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

Published online by Cambridge University Press:  01 February 2021

Tobias Dyckerhoff
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146Hamburg, Germany; E-mail:
Gustavo Jasso
Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115Bonn, Germany; E-mail:
Yankι Lekili
Department of Mathematics, King’s College London, Strand, LondonWC2R 2LS, United Kingdom; E-mail:


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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.

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