that \(A\) has a monoidal categorification \(M\) and also that each object \(B\) in \(M\) has unique finite composition series, (i.e., find simple subobject \(A_1\), then simple subobject of \(A_2\) of \(B/A_1\), etc ... composition series if colleciton of all \(A\)'s)
  • Then
  1. each cluster variable of a has positivie Laurent expansion with respect to any cluster
  2. cluster monomials are linearly independent



history



related items


computational resource


expositions

articles

  • Kang, Seok-Jin, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. ‘Monoidal Categorification of Cluster Algebras II’. arXiv:1502.06714 [math], 24 February 2015. http://arxiv.org/abs/1502.06714.
  • Kang, Seok-Jin, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. “Monoidal Categorification of Cluster Algebras.” arXiv:1412.8106 [math], December 27, 2014. http://arxiv.org/abs/1412.8106.
  • Ma, Hai-Tao, Yan-Min Yang, and Zhu-Jun Zheng. “Quantum Cluster Algebra Structure on the Finite Dimensional Representations of \(U_q(\widehat{sl_{2}})\).” arXiv:1406.2452 [math-Ph], June 10, 2014. http://arxiv.org/abs/1406.2452.
  • Yang, Yan-Min, and Zhu-Jun Zheng. 2014. “Cluster Algebra Structure on the Finite Dimensional Representations of \(U_q(\widehat{A_{3}})\) for \(l\)=2.” arXiv:1403.5124 [math-Ph], March. http://arxiv.org/abs/1403.5124.
  • Hernandez, David, and Bernard Leclerc. ‘Monoidal Categorifications of Cluster Algebras of Type A and D’. arXiv:1207.3401 [math], 14 July 2012. http://arxiv.org/abs/1207.3401.
  • Nakajima, Hiraku. 2011. “Quiver varieties and cluster algebras”. Kyoto Journal of Mathematics 51 (1): 71-126. doi:10.1215/0023608X-2010-021.
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