"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
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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 | ||
2012년 10월 28일 (일) 16:45 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
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
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.
- http://www.ams.org/mathscinet
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- http://arxiv.org/
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- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://mathoverflow.net/search?q=
- http://math.stackexchange.com/search?q=
- http://math.stackexchange.com/search?q=
blogs
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- http://ncatlab.org/nlab/show/HomePage
experts on the field