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Feb 20, 2021 at 17:50 comment added მამუკა ჯიბლაძე Thanks! I actually meant to ask about the part that you call "compute in the usual way" but this is also straightforward: $[x,-]$ acts on the PBW basis as $2\frac\partial{\partial y}$ and $[-,y]$ as $2\frac\partial{\partial x}$...
Feb 20, 2021 at 17:14 comment added David E Speyer @მამუკაჯიბლაძე You can decide whether or not it is obvious, I added the details.
Feb 20, 2021 at 17:14 history edited David E Speyer CC BY-SA 4.0
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Feb 20, 2021 at 16:34 comment added მამუკა ჯიბლაძე Is it obvious that the argument about reducing the center to scalars works without invertibility of $[x,y]$?
Feb 20, 2021 at 14:40 history answered David E Speyer CC BY-SA 4.0