Monoidal categorifications of cluster algebras

introduction

• replace cluster variables by modules of quantum groups
• motivation comes from Positivity conjecture on cluster algebras

main results

• Hernandez-Leclerc and Nakajima categorified cluster algebras of finite type using Kirillov-Reshetikhin (KR) modules of Quantum affine algebra

monoidal categorification

• $A$ : cluster algebra
• $M$ : monoidal categorify
• $M$ is a monoidal categorification of $A$ if the Grothendieck ring $K_0(M)$ of $M$ is isomorphic to $A$ which induces bijection between

1. cluster monomials of $A$
2. the classes of real simple objects of $M$ where $V$ is real if $V\otimes V$ is simple

• cluster variables of $A$ (including coefficients) corresponds to 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

Caldero-Chapoton formula

• $CC(V) =\chi_{V}$

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
• Coordinate ring of maximal unipotent subgroup

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.