We show that any Abelian module category over the (degenerate or quantum) Heisenberg category satisfying suitable finiteness conditions may be viewed as a 2-representation over a corresponding Kac–Moody 2-category (and vice versa). This gives a way to construct Kac–Moody actions in many representation-theoretic examples which is independent of Rouquier’s original approach via “control by \(K_0\).” As an application, we prove an isomorphism theorem for generalized cyclotomic quotients of these categories, extending the known isomorphism between cyclotomic quotients of type A affine Hecke algebras and quiver Hecke algebras.

中文翻译：

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Heisenberg和Kac-Moody的分类

我们证明，在（简并的或量子的）Heisenberg类别上满足适当有限条件的任何Abelian模件类别都可以视为对应的Kac-Moody 2类别的2表示（反之亦然）。这提供了一种在许多表示理论示例中构造Kac-Moody动作的方法，该方法独立于Rouquier的原始方法，即“通过\（K_0 \）控制”。作为一种应用，我们证明了这些类别的广义环商的一个同构定理，扩展了A型仿射Hecke代数和颤动Hecke代数的环商之间的已知同构。