We show that actions of the odd categorification of sl 2 induce derived superequivalence analogous to those introduced by Chuang and Rouquier. Using Kang, Kashiwara and Oh's action of the odd 2-category on blocks of the cyclotomic affine Hecke-Clifford algebra, our equivalences imply that blocks related by a certain affine Weyl group action are derived equivalent. By recent results of Kleshchev and Livesey, we show this implies Broué's abelian defect conjecture for the modular representations of the spin symmetric group.
Let U be a quantized enveloping algebra. We consider the adjoint action of an sl 2 -subalgebra of U on a subalgebra of U + that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with sl 2 on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara [KK12] for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the categorical action with the monoidal structure, categorifying the notion of derivation on an algebra. As an application of some of our results, we categorify the higher order quantum Serre relations, extending results of Stošić [Sto15] to the non simply-laced case.
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