Heisenberg and Kac-Moody categorification

TL;DR

This paper establishes a correspondence between Abelian modules over Heisenberg categories and 2-representations of Kac-Moody 2-categories, enabling new constructions of Kac-Moody actions and extending isomorphism results in categorification.

Contribution

It introduces a novel framework linking Heisenberg and Kac-Moody categorifications, independent of previous methods, and proves an isomorphism theorem for cyclotomic quotients.

Findings

Provides a new method to construct Kac-Moody actions in representation theory.

Extends the isomorphism between cyclotomic quotients of affine Hecke algebras and quiver Hecke algebras.

Establishes a correspondence between Abelian modules over Heisenberg categories and Kac-Moody 2-representations.

Abstract

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.