The degenerate Heisenberg category and its Grothendieck ring

Abstract

The degenerate Heisenberg category $\mathcal{H}ei{s}_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain $\mathbb{Z}$-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge $-1$. We prove this conjecture and extend it to arbitrary central charge $k\in \mathbb{Z}$. We also explain how to categorify the comultiplication (generically).

arXiv commentv3: small corrections based on a referee report

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Cited by in corpus (5)

• On the definition of quantum Heisenberg category
• Frobenius Heisenberg categorification
• Foundations of Frobenius Heisenberg categories
• String diagrams and categorification
• Embedding Deligne's category $\underline{\mathrm{Re}}\mathrm{p}(S_t)$ in the Heisenberg category