Drinfeld centralizers and Rouquier complexes

TL;DR

This paper explores the properties of Drinfeld centralizers in monoidal categories and their homotopy categories, providing new lifting lemmas and homotopy coherent results related to Rouquier complexes and braid group elements.

Contribution

It introduces a lifting lemma connecting Drinfeld centralizers in categories and their homotopy categories, and develops homotopy coherent versions involving the $A_{ abla}$-Drinfeld centralizer.

Findings

Proves the centrality of the full twist in the Hecke category.

Establishes properties of half twists and Coxeter braids.

Introduces the $A_{ abla}$-Drinfeld centralizer for homotopy coherence.

Abstract

The Drinfeld centralizer of a monoidal category $\mathcal{A}$ in a bimodule category $\mathcal{M}$ is the category $\mathcal{Z}(\mathcal{A},\mathcal{M})$ of objects in $\mathcal{M}$ for which the left and right actions by objects of $\mathcal{A}$ coincide, naturally. In this paper we study the interplay between Drinfeld centralizers of $\mathcal{A}$ and its homotopy category $\mathcal{K}^b(\mathcal{A})$, culminating with our ``lifting lemma,'' which provides a sufficient condition for an object of $\mathcal{Z}(\mathcal{A}, \mathcal{K}^b(\mathcal{M}))$ to lift to an object of $\mathcal{Z}(\mathcal{K}^b(\mathcal{A}), \mathcal{K}^b(\mathcal{M}))$. The central application of this lifting lemma is a proof of some folklore facts about conjugation by Rouquier complexes in the Hecke category: the centrality of the full twist, and related properties of half twists and Coxeter braids. We also prove stronger, homotopy coherent versions of these statements, stated using the notion of the $A_{\infty}$-Drinfeld centralizer, which we believe is new.