Algebras of Non-Local Screenings and Diagonal Nichols Algebras
In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a d...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , |
| 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/123456789/211527 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Algebras of Non-Local Screenings and Diagonal Nichols Algebras. Ilaria Flandoli and Simon D. Lentner. SIGMA 18 (2022), 018, 81 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862725889794506752 |
|---|---|
| author | Flandoli, Ilaria Lentner, Simon D. |
| author_facet | Flandoli, Ilaria Lentner, Simon D. |
| citation_txt | Algebras of Non-Local Screenings and Diagonal Nichols Algebras. Ilaria Flandoli and Simon D. Lentner. SIGMA 18 (2022), 018, 81 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center.
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| first_indexed | 2026-03-21T08:48:00Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211527 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T08:48:00Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Flandoli, Ilaria Lentner, Simon D. 2026-01-05T12:26:00Z 2022 Algebras of Non-Local Screenings and Diagonal Nichols Algebras. Ilaria Flandoli and Simon D. Lentner. SIGMA 18 (2022), 018, 81 pages 1815-0659 2020 Mathematics Subject Classification: 16T05;17B69 arXiv:1911.11040 https://nasplib.isofts.kiev.ua/handle/123456789/211527 https://doi.org/10.3842/SIGMA.2022.018 In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center. IF and SL are partially supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”. Many thanks to Christian Reiher for suggesting the analytic continuation by partial integration in Section 4.3, to Ivan Angiono for explaining to us Proposition 2.20, to Sven Ole Warnaar for answering questions on the -Selberg integral formula, and to the anonymous referees for many helpful comments on the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Algebras of Non-Local Screenings and Diagonal Nichols Algebras Article published earlier |
| spellingShingle | Algebras of Non-Local Screenings and Diagonal Nichols Algebras Flandoli, Ilaria Lentner, Simon D. |
| title | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_full | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_fullStr | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_full_unstemmed | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_short | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_sort | algebras of non-local screenings and diagonal nichols algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211527 |
| work_keys_str_mv | AT flandoliilaria algebrasofnonlocalscreeningsanddiagonalnicholsalgebras AT lentnersimond algebrasofnonlocalscreeningsanddiagonalnicholsalgebras |