Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure <(B(1))over bar> of type A, and study relations between an NCCR and crepant resolutions Y and Y+ of <(B(1))over bar>. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions Y and Y+ of <(B(1))over bar> as moduli spaces of representations of the quiver. We also study the Kawamata Namikawa's derived equivalence between crepant resolutions Y and Y+ of <(B(1))over bar> in terms of an NCCR. We also show that the P-twist on the derived category of Y corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama Wemyss's mutations. (C) 2017 Elsevier Inc. All rights reserved.