Questions tagged [heisenberg-groups]
The heisenberg-groups tag has no usage guidance.
42
questions
0
votes
1
answer
131
views
Action of the Haar measure on the Heisenberg group
The Heisenberg group is $\mathbb{H}^N=\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the group operation
\begin{equation}
(...
- 479
1
vote
0
answers
59
views
A question about semigroups in a Heisenberg group
I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
- 479
1
vote
0
answers
32
views
regularity theory of parabolic equations in Heisenberg group
I'm trying to understand if there are regularity results for mild solutions of partial differential equations in Heisenberg group. In this paper (Theorem 1.3 (iii) and proof of Theorem 1.1) the author ...
- 479
0
votes
0
answers
39
views
Are there results of parabolic regularity in Heisenberg groups?
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 ...
- 479
1
vote
0
answers
55
views
Sub Laplacian on the quaternion Heisenberg group $\mathbb{H}$
The sublaplacian is defined by $\mathcal{L}=-\left(X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, which is independent of the choice of the orthonormal basis of $\mathbb{H}$. It is well known that ...
- 469
0
votes
0
answers
144
views
Heisenberg group
Let
$X_{j}=\frac{\partial}{\partial x_{j}}-\frac{1}{2}y_{j}\frac{\partial}{\partial t}$, $j=1,2,\dots,n$
$Y_{j}=\frac{\partial}{\partial y_{j}}+\frac{1}{2}x_{j}\frac{\partial}{\partial t},j=1,2,\dots,...
- 469
2
votes
0
answers
185
views
Schrödinger representation of the Heisenberg group
Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 469
2
votes
0
answers
101
views
Finite dimensional unitary representations of the discrete Heisenberg group
Let $H(\mathbb{Z})$ be the discrete Heisenberg group. What are the finite dimensional irreducible unitary representations of $H(\mathbb{Z})$? Do they all arise from the coordinate-wise quotient map to ...
1
vote
0
answers
251
views
Classification of the normal subgroups of the discrete Heisenberg group
Let $H$ be the discrete Heisenberg group, i.e., the set of matrices of the form
$\begin{bmatrix}
1 & x & z \\
0 & 1 & y \\
0 & 0 & 1
\end{bmatrix}$
where $x,y,z \in \mathbb{Z}$...
1
vote
0
answers
424
views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
- 11
3
votes
0
answers
76
views
Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\...
- 3,016
4
votes
0
answers
185
views
What is the analogue of the Jacobi theta function in the Weyl representation?
It is known (see https://en.wikipedia.org/wiki/Theta_representation) that the Jacobi theta function
$$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$
arises from a certain ...
- 141
1
vote
1
answer
160
views
Ratio limit results for restricted partition functions
This concerns difference/limit ratio results for special restricted partitions.
Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...
- 4,629
1
vote
0
answers
303
views
Canonical commutation relations-bounded vs. unbounded picture
Suppose that $Q,P$ are self-adjoint operators which satisfy the relation $$(1) \ \ \ \ \ [Q,P]=iI$$ One can easily show that in this case $P,Q$ cannot be bounded. However one can find unbounded ...
- 9,150
2
votes
1
answer
184
views
Semidirect product of metaplectic group and Heisenberg group
I know that Symplectic group has an action on Heisenberg group.
I am wondering how to extend this to non-trivial two fold metaplectic covering?
Thanks in advance!
- 1,739
20
votes
1
answer
1k
views
Carnot-Carathéodory metric
The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric.
Question 1. What is the origin of this name? Who was the first to introduce it?
I believe that the "...
- 26.4k
4
votes
1
answer
342
views
Heisenberg groups, Carnot groups and contact forms
The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
$$
\alpha = dt + 2 \sum_{j=1}^n (x_j \, dy_j - y_j \, dx_j).
$$
Question. Can one describe ...
- 26.4k
1
vote
0
answers
55
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 ...
- 5,001
8
votes
1
answer
459
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 ...
- 64.8k
5
votes
0
answers
195
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 ...
- 26.4k
4
votes
1
answer
205
views
Legendre's symbol in Schrödinger model for the Weil representation
I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$.
The way to present the action of the Weil representation $\omega$ of $Sp(2n,\...
11
votes
3
answers
699
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 ...
- 26.4k
10
votes
0
answers
389
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
...
- 26.4k
5
votes
0
answers
129
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 ...
- 613
1
vote
0
answers
88
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 &...
user6671
3
votes
0
answers
125
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 ...
- 630
1
vote
0
answers
180
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 ...
- 1,662
10
votes
1
answer
544
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 ...
- 411
1
vote
0
answers
164
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 ...
- 630
-1
votes
1
answer
292
views
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, \...
- 231
6
votes
1
answer
2k
views
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 ...
- 630
6
votes
0
answers
255
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). $$
For $z=x+ i y \in \mathbb C$ ...
- 630
5
votes
3
answers
522
views
Show that the Laplacian operator on the Heisenberg group is negative
The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...
- 51
4
votes
1
answer
211
views
dirichlet problem in the heisenberg group
Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...
- 277
4
votes
1
answer
334
views
Horizontal Sobolev space on Carnot group
This question is connected with my previous: Heisenberg group: function without vertical derivative.
Here I am trying to look from another side: what is a difference between Sobolev space and ...
- 399
8
votes
0
answers
407
views
Heisenberg group: function without vertical derivative
Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...
- 399
4
votes
1
answer
1k
views
Irreducible representation of Heisenberg group with characteristic 2?
As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than $2$....
- 347
2
votes
1
answer
246
views
totally geodesic submanifold of Heisenberg group
Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold ...
- 270
6
votes
0
answers
164
views
Joint representation of the semi-direct product of the metaplectic group and Heisenberg group
Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the semi-...
- 61
3
votes
2
answers
400
views
Hesse pencil and Schrodinger representation of Heisenberg group
Let $E$ be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let $\mathcal{L}$ be a line bundle of degree $3$. Heisenberg group $H_3$ acts on global sections of $\...
- 1,525
2
votes
0
answers
270
views
Finite Heisenberg groups action on cohomology of line bundles
Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
- 1,525
3
votes
0
answers
465
views
Harmonic analysis on the Heisenberg group
It is well known that:
Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion
$$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{\infty} \int_{0}^{\infty} f \...
- 1,051