"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 3번째 줄: | 3번째 줄: | ||
* replace cluster variables by modules | * replace cluster variables by modules | ||
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==notions== | ==notions== | ||
| 12번째 줄: | 12번째 줄: | ||
* representation of a quiver : collection of vector space and linear maps between them | * representation of a quiver : collection of vector space and linear maps between them | ||
* homomorphism of 2 quiver representations | * homomorphism of 2 quiver representations | ||
| − | * path algebra of a quiver | + | * path algebra of a quiver |
** given a quiver Q, a path p is a sequence of arrows with some conditions | ** given a quiver Q, a path p is a sequence of arrows with some conditions | ||
** path algebra : set of all k-linear combinations of all paths (including e_i's) | ** path algebra : set of all k-linear combinations of all paths (including e_i's) | ||
| − | ** p_1p_2 will correspond to a | + | ** 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>) |
* quiver representation is in fact, a representaion of path algebra of a quiver | * quiver representation is in fact, a representaion of path algebra of a quiver | ||
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==Caldero-Chapoton formula== | ==Caldero-Chapoton formula== | ||
| 26번째 줄: | 26번째 줄: | ||
* CC(V) =\chi_{V} | * CC(V) =\chi_{V} | ||
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==monoidal categorification== | ==monoidal categorification== | ||
| 40번째 줄: | 40번째 줄: | ||
(ii) cluster variables' of a (including coefficients) are classes of real prime simple objects | (ii) cluster variables' of a (including coefficients) are classes of real prime simple objects | ||
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\prop | \prop | ||
| 56번째 줄: | 56번째 줄: | ||
(ii) cluster monomials are linearly independent | (ii) cluster monomials are linearly independent | ||
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==periodicity conjecture== | ==periodicity conjecture== | ||
| 66번째 줄: | 66번째 줄: | ||
* outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams | * outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams | ||
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==history== | ==history== | ||
| 76번째 줄: | 73번째 줄: | ||
* 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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==related items== | ==related items== | ||
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* [[quiver representations|Quiver representations]] | * [[quiver representations|Quiver representations]] | ||
* [[categorification of quantum groups]] | * [[categorification of quantum groups]] | ||
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==expositions== | ==expositions== | ||
| 120번째 줄: | 91번째 줄: | ||
* [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== | ||
| 130번째 줄: | 99번째 줄: | ||
* 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. | * 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. | ||
* 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. | * 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. | ||
| − | * 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. | + | * 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:cluster algebra]] | [[분류:cluster algebra]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 6월 24일 (월) 13:44 판
introduction
- replace cluster variables by modules
notions
- quiver : oriented graph
- representation of a quiver : collection of vector space and linear maps between them
- homomorphism of 2 quiver representations
- path algebra of a quiver
- given a quiver Q, a path p is a sequence of arrows with some conditions
- path algebra : set of all k-linear combinations of all paths (including e_i's)
- p_1p_2 will correspond to a composition \(p_2\circ p_1\) of two maps (\(U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W\))
- quiver representation is in fact, a representaion of path algebra of a quiver
Caldero-Chapoton formula
- CC(V) =\chi_{V}
monoidal categorification
M : monoidal categorification
M is a monoidal categorification of A if the Grothendieck ring of M is isomorphic to A and if
(i) cluster monomials' of A are the classes of real simple objects of M
(ii) cluster variables' of a (including coefficients) are classes of real prime simple objects
\prop
Suppose that A has a monoidal categorification M and also that each object B in M has unique finite composition series
(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
(i) each cluster variable of a has positivie Laurent expansion with respect to any cluster
(ii) cluster monomials are linearly independent
periodicity conjecture
- outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
history
expositions
- 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
- David Hernandez, Bernard Leclerc , Monoidal categorifications of cluster algebras of type A and D 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.
- Rupel, Dylan. 2010. “On Quantum Analogue of The Caldero-Chapoton Formula”. 1003.2652 (3월 12). doi:doi:10.1093/imrn/rnq192. http://arxiv.org/abs/1003.2652.
- Caldero, Philippe, 와/과Andrei Zelevinsky. 2006. “Laurent expansions in cluster algebras via quiver representations”. math/0604054 (4월 3). http://arxiv.org/abs/math/0604054.
- Caldero, Philippe, 와/과Frederic Chapoton. 2004. “Cluster algebras as Hall algebras of quiver representations”. math/0410187 (10월 7). http://arxiv.org/abs/math/0410187.