01/11 2008, 星期五
Brief Summary on quiver interpretation on Cluster algebra
I recognized that recent researches on Cluster algebras and representation theory are mostly linked with the methods arising from geometry and other field, such as Cluster algebra and Ringel Hall algebra, the coordinate rings of double Bruhat cells and Cluster algebra, relation between cluster algebra and higher Techimuller space, Cluster algebras and representation of preprojective algebra, semicanonical bases of Luzstig, etc. I am quite interested in part of these researches, especially the Ringel Hall algebra and Cluster algebra. I recalled Claus Michael Ringel ever used Hall algebra to realize positive part of Quantum Groups and J.Xiao realized negative part in the similar way. A natural question arose motivated by these works. Can we realize a cluster algebra of finite type as a “Hall algebra” of the corresponding Cluster category? Caldero and Chapoton obtained a cluster variable formula which gave an explicit expression for the cluster variable associated with a positive root corresponding to an indecomposable module. Then B.Keller gave a natural basis for Cluster algebra labeled by the set of exceptional objects of corresponding Cluster category. Recently, J.Xiao deduced projective version of Green’s formula and applied it to prove that Caldero-Keller’s multiplication formula holds for acyclic cluster algebra of arbitrary type. Now, I am reading and checking this paper. Meanwhile, Buan, Marsh and Reiten gave a quiver interpretation of mutation between Cluster algebra, a one to one corresponding between tilting seeds and initial seeds for an acyclic Cluster algebra was derived. Thus, it seems that Cluster category is a successful model for Cluster algebra and back in turn, Cluster algebra is combinatorial invariant of Cluster category which was proved by B.Keller. On learning these exciting results, I am anxious to enter this field and decide to choose Cluster algebra and related topics for my major research area. As mentioned above, Cluster algebra is invariant of Cluster categories. In another view, we can obtain this result by categorifying Cluster algebra, isomorphism between two algebras becomes triangulated equivalence between two Cluster categories. Maybe another hopeful research direction is generalizing Cluster category to higher dimension for professor Iyama developed higher dimensional Auslander Reiten theory, it gave some interesting combinatorial structures for Higher Auslander Reiten quiver.
很久没来了，恩，你这里的东西我都看不懂，呵呵