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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2012
    • (edited Mar 1st 2012)

    I have been expanding and polishing the entry Heisenberg group.

    This had existed in bad shape for quite a while, but now it’s maybe getting into better shape.

    I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that

    • there are different Lie groups for a given Heisenberg Lie algebra,

    • and the appearance of an β€œii” in [q,p]=i[q,p] = i may be all understood as not picking the simply conncted ones of these;

    I also added remarks on the relation to Poisson brackets, and symplectomorphisms.

    In this context: either I am dreaming, or there is a mistake in the Wikipedia entry Poisson bracket - Lie algebra.

    There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2023

    added pointer to:

    • Ernst Binz, Sonja Pods, The geometry of Heisenberg groups β€” With Applications in Signal Theory, Optics, Quantization, and Field Quantization, Mathematical Surveys and Monographs 151, American Mathematical Society (2008) [ams:surv-151]

    (here and elsewhere, such as at Heisenberg Lie algebra)

    diff, v17, current