Heisenberg and Kac–Moody categorification

Abstract

We show that any Abelian module category over the (degenerate or quantum) Heisenberg category satisfying suitable finiteness conditions may be viewed as a 2representation 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.

The nil-Brauer category

“…To establish additional useful relations, we adopt a generating function formalism which is a slight refinement of the setup introduced in [8]. We denote the rth power of • under vertical composition simply by labeling the dot with r. More generally, given a polynomial by attaching what we call a pin to the dot, labeling the node at the head of the pin by f (x):…”

A new approach to the representation theory of the partition category

“…It would be hard to formulate this without the aid of generating functions. Heis, and puq is the formal Laurent series from [BSW,(3.13)]. Under the embedding of APar into Heis, we have that…”

“…The following bubble slide relations hold in APar : The two equations are equivalent, so we just prove the first one. When working with Heis, we adopt the notation of[BSW, §3.1]: an open dot labelled by x r means the rth power of the open dot in…”