# Questions tagged [heisenberg-groups]

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14
questions with no upvoted or accepted answers

**8**

votes

**0**answers

270 views

### 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
...

**8**

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374 views

### 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 ...

**6**

votes

**0**answers

210 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 $z=x+ i y \in \mathbb C$ ...

**6**

votes

**0**answers

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-...

**5**

votes

**0**answers

188 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 ...

**5**

votes

**0**answers

114 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 ...

**3**

votes

**0**answers

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 ...

**3**

votes

**0**answers

433 views

### Harmonic analysis on the Heisenberg group

It is well known that:
Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion
$$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{\infty} \int_{0}^{\infty} f \...

**2**

votes

**0**answers

245 views

### Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...

**1**

vote

**0**answers

186 views

### Canonical commutation relations-bounded vs. unbounded picture

Suppose that $Q,P$ are self-adjoint operators which satisfy the relation $$(1) \ \ \ \ \ [Q,P]=iI$$ One can easily show that in this case $P,Q$ cannot be bounded. However one can find unbounded ...

**1**

vote

**0**answers

42 views

### 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 ...

**1**

vote

**0**answers

75 views

### Factoring in discrete Heisenberg group $H_3(\mathbb{Z})$

Let $H_3(\mathbb{Z})$ be the discrete Heisenberg group generated by $x=\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix},\ \ y=\begin{pmatrix}
1 & 0 &...

**1**

vote

**0**answers

171 views

### Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...

**1**

vote

**0**answers

131 views

### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...