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The symplectic geometry of higher Auslander algebras: Symmetric products of disks
Forum of Mathematics, Sigma ( IF 1.2 ) Pub Date : 2021-02-01 , DOI: 10.1017/fms.2021.2 Tobias Dyckerhoff , Gustavo Jasso , Yankι Lekili
Forum of Mathematics, Sigma ( IF 1.2 ) Pub Date : 2021-02-01 , DOI: 10.1017/fms.2021.2 Tobias Dyckerhoff , Gustavo Jasso , Yankι Lekili
We show that the perfect derived categories of Iyama’sd -dimensional Auslander algebras of type${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of thed -fold symmetric product of the$2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to thed -fold symmetric product of the disk and those of its$(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to thed -fold symmetric product of the disk organise into a paracyclic object equivalent to thed -dimensional Waldhausen$\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides thed -fold delooping of the connective algebraicK -theory space of the ring of coefficients.
中文翻译:
高等 Auslander 代数的辛几何:圆盘的对称积
我们证明了 Iyama 的完美派生类别d 维 Auslander 类型的代数${\mathbb {A}}$ 相当于部分包裹的深谷类别d - 的折叠对称乘积$2$ 维单位圆盘,在其边界上具有有限多个停靠点。此外,我们观察到 Koszul 对偶性在与d - 盘的对称乘积和它的$(nd)$ -折叠对称产品;这一观察导致了贝克特定理的辛证明,该定理涉及在相应的较高 Auslander 类型代数之间导出的 Morita 等价性${\mathbb {A}}$ . 作为我们结果的副产品,我们推断部分包裹的 Fukaya 类别与d - 圆盘的折叠对称积组织成一个等效于d 维瓦尔德豪森$\text {S}_{\bullet }$ -构造,一个单纯的空间,其几何实现提供了d - 连接代数的折叠去环ķ - 系数环的理论空间。
更新日期:2021-02-01
中文翻译:
高等 Auslander 代数的辛几何:圆盘的对称积
我们证明了 Iyama 的完美派生类别