A general approach to Heisenberg categorification via wreath product algebras

TL;DR

This paper introduces a monoidal category associated with any graded Frobenius superalgebra that acts on wreath product modules and connects to a quantum lattice Heisenberg algebra, generalizing and unifying previous results.

Contribution

It constructs a universal categorification framework for Heisenberg algebras via wreath product algebras, extending known cases and proving an open conjecture.

Findings

Grothendieck group of the category is isomorphic to the Heisenberg algebra

Contains generalizations of the degenerate affine Hecke algebra

Proves an open conjecture of Cautis--Licata

Abstract

We associate a monoidal category $\mathcal{H}_B$, defined in terms of planar diagrams, to any graded Frobenius superalgebra $B$. This category acts naturally on modules over the wreath product algebras associated to $B$. To $B$ we also associate a (quantum) lattice Heisenberg algebra $\mathfrak{h}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of $\mathcal{H}_B$ is isomorphic, as an algebra, to $\mathfrak{h}_B$. For specific choices of Frobenius algebra $B$, we recover existing results, including those of Khovanov and Cautis--Licata. We also prove that certain morphism spaces in the category $\mathcal{H}_B$ contain generalizations of the degenerate affine Hecke algebra. Specializing $B$, this proves an open conjecture of Cautis--Licata.