A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmül...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211525 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862685956952293376 |
|---|---|
| author | Morand, Kevin |
| author_facet | Morand, Kevin |
| citation_txt | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their ''quasi'' generalisations. Using results due to T. Willwacher and M. Zivković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞-algebra structure non-trivially deforming the ''big bracket'' for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in = 3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in = 2). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.
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| first_indexed | 2026-03-17T12:02:09Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211525 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T12:02:09Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Morand, Kevin 2026-01-05T12:24:58Z 2022 A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages 1815-0659 2020 Mathematics Subject Classification: 53D55; 18G85; 17B62 arXiv:2102.07593 https://nasplib.isofts.kiev.ua/handle/123456789/211525 https://doi.org/10.3842/SIGMA.2022.020 Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their ''quasi'' generalisations. Using results due to T. Willwacher and M. Zivković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞-algebra structure non-trivially deforming the ''big bracket'' for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in = 3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in = 2). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids. The author is grateful to T. Basile, D. Lejay, H. Y. Liao, and P. Xu for discussions as well as to Y. Kosmann–Schwarzbach, S. Merkulov, and T. Willwacher for correspondence. The author would also like to thank J. H. Park for support. Finally, the author is grateful to anonymous referees whose suggestions greatly helped to improve the quality of the present paper. This work was supported by the Brain Pool Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science and ICT (2018H1D3A1A01030137), and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A1A03047877). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids Article published earlier |
| spellingShingle | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids Morand, Kevin |
| title | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids |
| title_full | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids |
| title_fullStr | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids |
| title_full_unstemmed | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids |
| title_short | A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids |
| title_sort | note on multi-oriented graph complexes and deformation quantization of lie bialgebroids |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211525 |
| work_keys_str_mv | AT morandkevin anoteonmultiorientedgraphcomplexesanddeformationquantizationofliebialgebroids AT morandkevin noteonmultiorientedgraphcomplexesanddeformationquantizationofliebialgebroids |