Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Dranowski, Anne Guo, Meng Lauda, Aaron Manion, Andrew |
| author_facet | Dranowski, Anne Guo, Meng Lauda, Aaron Manion, Andrew |
| contents | Leveraging skew Howe duality, we show that Lawson-Lipshitz-Sarkar's spectrification of Khovanov's arc algebra gives rise to 2-representations of categorified quantum groups over $\mathbb{F}_2$ that we call spectral 2-representations. These spectral 2-representations take values in the homotopy category of spectral bimodules over spectral categories. We view this as a step toward a higher representation theoretic interpretation of spectral enhancements in link homology. A technical innovation in our work is a streamlined approach to spectrifying arc algebras, using a set of canonical cobordisms that we call frames, that may be of independent interest. As a step towards extending these spectral 2-representations to integer coefficients, we also work in the $\mathfrak{gl}_2$ setting and lift the Blanchet-Khovanov algebra to a multifunctor into a multicategory version of Sarkar-Scaduto-Stoffregen's signed Burnside category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras Dranowski, Anne Guo, Meng Lauda, Aaron Manion, Andrew Quantum Algebra Algebraic Topology Geometric Topology Representation Theory 57K18, 55P43 (primary), 17B37, 18N25 (secondary) Leveraging skew Howe duality, we show that Lawson-Lipshitz-Sarkar's spectrification of Khovanov's arc algebra gives rise to 2-representations of categorified quantum groups over $\mathbb{F}_2$ that we call spectral 2-representations. These spectral 2-representations take values in the homotopy category of spectral bimodules over spectral categories. We view this as a step toward a higher representation theoretic interpretation of spectral enhancements in link homology. A technical innovation in our work is a streamlined approach to spectrifying arc algebras, using a set of canonical cobordisms that we call frames, that may be of independent interest. As a step towards extending these spectral 2-representations to integer coefficients, we also work in the $\mathfrak{gl}_2$ setting and lift the Blanchet-Khovanov algebra to a multifunctor into a multicategory version of Sarkar-Scaduto-Stoffregen's signed Burnside category. |
| title | Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras |
| topic | Quantum Algebra Algebraic Topology Geometric Topology Representation Theory 57K18, 55P43 (primary), 17B37, 18N25 (secondary) |
| url | https://arxiv.org/abs/2402.11368 |