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)‾ of type A, and study relations between an NCCR and crepant resolutions Y and Y+ of B(1)‾. 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)‾ 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)‾ 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.