On the definition of quantum Heisenberg category

Abstract

We introduce a diagrammatic monoidal category $\mathcal{H}ei{s}_k(z,t)$ 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 $\mathcal{H}ei{s}_0(z,t)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{H}ei{s}_k(z,t)$.

arXiv commentv1: preliminary version; v2: minor corrections, published version; v3: contains corrections of some errors present in the published version

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• Hecke algebras, finite general linear groups, and Heisenberg categorification
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Cited by in corpus (14)

• Frobenius Heisenberg categorification
• The degenerate Heisenberg category and its Grothendieck ring
• Heisenberg and Kac-Moody categorification
• Foundations of Frobenius Heisenberg categories
• Quantum Frobenius Heisenberg categorification
• Unfurling Khovanov-Lauda-Rouquier algebras
• Affinization of monoidal categories
• The quantum isomeric supercategory
• String diagrams and categorification
• Quantum affine wreath algebras
• A basis theorem for the affine Kauffmann category and its cyclotomic quotients
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Frobenius W-algebras and traces of Frobenius Heisenberg categories
• Planar algebras for the Young graph and the Khovanov Heisenberg category