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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • (edited Mar 1st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have been expanding and polishing the entry _[[Heisenberg group]]_. This had existed in bad shape for quite a while, but now it's maybe getting into better shape. I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that * there are different Lie groups for a given Heisenberg Lie algebra, * and the appearance of an "$i$" in $[q,p] = i$ may be all understood as not picking the simply conncted ones of these; I also added remarks on the relation to Poisson brackets, and symplectomorphisms. In this context: either I am dreaming, or there is a mistake in the Wikipedia entry _[Poisson bracket - Lie algebra](http://en.wikipedia.org/wiki/Poisson_bracket#Lie_algebra)_. There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.

   I have been expanding and polishing the entry Heisenberg group.

   This had existed in bad shape for quite a while, but now it’s maybe getting into better shape.

   I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that

   • there are different Lie groups for a given Heisenberg Lie algebra,

   • and the appearance of an “$i$” in $[q,p] = i$ may be all understood as not picking the simply conncted ones of these;

   I also added remarks on the relation to Poisson brackets, and symplectomorphisms.

   In this context: either I am dreaming, or there is a mistake in the Wikipedia entry Poisson bracket - Lie algebra.

   There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Ernst Binz]], Sonja Pods, *The geometry of Heisenberg groups --- With Applications in Signal Theory, Optics, Quantization, and Field Quantization*, Mathematical Surveys and Monographs **151**, American Mathematical Society (2008) &lbrack;[ams:surv-151](https://bookstore.ams.org/surv-151)&rbrack; (here and elsewhere, such as at *[[Heisenberg Lie algebra]]*) <a href="https://ncatlab.org/nlab/revision/diff/Heisenberg+group/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/Heisenberg+group/17">v17</a>, <a href="https://ncatlab.org/nlab/show/Heisenberg+group">current</a>

   added pointer to:

   • Ernst Binz, Sonja Pods, The geometry of Heisenberg groups — With Applications in Signal Theory, Optics, Quantization, and Field Quantization, Mathematical Surveys and Monographs 151, American Mathematical Society (2008) [ams:surv-151]

   (here and elsewhere, such as at Heisenberg Lie algebra)

   diff, v17, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 2nd 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a couple more references on the "integer Heisenberg group": * Roman Budylin: *Conjugacy classes in discrete Heisenberg groups*, Sbornik: Mathematics **205** 8 (2014) 1069–1079 &lbrack;[arXiv:1405.5499](https://arxiv.org/abs/1405.5499), [doi:10.1070/SM2014v205n08ABEH004410](https://doi.org/10.1070/SM2014v205n08ABEH004410)&rbrack; * Jayadev S. Athreya, Ioannis Konstantoulas: *Lattice deformations in the Heisenberg group*, Groups, Geometry and Dynamics **14** 3 (2020) 1007–1022 &lbrack;[arXiv:1510.01433](https://arxiv.org/abs/1510.01433), [doi:10.4171/ggd/572](https://doi.org/10.4171/ggd/572)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Heisenberg+group/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/Heisenberg+group/22">v22</a>, <a href="https://ncatlab.org/nlab/show/Heisenberg+group">current</a>

   added a couple more references on the “integer Heisenberg group”:

   • Roman Budylin: Conjugacy classes in discrete Heisenberg groups, Sbornik: Mathematics 205 8 (2014) 1069–1079 [arXiv:1405.5499, doi:10.1070/SM2014v205n08ABEH004410]

   • Jayadev S. Athreya, Ioannis Konstantoulas: Lattice deformations in the Heisenberg group, Groups, Geometry and Dynamics 14 3 (2020) 1007–1022 [arXiv:1510.01433, doi:10.4171/ggd/572]

   diff, v22, current