Frobenius Heisenberg categorification

TL;DR

This paper introduces a unified framework for categorifying lattice Heisenberg algebras using graded Frobenius superalgebras, generalizing and improving upon existing Heisenberg categories with new presentations and variants.

Contribution

It constructs a broad class of Heisenberg categorifications associated with graded Frobenius superalgebras, unifying previous categories and providing more efficient presentations.

Findings

Recover many known Heisenberg categories as special cases

Provide new, more efficient presentations of existing categories

Introduce new versions of affine oriented Brauer categories

Abstract

We associate a graded monoidal supercategory $\mathcal{H}\mathit{eis}_{F,k}$ to every graded Frobenius superalgebra $F$ and integer $k$. These categories, which categorify a broad range of lattice Heisenberg algebras, recover many previously defined Heisenberg categories as special cases. In this way, the categories $\mathcal{H}\mathit{eis}_{F,k}$ serve as a unifying and generalizing framework for Heisenberg categorification. Even in the case of previously defined Heisenberg categories, we obtain new, more efficient, presentations of these categories, based on an approach of Brundan. When $k=0$, our construction yields new versions of the affine oriented Brauer category depending on a graded Frobenius superalgebra.