"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 잔글 (Pythagoras0 사용자가 Categorifications of cluster algebras 문서를 Monoidal categorifications of cluster algebras 문서로 옮겼습니다.) |
(차이 없음)
| |
2013년 10월 20일 (일) 10:11 판
introduction
- replace cluster variables by modules
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
- cluster monomials' of A are the classes of real simple objects of M
- cluster variables' of a (including coefficients) are 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
periodicity conjecture
- outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
history
expositions
- collection of notes http://mathserver.neu.edu/~sstella/seminars/cac-2011.shtml
- 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.