The degenerate Heisenberg category and its Grothendieck ring

Jonathan Brundan; Alistair Savage; Ben Webster

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arXiv · 1812.03255

The degenerate Heisenberg category and its Grothendieck ring

Jonathan Brundan·Alistair Savage·Ben Webster

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Abstract

The degenerate Heisenberg category $\mathcal{H}eis_k$ 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).

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Jonathan Brundan, Alistair Savage, Ben Webster. 2018-12-08. The degenerate Heisenberg category and its Grothendieck ring. https://arxiv.org/abs/1812.03255

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