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under “Relation to the Weyl algebra” (here) it used to say (I wrote this, some time ago) without qualification that the universal envelope of the Heisenberg algeba becomes the Weyl algebra only after identifying the extra central generator with the unit in the ground field.
But this depends on convention: If the Weyl algebra is regarded as the formal deformation quantization of the given symplectic vector space, then the central element is the formal parameter of the deformation and not identified with the unit.
I have adjusted the wording and added a couple of (so far somewhat random) references whose authors regard the situation in this second sense.
The same discussion should be had at Weyl algebra and under “Examples” at universal enveloping algebra. Maybe it is worth splitting this example off as a separate page and re-!include
it in these entries.
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