"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
* replace cluster variables by modules of quantum groups | * replace cluster variables by modules of quantum groups | ||
| − | * [[ | + | * motivation comes from [[Positivity conjecture on cluster algebras]] |
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| − | * | + | ==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]] | ||
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==monoidal categorification== | ==monoidal categorification== | ||
| − | * M : monoidal | + | * $A$ : cluster algebra |
| − | * M is a monoidal categorification of A if the Grothendieck ring of M is isomorphic to A | + | * $M$ : monoidal categorify |
| − | # cluster monomials | + | * $M$ 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=== | ===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) | + | * 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 | * 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 | ||
# cluster monomials are linearly independent | # cluster monomials are linearly independent | ||
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| + | ==Caldero-Chapoton formula== | ||
| + | * $CC(V) =\chi_{V}$ | ||
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2013년 10월 21일 (월) 07:00 판
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
Caldero-Chapoton formula
- $CC(V) =\chi_{V}$
periodicity conjecture
- outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
history
- Quiver representations
- categorification of quantum groups
- 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
- 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.