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The degenerate Heisenberg category and its Grothendieck ring

Jonathan Brundan, Alistair Savage, Ben Webster

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).

The degenerate Heisenberg category and its Grothendieck ring

Abstract

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

Paper Structure

This paper contains 8 sections, 35 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. A general result about Grothendieck groups
  3. Categorification of symmetric functions
  4. The degenerate affine Hecke category
  5. The degenerate Heisenberg category
  6. A new proof of the basis theorem
  7. Proofs of Theorems \ref{['t1']} and \ref{['t2']}
  8. Proof of Theorem \ref{['t3']}

Key Result

Theorem 1.1

There is a ring isomorphism $\gamma_k:\mathrm{Heis}_k \stackrel{\sim}{\rightarrow} K_0(\operatorname{Kar}({\mathcal{H}eis}_k))$ such that $s_\lambda^{\pm}\mapsto [S_\lambda^{\pm}]$ for each $\lambda \in \mathcal{P}$. In particular, $h_n^{\pm} \mapsto [H_n^{\pm}]$ and $e_n^{\pm} \mapsto [E_n^{\pm}]$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 65 more