Vol. 14, No. 2, 2020

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On the definition of quantum Heisenberg category

Jonathan Brundan, Alistair Savage and Ben Webster

Vol. 14 (2020), No. 2, 275–321
Abstract

We introduce a diagrammatic monoidal category $\mathsc{ℋ}ei{s}_{k}\left(z,t\right)$ which we call the quantum Heisenberg category; here, $k\in ℤ$ 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 $\mathsc{ℋ}ei{s}_{0}\left(z,t\right)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathsc{ℋ}ei{s}_{k}\left(z,t\right)$.

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