Symmetries of the Space of Linear Symplectic Connections
There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The cr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148602 |
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| Zitieren: | Symmetries of the Space of Linear Symplectic Connections / D.J.F. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 20 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862696481763360768 |
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| author | Fox, D.J.F. |
| author_facet | Fox, D.J.F. |
| citation_txt | Symmetries of the Space of Linear Symplectic Connections / D.J.F. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 20 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.
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| first_indexed | 2025-12-07T16:27:45Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148602 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:27:45Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fox, D.J.F. 2019-02-18T16:29:17Z 2019-02-18T16:29:17Z 2017 Symmetries of the Space of Linear Symplectic Connections / D.J.F. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D20; 53D05; 53C05; 17B99 DOI:10.3842/SIGMA.2017.002 https://nasplib.isofts.kiev.ua/handle/123456789/148602 There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term. I thank the anonymous referees for their thoughtful criticisms and detailed
 corrections which helped improve the article, particularly the exposition in Section 6. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Symmetries of the Space of Linear Symplectic Connections Article published earlier |
| spellingShingle | Symmetries of the Space of Linear Symplectic Connections Fox, D.J.F. |
| title | Symmetries of the Space of Linear Symplectic Connections |
| title_full | Symmetries of the Space of Linear Symplectic Connections |
| title_fullStr | Symmetries of the Space of Linear Symplectic Connections |
| title_full_unstemmed | Symmetries of the Space of Linear Symplectic Connections |
| title_short | Symmetries of the Space of Linear Symplectic Connections |
| title_sort | symmetries of the space of linear symplectic connections |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148602 |
| work_keys_str_mv | AT foxdjf symmetriesofthespaceoflinearsymplecticconnections |