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Multiplicative preprojective algebras are 2-Calabi–Yau

Daniel Kaplan and Travis Schedler

Vol. 17 (2023), No. 4, 831–883

We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi–Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and hence the Calabi–Yau structure is unique. If the quiver is a cycle, we show that the algebra is a noncommutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the dg versions of these algebras (arising as certain Fukaya categories) are formal. We conjecture that the same properties hold for all non-Dynkin quivers, with respect to any extended Dynkin subquiver (note that the cycle is the type A case). Finally, we prove that multiplicative quiver varieties — for all quivers — are formally locally isomorphic to ordinary quiver varieties. In particular, they are all symplectic singularities (which implies they are normal and have rational Gorenstein singularities). This includes character varieties of Riemann surfaces with punctures and monodromy conditions. We deduce this from a more general statement about 2-Calabi–Yau algebras (following Bocklandt, Galluzzi, and Vaccarino).

multiplicative preprojective algebra, Calabi\kern0.05em–\kern-0.1em Yau algebra, NCCR, Ginzburg dg algebra, wrapped Fukaya category, quiver variety, symplectic singularity
Mathematical Subject Classification
Primary: 16G20, 16S38
Secondary: 16E05, 16E65, 53D30
Received: 15 October 2020
Revised: 29 September 2021
Accepted: 10 June 2022
Published: 2 May 2023
Daniel Kaplan
Hasselt University
Travis Schedler
Department of Mathematics
Imperial College London
South Kensington Campus
United Kingdom

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