Reductions of invariant bi-Poisson structures and locally free actions
Authors:
- Ihor Mykytyuk
Abstract
Let (X, G, ω1 , ω2, {η t}) be a manifold with a bi-Poisson structure {η t} generated by a pair of G-invariant symplectic structures ω1 and ω2, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds (RL i , G i , ωi 1 , ωi 2 , {η t i }), i = 1, 2 such that (1) RL i is a submanifold of an open dense subset X(H) ⊂ X; (2) symplectic structures ωi 1 and ωi 2 , generating a bi-Poisson structure {η t i }, are G i - invariant and coincide with restrictions ω1 |RL i and ω2|RL i ; (3) the canonically defined group G i acts properly and locally freely on RL i ; (4) orbit spaces X(H)/G and RL i/G i are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on X(H) and G i -invariant functions on RL i are isomorphic as Poisson algebras with the bi-Poisson structures {η t} and {η t i } respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group.
- Record ID
- CUTe2131faffdb14a008771bb125b0c9b0e
- Publication categories
- ;
- Author
- Journal series
- Symmetry, ISSN , e-ISSN 2073-8994, Monthly
- Issue year
- 2021
- Vol
- 13
- No
- 11
- Pages
- [1-14]
- Article number
- 2043
- Other elements of collation
- Bibliografia (na s.) - 14; Bibliografia (liczba pozycji) - 18; Oznaczenie streszczenia - Abstr.; Numeracja w czasopiśmie - Vol. 13, Iss. 11, Spec. Iss.
- Substantive notes
- Special Issue: Symmetry of Hamiltonian Systems: Classical and Quantum Aspects
- Keywords in English
- bi-Poisson structure, reduction, proper action
- DOI
- DOI:10.3390/sym13112043 Opening in a new tab
- URL
- https://www.mdpi.com/2073-8994/13/11/2043 Opening in a new tab
- Language
- eng (en) English
- License
- Score (nominal)
- 70
- Additional fields
- Indeksowana w: Web of Science, Scopus
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUTe2131faffdb14a008771bb125b0c9b0e/
- URN
urn:pkr-prod:CUTe2131faffdb14a008771bb125b0c9b0e
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