This is the first of four articles studying some slight generalisations H n m of Khovanov's diagram algebra, as well as quasi-hereditary covers K n m of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions K ∞ m , K ±∞
This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined "projective functors" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.
Abstract. We prove that integral blocks of parabolic category O associated to the subalgebra gl m (C) ⊕ gl n (C) of gl m+n (C) are Morita equivalent to quasihereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A.
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