The symplectic geometry of higher Auslander algebras: Symmetric products of disks

• Published in: Forum of mathematics. Sigma Vol. 9
• Main Authors: Dyckerhoff, Tobias, Jasso, Gustavo, Lekili, Yankι
• Format: Journal Article
• Language: English
• Published: Cambridge University Press 01.01.2021
• Subjects: 16G70; 19D99; 53D37
• ISSN: 2050-5094; 2050-5094
• DOI: 10.1017/fms.2021.2

We show that the perfect derived categories of Iyama’s d -dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d -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 the d -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 the d -fold symmetric product of the disk organise into a paracyclic object equivalent to the d -dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d -fold delooping of the connective algebraic K -theory space of the ring of coefficients.