Non-kissing and non-crossing complexes for locally gentle algebras
Non-kissing and non-crossing complexes for locally gentle algebras
复制标题
局部温和代数的非接吻和非交叉复合体
DOI：
10.4171/jca/35
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
Pierre
中科院分区：
文献类型：
--
作者：
Yann Palu;Vincent Pilaud;Pierre
从一个局部柔和的边界箭图出发，我们一方面定义了一个单纯的情结，称为不接吻情结。另一方面，我们构造了一个带有边界的有穿孔、有标记、有方向的曲面，并赋予了一对对偶剖分。根据这些几何数据，我们定义了两个单纯复形：手风琴复形和激流复形，这是A.Garver和T.McConville在圆盘情况下的推广工作。我们证明了这三个单纯复形是同构的，并且它们是纯的和薄的。特别是，在它们的方面有一个突变的概念，类似于$\tau$-倾斜突变。在这一过程中，我们还构造了局部柔界箭图的同构类集合与带有边界的穿孔、标记、定向曲面的同构类集合之间的逆双射，并赋予了一对偶剖分。
Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to $\tau$-tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.