For a symplectic manifold, the symplectomorphism group
Analogous constructions apply when symplectic manifolds are generalized to n-plectic infinity-groupoids: for an n-plectic manifold, and -plectomorphism is a diffeomorphism that preserves the -plectic form .
A further subgroup is that of Hamiltonian symplectomorphisms. The group extension of that whose elements are pairs consisting of a Hamiltonian diffeomorphism and a choice of Hamiltonian for this is the quantomorphism group.
higher and integrated Kostant-Souriau extensions
| | higher prequantum geometry | cohesive ∞-group | Hamiltonian symplectomorphism ∞-group | moduli ∞-stack of -flat ∞-connections on | quantomorphism ∞-group | 1 | symplectic geometry | Lie algebra | Hamiltonian vector fields | real numbers | Hamiltonians under Poisson bracket | | 1 | | Lie group | Hamiltonian symplectomorphism group | circle group | quantomorphism group | | 2 | 2-plectic geometry | Lie 2-algebra | Hamiltonian vector fields | line Lie 2-algebra | Poisson Lie 2-algebra | | 2 | | Lie 2-group | Hamiltonian 2-plectomorphisms | circle 2-group | quantomorphism 2-group | | | n-plectic geometry | Lie n-algebra | Hamiltonian vector fields | line Lie n-algebra | Poisson Lie n-algebra | | | | smooth n-group | Hamiltonian n-plectomorphisms | circle n-group | quantomorphism n-group |
(extension are listed for sufficiently connected )