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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 30th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexunder "Relation to the Weyl algebra" ([here](https://ncatlab.org/nlab/show/Heisenberg+Lie+algebra#RelationToTheWeylAlgebra)) 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 $\hbar$ 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. <a href="https://ncatlab.org/nlab/revision/diff/Heisenberg+Lie+algebra/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Heisenberg+Lie+algebra/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Heisenberg+Lie+algebra">current</a>

   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 $\hbar$ 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.

   diff, v12, current