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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 2nd 2012
   • (edited Mar 2nd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am experimenting with a notion of _Heisenberg Lie $n$-algebras_, for all $n \in \mathbb{N}$. I have made an experimental note on this [here](http://ncatlab.org/nlab/show/Heisenberg+Lie+algebra#InnplecticGeometry) in the entry _[[Heisenberg Lie algebra]]_. It's explicitly marked as "experimental". If it turns out to be a _bad idea_, I'll remove it again. Please try to shoot it down to see if I can rescue it! :-) I mean, the definition in itself is elementary and very simple. The question is if this is "the right notion" to consider. The reasoning here is: by the arguments as mentioned on the nCafé [here](http://golem.ph.utexas.edu/category/2012/02/prequantization_in_homotopy_ty.html) we may feel sure that Chris Rogers's notion of [[Poisson Lie n-algebra]] is correct. (Not that there were any particular doubts, but the fact that we can derive it from very general abstract homotopy theoretic constructions reinforces belief in it.) But the ordinary Heisenberg Lie algebra is just the sub-Lie algebra of the Poisson Lie algebra on the constant and the linear functions. Therefore it makes sense to look at the sub-Lie $n$-algebra of the Poisson Lie $n$-alhebra on the constant and linear differential forms That's what my experimental definition does.

   I am experimenting with a notion of Heisenberg Lie $n$-algebras, for all $n \in \mathbb{N}$.

   I have made an experimental note on this here in the entry Heisenberg Lie algebra.

   It’s explicitly marked as “experimental”. If it turns out to be a bad idea, I’ll remove it again. Please try to shoot it down to see if I can rescue it! :-)

   I mean, the definition in itself is elementary and very simple. The question is if this is “the right notion” to consider. The reasoning here is:

   by the arguments as mentioned on the nCafé here we may feel sure that Chris Rogers’s notion of Poisson Lie n-algebra is correct. (Not that there were any particular doubts, but the fact that we can derive it from very general abstract homotopy theoretic constructions reinforces belief in it.)

   But the ordinary Heisenberg Lie algebra is just the sub-Lie algebra of the Poisson Lie algebra on the constant and the linear functions. Therefore it makes sense to look at the sub-Lie $n$-algebra of the Poisson Lie $n$-alhebra on the constant and linear differential forms That’s what my experimental definition does.