2010/06/01 by Gregoire Dupont, Dupont, Gregoire
Mathematics · #13F60 #16G20 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:13F60 #msc:16G20
paper · pdf · doi:10.48550/arxiv.1006.0166
20 pages. This is an adaptation of the first part of the preprint arXiv:0811.2909. To appear in the Journal of Pure and Applied Algebra
arxiv created 2010/06/01 · arxiv updated 2010/06/02
Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra \mathcal A(Q). We prove that the set \mathcal G(Q) of generic variables contains naturally the set \mathcal M(Q) of cluster monomials in \mathcal A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set \mathcal G(Q) is a \mathbb Z-basis of \mathcal A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.