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What are examples of non-equivariant momentum maps?

Off the top of my hat, I know about the following examples:

  • the action of translations of a symplectic vector space (yielding the Heisenberg group as a central extension),
  • the action of the group of Galilean transformations on $\mathbb R^6$ and
  • certain actions of diffeomorphism groups (e.g. the one constructing the Virasoro group, or in fluid dynamics).

Are there more interesting classes of examples?

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  • $\begingroup$ Are you asking for examples of Hamiltonian actions that are not Poisson (i.e., Hamiltonian actions for which there is no choice of an equivariant momentum mapping), or are you asking for examples of Poisson actions for which there is an 'interesting' momentum mapping that happens not to be equivariant? $\endgroup$
    – Robert Bryant
    Commented May 15, 2023 at 17:18
  • $\begingroup$ I'm mostly looking for examples of Hamiltonian actions whose non-equivariance cocycle is non-trivial in cohomology (i.e. those that cannot be made equivariant by adding a suitable constant). $\endgroup$
    – Tobias Diez
    Commented May 16, 2023 at 8:50
  • $\begingroup$ @TobiasDiez where can I read more about these examples, specially the one related to diffeomorphism groups? $\endgroup$
    – QGravity
    Commented Sep 9, 2023 at 6:00

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