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

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<h5>monoidal categorification</h5>
 
<h5>monoidal categorification</h5>
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M : monoidal categorification
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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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(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
 +
 +
 
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\prop
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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
 +
 +
(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
 +
 +
 
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 +
 
  
 
 
 
 
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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.<br>  <br>
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* 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.
 
* 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.
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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.<br>  <br>
 
* http://www.ams.org/mathscinet
 
* http://www.ams.org/mathscinet
 
* http://www.zentralblatt-math.org/zmath/en/
 
* http://www.zentralblatt-math.org/zmath/en/

2011년 4월 13일 (수) 05:33 판

introduction
  • replace cluster variables by modules

 

 

notions
  • quiver : oriented graph
  • represetation 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

 

 

finite type quiver classfication
  • quiver has finite type of there are finitely many indecomposables

 

 

\thm (Gabriel)

A connected quiver Q has finite type iff corresponding graph is Dynking diagram (A,D,E)

 

 

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

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions

 

 

articles

 

 

question and answers(Math Overflow)

 

blogs

 

 

experts on the field

 

 

links