# Questions tagged [heisenberg-groups]

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### Carnot-Carathéodory metric

The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric. Question 1. What is the origin of this name? Who was the first to introduce it? I believe that the "...
3answers
576 views

### Non embedding of the Heisenberg group

It is well known that Heisenberg groups cannot be bi-Lipschitz embedded into Euclidean spaces. A standard proof uses the fact that a Lipschitz mapping from a Heisenberg group into a Euclidean space is ...
1answer
519 views

### Non-isomorphic Heisenberg groups over rings

Suppose $R_1,R_2$ are finite unital commutative rings. Consider Heisenberg groups $H_3(R_1)$ and $H_3(R_2)$ (upper unitriangular marticies $3 \times 3$). Proposition. If $R_1 \not\cong R_2$ (as ...
1answer
328 views

### The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
0answers
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### Lipschitz homotopy groups

There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find ...
0answers
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### Heisenberg group: function without vertical derivative

Let $\mathbb H$ be Heisenberg group with vector fields $$X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t$$ and $U\subset\mathbb H$ is an open set. I am ...
1answer
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### Difference between the Laplacian and the sub-Laplacian of a Lie group

Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...
0answers
214 views

### How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For $z=x+ i y \in \mathbb C$ ...
0answers
162 views

### Joint representation of the semi-direct product of the metaplectic group and Heisenberg group

Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the semi-...
3answers
456 views

### Show that the Laplacian operator on the Heisenberg group is negative

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For $z=x+ i y \in \mathbb C$ ...
0answers
189 views

### Contact geometry: approximation of Legendrian mappings

Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$. Question 1. Is it true that a $C^1$-contact ...
0answers
115 views

### Modules of algebras with idempotents and the Stone-von Neumann theorem

The Stone-von Neumann theorem tells us that all unitary irreducible representations of the integrated/exponentiated/Weyl form of the canonical commutation relations (CCR) algebra in finite dimensions ...
1answer
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1answer
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### Heisenberg groups, Carnot groups and contact forms

The horizontal distribution in the Heisenberg group is the kernel of the standard contact form: $$\alpha = dt + 2 \sum_{j=1}^n (x_j \, dy_j - y_j \, dx_j).$$ Question. Can one describe ...
1answer
184 views

### dirichlet problem in the heisenberg group

Good morning everybody. I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...
0answers
121 views

### About the purpose of introducing '"groups of Heisenberg type"

I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
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### Explicit Quasisymmetric embedding into Euclidean space

It is known that every doubling metric space admits quasisymmetric map into Euclidean space. My question is, is there a known explicit (closed-form) quasisymmetry from the Heisenberg group into a ...
0answers
75 views

1answer
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### Representations of the $3\times 3$ Heisenberg group [closed]

I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations. Following this article given a symplectic bilinear form \$\langle, \...