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