Monoidal categorifications of cluster algebras

introduction

• replace cluster variables by modules of quantum groups
• Kirillov-Reshetikhin (KR) modules of Quantum affine algebra

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

1. cluster monomials' of A are the classes of real simple objects of M
2. 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

1. each cluster variable of a has positivie Laurent expansion with respect to any cluster
2. cluster monomials are linearly independent

periodicity conjecture

• outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Quiver representations
• categorification of quantum groups

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxbXM4aFBiRWotdk0/edit?usp=drivesdk

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.