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

수학노트
둘러보기로 가기 검색하러 가기
(피타고라스님이 이 페이지를 개설하였습니다.)
 
 
(사용자 2명의 중간 판 51개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 +
==introduction==
 +
* replace cluster variables by modules of quantum groups
 +
* motivation comes from [[Positivity conjecture on cluster algebras]]
  
 +
 +
==main results==
 +
* 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]]
 +
 +
 +
 +
 +
==monoidal categorification==
 +
* <math>A</math> : cluster algebra
 +
* <math>M</math> : monoidal categorify
 +
* <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
 +
# cluster monomials of <math>A</math>
 +
# the classes of real simple objects of <math>M</math> where <math>V</math> is ''real'' if <math>V\otimes V</math> is simple
 +
* cluster variables of <math>A</math> (including coefficients) corresponds to classes of real prime simple objects
 +
 +
 +
 +
===proposition===
 +
* 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
 +
# each cluster variable of a has positivie Laurent expansion with respect to any cluster
 +
# cluster monomials are linearly independent
 +
 +
 +
 +
 +
==history==
 +
 +
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 +
 +
 +
 +
 +
 +
==related items==
 +
* [[Additive categorifications of cluster algebras]]
 +
* [[categorification of quantum groups]]
 +
* [[Quiver Hecke algebras]]
 +
* [[Coordinate ring of maximal unipotent subgroup]]
 +
 +
 +
==computational resource==
 +
* https://docs.google.com/file/d/0B8XXo8Tve1cxbXM4aFBiRWotdk0/edit?usp=drivesdk
 +
 +
 +
 +
==expositions==
 +
* collection of notes http://mathserver.neu.edu/~sstella/seminars/cac-2011.shtml
 +
* 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.
 +
* 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.mpim-bonn.mpg.de/node/365 Total positivity, cluster algebras and categorification]
 +
 +
==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 <math>U_q(\widehat{sl_{2}})</math>.” 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 <math>U_q(\widehat{A_{3}})</math> for <math>l</math>=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”. <em>Kyoto Journal of Mathematics</em> 51 (1): 71-126. doi:10.1215/0023608X-2010-021.
 +
 +
 +
 +
[[분류:cluster algebra]]
 +
[[분류:math and physics]]
 +
[[분류:math]]
 +
[[분류:quantum groups]]
 +
[[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.
원본 주소 "https://wiki.mathnt.net/index.php?title=Monoidal_categorifications_of_cluster_algebras&oldid=44407"