Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds
We develop a reduction scheme for the ∞-algebra of observables on a premultisymplectic manifold (, ) in the presence of a compatible Lie algebra action ↷ and a subset ⊂ . This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212359 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds. Casey Blacker, Antonio Michele Miti and Leonid Ryvkin. SIGMA 20 (2024), 061, 47 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862626368124092416 |
|---|---|
| author | Blacker, Casey Miti, Antonio Michele Ryvkin, Leonid |
| author_facet | Blacker, Casey Miti, Antonio Michele Ryvkin, Leonid |
| citation_txt | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds. Casey Blacker, Antonio Michele Miti and Leonid Ryvkin. SIGMA 20 (2024), 061, 47 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We develop a reduction scheme for the ∞-algebra of observables on a premultisymplectic manifold (, ) in the presence of a compatible Lie algebra action ↷ and a subset ⊂ . This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization.
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| first_indexed | 2026-03-14T15:44:39Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212359 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T15:44:39Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Blacker, Casey Miti, Antonio Michele Ryvkin, Leonid 2026-02-05T09:57:08Z 2024 Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds. Casey Blacker, Antonio Michele Miti and Leonid Ryvkin. SIGMA 20 (2024), 061, 47 pages 1815-0659 2020 Mathematics Subject Classification: 53D05; 53D20; 70S05; 70S10 arXiv:2206.03137 https://nasplib.isofts.kiev.ua/handle/123456789/212359 https://doi.org/10.3842/SIGMA.2024.061 We develop a reduction scheme for the ∞-algebra of observables on a premultisymplectic manifold (, ) in the presence of a compatible Lie algebra action ↷ and a subset ⊂ . This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization. C.B. would like to acknowledge the support of the Leonhard Euler International Mathematical Institute in Saint Petersburg, the Saint Petersburg State University, and the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2022-287. A.M.M. thanks the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the grant agreement no. 101034324 and has been partially supported by the Italian Group for Algebraic and Geometric Structures and their Application (GNSAGA–INdAM). L.R. is supported by the CNRS project GraNum and by the RTG2491. The authors thank Janina Bernardy, Christian Blohmann, Luca Vitagliano, and Marco Zambon for helpful and motivating conversations, and Tatyana Barron for indicating that the premultisymplectic form ω needs only be g-invariant, and not reducible, for the reduced space of observables to be defined. The authors would also like to thank the anonymous referees for their helpful and insightful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds Article published earlier |
| spellingShingle | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds Blacker, Casey Miti, Antonio Michele Ryvkin, Leonid |
| title | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds |
| title_full | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds |
| title_fullStr | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds |
| title_full_unstemmed | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds |
| title_short | Reduction of ∞-Algebras of Observables on Multisymplectic Manifolds |
| title_sort | reduction of ∞-algebras of observables on multisymplectic manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212359 |
| work_keys_str_mv | AT blackercasey reductionofalgebrasofobservablesonmultisymplecticmanifolds AT mitiantoniomichele reductionofalgebrasofobservablesonmultisymplecticmanifolds AT ryvkinleonid reductionofalgebrasofobservablesonmultisymplecticmanifolds |