Monoidal categorifications of cluster algebras

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

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

related items

• Quiver representations
• categorification of quantum groups

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.