Reductions of invariant bi-Poisson structures and locally free actions / Ihor Mykytyuk // Symmetry [Dokument elektroniczny]. – 2021, Vol. 13, Iss. 11, Spec. Iss., 14 s. – Tryb dostępu: https://doi.org/10.3390/sym13112043. ... więcej

Reductions of invariant bi-Poisson structures and locally free actions / Ihor Mykytyuk // Symmetry [Dokument elektroniczny]. – 2021, Vol. 13, Iss. 11, Spec. Iss., 14 s. – Tryb dostępu: https://doi.org/10.3390/sym13112043. – Oznaczenie ref./art.: 2043. – Bibliogr. 18 poz., Abstr. – Special Issue: Symmetry of Hamiltonian Systems: Classical and Quantum Aspects. – doi: 10.3390/sym13112043. – ISSN 2073-8994

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 ... więcej

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.