Relating Stated Skein Algebras and Internal Skein Algebras
We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211626 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862666028735004672 |
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| author | Haïoun, Benjamin |
| author_facet | Haïoun, Benjamin |
| citation_txt | Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges, and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edge internal skein algebras using excision properties of skein categories, and agree with excision properties of stated skein algebras when = q²(₂)-modᶠⁱⁿ. Our proofs are mostly based on skein theory, and we do not require the reader to be familiar with the formalism of higher categories.
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| first_indexed | 2026-03-16T09:28:56Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211626 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T09:28:56Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Haïoun, Benjamin 2026-01-07T13:40:29Z 2022 Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages 1815-0659 2020 Mathematics Subject Classification: 57K16; 18M15 arXiv:2104.13848 https://nasplib.isofts.kiev.ua/handle/123456789/211626 https://doi.org/10.3842/SIGMA.2022.042 We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges, and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edge internal skein algebras using excision properties of skein categories, and agree with excision properties of stated skein algebras when = q²(₂)-modᶠⁱⁿ. Our proofs are mostly based on skein theory, and we do not require the reader to be familiar with the formalism of higher categories. I would like to thank my three advisors: Francesco Costantino for his guidance throughout the discovery of this subject and the editing of this article; Joan Bellier-Milles for all the time he spent on explanations, and David Jordan for his very helpful conversations and comments. I am grateful to Patrick Kinnear for his remarks and advice. I would also like to thank the anonymous referees for their exceptionally detailed feedback. This research took place in the Institut Mathématique de Toulouse and was supported by the École Normale Supérieure de Lyon. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Relating Stated Skein Algebras and Internal Skein Algebras Article published earlier |
| spellingShingle | Relating Stated Skein Algebras and Internal Skein Algebras Haïoun, Benjamin |
| title | Relating Stated Skein Algebras and Internal Skein Algebras |
| title_full | Relating Stated Skein Algebras and Internal Skein Algebras |
| title_fullStr | Relating Stated Skein Algebras and Internal Skein Algebras |
| title_full_unstemmed | Relating Stated Skein Algebras and Internal Skein Algebras |
| title_short | Relating Stated Skein Algebras and Internal Skein Algebras |
| title_sort | relating stated skein algebras and internal skein algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211626 |
| work_keys_str_mv | AT haiounbenjamin relatingstatedskeinalgebrasandinternalskeinalgebras |