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

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==introduction==
 
==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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==notions==
 
  
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* quiver : oriented graph
 
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* representation of a quiver : collection of vector space and linear maps between them
 
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* homomorphism of 2 quiver representations
 
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*  path algebra of a quiver
 
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** given a quiver Q, a path p is a sequence of arrows with some conditions
 
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** path algebra : set of all k-linear combinations of all paths (including e_i's)
 
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** p_1p_2 will correspond to a composition <math>p_2\circ p_1</math> of two maps (<math>U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W</math>)
 
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* quiver representation is in fact, a representaion of path algebra of a quiver
 
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==Caldero-Chapoton formula==
 
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* CC(V) =\chi_{V}
 
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==monoidal categorification==
 
==monoidal categorification==
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* M : monoidal categorification
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* <math>A</math> : cluster algebra
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* M is a monoidal categorification of A if the Grothendieck ring of M is isomorphic to A and if
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* <math>M</math> : monoidal categorify
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# cluster monomials' of A are the classes of real simple objects of M
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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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# cluster variables' of a (including coefficients) are classes of real prime simple objects
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# cluster monomials of <math>A</math>
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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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* cluster variables of <math>A</math> (including coefficients) corresponds to classes of real prime simple objects
  
 
   
 
   
  
 
===proposition===
 
===proposition===
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* 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)
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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)
 
* Then
 
* Then
 
# each cluster variable of a has positivie Laurent expansion with respect to any cluster
 
# each cluster variable of a has positivie Laurent expansion with respect to any cluster
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==periodicity conjecture==
 
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* outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
 
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==history==
 
==history==
62번째 줄: 38번째 줄:
  
 
==related items==
 
==related items==
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* [[Quiver representations]]
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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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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxbXM4aFBiRWotdk0/edit?usp=drivesdk
  
 
   
 
   
  
 
==expositions==
 
==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
 
* Leclerc, Bernard. 2011. “Quantum loop algebras, quiver varieties, and cluster algebras”. <em>1102.1076</em> (2월 5). http://arxiv.org/abs/1102.1076.
 
* 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.
 
* [http://www.math.jussieu.fr/%7Ekeller/publ/Reisensburg.pdf Cluster algebras and quiver representations], Keller, Bernhard, 2006
 
* [http://www.math.jussieu.fr/%7Ekeller/publ/Reisensburg.pdf Cluster algebras and quiver representations], Keller, Bernhard, 2006
 
* [http://www.mpim-bonn.mpg.de/node/365 Total positivity, cluster algebras and categorification]
 
* [http://www.mpim-bonn.mpg.de/node/365 Total positivity, cluster algebras and categorification]
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==articles==
 
==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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* David Hernandez, Bernard Leclerc , Monoidal categorifications of cluster algebras of type A and D http://arxiv.org/abs/1207.3401
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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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* 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.
 
* Nakajima, Hiraku. 2011. “Quiver varieties and cluster algebras”. <em>Kyoto Journal of Mathematics</em> 51 (1): 71-126. doi:10.1215/0023608X-2010-021.
 
* 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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* Rupel, Dylan. 2010. “On Quantum Analogue of The Caldero-Chapoton Formula”. <em>1003.2652</em> (3월 12). doi:doi:10.1093/imrn/rnq192. http://arxiv.org/abs/1003.2652.
 
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* Caldero, Philippe, 와/과Andrei Zelevinsky. 2006. “Laurent expansions in cluster algebras via quiver representations”. <em>math/0604054</em> (4월 3). http://arxiv.org/abs/math/0604054.
 
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*  Caldero, Philippe, 와/과Frederic Chapoton. 2004. “Cluster algebras as Hall algebras of quiver representations”. <em>math/0410187</em> (10월 7). http://arxiv.org/abs/math/0410187. 
 
  
  
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[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:math]]
 
[[분류:math]]
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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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