On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories
On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories
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DOI：
10.1093/imrn/rnn029
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发表时间：
2007-09
影响因子：
1
通讯作者：
R. Dehy;B. Keller
中科院分区：
文献类型：
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作者：
R. Dehy;B. Keller
给定一个三角化的2-Calabi-Yau范畴C和一个簇倾斜子范畴T，C的对象X的指数是加范畴T的Grothendieck群的某个元素。本文证明了C的刚性对象由它的指数决定，倾斜簇子范畴T′的不可分解项的指数构成T的Grothendieck群的基，并且如果T和T′通过一个突变相联系，然后关于T和T′的指数通过Fomin和Zelevinsky在他们对簇代数的研究中引入的某种分段线性变换联系起来，系数这使我们能够给出从给定的簇倾斜子范畴T可达的所有刚性对象的指数的组合构造。这些指数与Fomin-Zelevinsky的g向量相一致。
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T′ form a basis of the Grothendieck group of T and that, if T and T′ are related by a mutation, then the indices with respect to T and T′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin-Zelevinsky's g-vectors.