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I have been expanding and polishing the entry Heisenberg group.
This had existed in bad shape for quite a while, but now itβs 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$i$β in $[q,p] = i$ 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.
added pointer to:
(here and elsewhere, such as at Heisenberg Lie algebra)
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