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added a bit to Heisenberg Lie algebra.
Mostly, I wrote a section Relation to Poisson algebra with a discussion of how the Heisenberg algebra naturally sits inside the Lie algebra underlying the Poisson algebra.
Hm, right after saving this, the Lab went down…
But, there seems no entry Heisenberg algebra.
Darn. Sorry. I meant to point to Heisenberg Lie algebra.
[edit: I have made Heisenberg algebra a redirect now. ]
Oh, now I see I even contributed to it (version 1)… :) It is growing nicely :)
Okay, thanks!
I added a paragraph on the relation of Heisenberg Lie algebra to certain construction of associative algebras, so called Heisenberg double, which generalized the Weyl algebra. In fact there are several associative algebras related to Heisenberg algebra. One can look simply at its universal enveloping algebra. Then one can extend it by a central element and/or with the “number operator”. One obtains variants called Heisenbeg-Weyl algebra, Weyl algebra, CCR-algebra and alike, which for various authors mean the same or a different thing, depending if the central element is 1 or not, if the number operator $N$ is a separate generator or not and so on. I do not know how will $n$Lab solve this terminologically difficult point.
Thanks. I have split that up into two new subsections: Relation to Weyl algebra and Relation to Heisenberg double.
I am not sure if I understood your “if another central ement is added” correctly:
let’s see: the Weyl algebra is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra obtained by identifying the central elements with multiples of the identity (hence by removing a central element).
Agreed?
I agree; Weyl algebra is smaller. Heisenberg double generalizes Weyl algebra. CCR algebra (canonical commutation relation algebra, or boson oscillator algebra) is essentially the same as the Weyl algebra, possibly in infinite dimension, but the noncentral number operator is taken as central, however $N-a^\dagger a$ is central, so the question is if it is set to zero or not.
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