On the definition of quantum Heisenberg category

Abstract

We introduce a diagrammatic monoidal category $\mathcal{\mathscr{H}}ei{s}_k\left(z,t\right)$ 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 Savage by inverting its polynomial generator, while $\mathcal{\mathscr{H}}ei{s}_0\left(z,t\right)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{\mathscr{H}}ei{s}_k\left(z,t\right)$.