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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.20633 |
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| _version_ | 1866913629171875840 |
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| author | Elias, Ben Hogancamp, Matthew |
| author_facet | Elias, Ben Hogancamp, Matthew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Drinfeld centralizers and Rouquier complexes Elias, Ben Hogancamp, Matthew Representation Theory Category Theory Geometric Topology 18N25, 18M50, 20C08 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. |
| title | Drinfeld centralizers and Rouquier complexes |
| topic | Representation Theory Category Theory Geometric Topology 18N25, 18M50, 20C08 |
| url | https://arxiv.org/abs/2412.20633 |