"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
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==monoidal categorification== | ==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 | + | # cluster monomials of <math>A</math> |
| − | # the classes of real simple objects of | + | # 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 | + | * cluster variables of <math>A</math> (including coefficients) corresponds to classes of real prime simple objects |
===proposition=== | ===proposition=== | ||
| − | * Suppose that | + | * 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 | ||
| 60번째 줄: | 60번째 줄: | ||
* 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 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. | * 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 | + | * 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 | + | * 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. | * 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. | ||
| 70번째 줄: | 70번째 줄: | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
| + | [[분류:quantum groups]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 14일 (토) 01:54 기준 최신판
introduction
- replace cluster variables by modules of quantum groups
- motivation comes from Positivity conjecture on cluster algebras
main results
- Hernandez-Leclerc and Nakajima categorified cluster algebras of finite type using Kirillov-Reshetikhin (KR) modules of Quantum affine algebra
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
- cluster monomials of \(A\)
- 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
- each cluster variable of a has positivie Laurent expansion with respect to any cluster
- cluster monomials are linearly independent
history
- Additive categorifications of cluster algebras
- categorification of quantum groups
- Quiver Hecke algebras
- Coordinate ring of maximal unipotent subgroup
computational resource
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”. 1102.1076 (2월 5). http://arxiv.org/abs/1102.1076.
- Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. 0807.1960 (7월 12). http://arxiv.org/abs/0807.1960.
- Cluster algebras and quiver representations, Keller, Bernhard, 2006
- 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 \(U_q(\widehat{sl_{2}})\).” 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.