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

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==introduction</h5>
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==introduction==
  
 
* replace cluster variables by modules
 
* replace cluster variables by modules
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==notions</h5>
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==notions==
  
 
* quiver : oriented graph
 
* quiver : oriented graph
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==Caldero-Chapoton formula</h5>
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==Caldero-Chapoton formula==
  
 
* CC(V) =\chi_{V}
 
* CC(V) =\chi_{V}
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==monoidal categorification</h5>
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==monoidal categorification==
  
 
M : monoidal categorification
 
M : monoidal categorification
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==periodicity conjecture</h5>
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==periodicity conjecture==
  
 
* 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</h5>
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==history==
  
 
* 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</h5>
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==related items==
  
 
* [[quiver representations|Quiver representations]]
 
* [[quiver representations|Quiver representations]]
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* http://en.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/
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==books</h5>
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==books==
  
 
 
 
 
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==expositions</h5>
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==expositions==
  
 
* Leclerc, Bernard. 2011. “Quantum loop algebras, quiver varieties, and cluster algebras”. <em>1102.1076</em> (2월 5). http://arxiv.org/abs/1102.1076.
 
* Leclerc, Bernard. 2011. “Quantum loop algebras, quiver varieties, and cluster algebras”. <em>1102.1076</em> (2월 5). http://arxiv.org/abs/1102.1076.
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
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* David Hernandez, Bernard Leclerc , Monoidal categorifications of cluster algebras of type A and D http://arxiv.org/abs/1207.3401
 
* David Hernandez, Bernard Leclerc , Monoidal categorifications of cluster algebras of type A and D http://arxiv.org/abs/1207.3401
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==question and answers(Math Overflow)</h5>
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==question and answers(Math Overflow)==
  
 
* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
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==blogs</h5>
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==blogs==
  
 
*  구글 블로그 검색<br>
 
*  구글 블로그 검색<br>
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==experts on the field</h5>
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==experts on the field==
  
 
* http://arxiv.org/
 
* http://arxiv.org/
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==links</h5>
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==links==
  
 
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]

2012년 10월 28일 (일) 14:38 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

 

 

related items

 

 

encyclopedia==    

books

 

 

 

expositions

 

 

articles==    

question and answers(Math Overflow)

 

blogs

 

 

experts on the field

 

 

links

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