On the definition of quantum Heisenberg category

TL;DR

This paper introduces the quantum Heisenberg category, a diagrammatic monoidal category with parameters, unifying and extending previous constructions related to Khovanov's Heisenberg category and HOMFLY-PT skein category.

Contribution

It defines a new quantum Heisenberg category with parameters, generalizing known categories and establishing a basis theorem for its morphism spaces.

Findings

Special cases recover known categories like Khovanov's Heisenberg and HOMFLY-PT skein categories.

Introduces a basis theorem for morphism spaces in the quantum Heisenberg category.

Unifies various known quantum categories under a common framework.

Abstract

We introduce a diagrammatic monoidal category $\mathcal{H}eis_k(z,t)$ which we call the quantum Heisenberg category, here, $k \in \mathbb{Z}$ is "central charge" and $z$ and $t$ are invertible parameters. Special cases were known before: for central charge $k=-1$ and parameters $z = q-q^{-1}$ and $t = -z^{-1}$ our quantum Heisenberg category may be obtained from the deformed version of Khovanov's Heisenberg category introduced by Licata and the second author by inverting its polynomial generator, while $\mathcal{H}eis_0(z,t)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{H}eis_k(z,t)$.