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Action of the Witt algebra on categorified quantum groups

Jernej Grlj, Aaron D. Lauda

Abstract

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

Action of the Witt algebra on categorified quantum groups

Abstract

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on -foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

Paper Structure

This paper contains 12 sections, 12 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric functions
  4. Witt algebra
  5. The 2-category $\mathcal{U}_Q(\mathfrak{g})$
  6. Symmetric functions
  7. Action of the Witt algebra on $\mathcal{U}_Q$
  8. Defining action of the Witt algebra
  9. Proof of Theorem \ref{['main']}
  10. Specialization to $\mathfrak{sl}_2$ and $\mathcal{U}_Q(\mathfrak{sl}_2)$
  11. Relation to Witt algebra action on foams
  12. Witt actions and trace decategorifications

Key Result

Lemma 2.2

There is an injective map of Lie algebras $\iota \colon \mathfrak{sl}_2 \rightarrow {\mathfrak{W}}$ given by: whose image is in ${{\mathfrak{W}}_{-1}^{\infty}}$. This is a maximal finite-dimensional subalgebra of ${{\mathfrak{W}}_{-1}^{\infty}}$.

Figures (1)

  • Figure :

Theorems & Definitions (28)

  • Definition 2.1
  • Lemma 2.2: Lemma 3.6 qi2024symmetriesmathfrakglnfoams
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 18 more