# Questions tagged [heisenberg-group]

This tag is for the questions relating to Heisenberg group (or Weyl-Heisenberg group) which is a Lie group integrating a Heisenberg Lie algebra. It is another illustration of its perception as an extraneous object: physicists call it by the name of a mathematician, and mathematicians by the name of a physicists.

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### Riemannian structure on the Heisenberg group.

Currently I am reading about Heisenberg Group. And I understand that this group is one of the simplest examples of sub-Riemannian manifolds. I have read a lot about it structure, geodesics and e.t.c. ...
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1 vote
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### Change of coordinates in vector fields on Heisenberg group

In the book Geometric Analysis on the Heisenberg Group and Its Generalizations proposition 1.3 says: Under the change of coordinates $$y_1 = x_1 , \, y_2 = x_2, \, \tau = 4t -2x_1x_2,$$ the vector ...
• 2,950
1 vote
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### Growh generating function for finite Heisenberg groups

Take standard 2d-Heisenberg group over finite ring Z/p. Choose standard generators $x_i, y_i$. Consider generating polynomial for growth: $g(t) = \sum_i g_i t^i$ , where $g_i$ are the ball sizes. ...
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### I want to study the analysis on Heisenberg group, Please recommend me a comprehensive textbook.

I want to study the properties of Sobolev space on the Heisenberg group. For example, if the horizontal gradient of $f$ is bounded, why is $f$ lipschitz continuity?
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### Schrödinger representation for the Heisenberg group.

I'm studying the Schrodinger representation for the Heisenberg Group in Foundations of Harmonic Analysis by Rottensteiner (https://services.phaidra.univie.ac.at/api/object/o:1265111/get) Studying this ...
• 3,598
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### Representation of the Heisenberg group.

I am understanding the Heisenberg group interpreted as a group of operators. At the moment, I have been able to understand the following: The set of matrices \begin{align} A=\begin{bmatrix} 1 & a &...
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### Determine critical points of $f(x,y) = x^2-3xy+y^2$ using hessian

Let $f(x,y) = x^2-3xy+y^2$. Determine whether the point $(0,0)$ is a local maxima, local minima, or a saddle point using the eigenvalues of the Hessian of $f$ at the point $(0,0)$ or the eigenvalues ...
1 vote
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### Koranyi norm in Heisenberg group gives a Banach space structure?

The $(2N +1)-$dimensional Heisenberg group $\mathbb{H}^N$ is the space $\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the ...
• 2,950
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### Heat semigroup in Heisenberg groups properties

I'm trying to find the construction of the heat semigroup in Heisenberg groups, but I haven't found anything. In particular, I'm trying to find out if the semigroup in the Heisenberg group has the ...
• 2,950
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### Two different models, both called Nil-geometry: what is the precise relation between them?

Trying to understand a bit about the so called Nil-Geometry, I have found two models (are there more?), namely: The Heisenberg group: we identify the points $(x,y,z)\in \mathbb{R}^3$ with the ...
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1 vote
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### Lie Algebra with certain properties is of odd dimension

Let $L$ be a Lie Algebra such that $\dim Z(L)=1$ and $L/Z(L)$ is abelian. Prove that $L\cong H_{2n+1}$ (Heisenberg algebra) for some $n\in\mathbb{N}$. I was able to show that if $\dim L$ is odd, then ...
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### What does $Y(1,z)$ = id for vertex algebras mean?

I'm adding an update to this post here with my current understanding of the situation for context. I read some Wikipedia articles and two texts. I am having some trouble so I figure I would attempt to ...
• 581
1 vote
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### How does one construct the concrete Heisenberg vertex operator algebra for the case of 3d Heisenberg group and associated lie algebra?

It seems I am feeling my way down a blind alley. I'm trying to understand this Heisenberg vertex operator in the concrete case of 3 dimensional Heisenberg group. It would seem that generally vertex ...
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### Showing the Heisenberg group is the central extension of the additive group

Context and some work so far: I found out about the Heisenberg group on Youtube. I'm a Physics student. I wanted to know more about it, and I realized there was more to learn. Here is what I found out....
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### Computation (example) with vector fields in $\mathbb{R}^3$.

I'm trying to apply some results about metric measure spaces for my thesis but I'm not understanding how to go further. I have $\mathbb{R}^3$ with the two vector fields $X$ and $Y$ that define the ...
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### Upper bound of group order where $g^3=e \forall g\in G$

If $G$ is a group such that $g^3=e$ for every $g\in G$, what is the upper bound for its order? I am aware of the Heisenberg group, and I cannot find a group with greater order that has this property. ...
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• 71
1 vote
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### Exponential map on Heisenberg group is a diffeomorphism.

Assume that a Riemannian manifold $M$ is simply connected. In further, assume that there is an global orthnormal frame $e_i$. If $f_i$ is flow of $e_i$, and $M$ is homeomorphic to $\mathbb{R}^n$, then ...
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