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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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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 ...
Ilovemath's user avatar
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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. ...
Alexander Chervov's user avatar
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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?
weak solution's user avatar
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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 ...
eraldcoil's user avatar
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1 answer
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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 &...
eraldcoil's user avatar
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3 votes
4 answers
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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 ...
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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 ...
Ilovemath's user avatar
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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 ...
Ilovemath's user avatar
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4 votes
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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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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 ...
Math101's user avatar
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2 answers
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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 ...
cows's user avatar
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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 ...
cows's user avatar
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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....
cows's user avatar
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Compute the one-parameter subgroups and the exponential map of the Heisenberg group

For the Heisenberg group $ H = \{ \begin{bmatrix} 1 & x & y\\ 0 & 1 & z\\ 0 & 0 & 1 \end{bmatrix} | \:x,y,z ...
Paul Joh's user avatar
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How to find the discrete Heisenberg group in this group given by a small presentation?

Consider the group $G$ given by the following presentation: $$G=\langle x,y\mid x^{-1}y^2xy^2=x^{-2}yx^{-2}y^3=1\rangle.$$ In this slides it is noted that this is a torsion-free polycyclic group, ...
Bernie's user avatar
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1 answer
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Character table of modular Heisenberg groups

Let $p$ be a prime number and let $G$ be the modular Heisenberg group of order $p^3$ $$ G = \left\{\, \begin{bmatrix} 1 & b & c \\ 0 & 1 & a \\ 0 & 0 & 1 \end{bmatrix} : a, b, ...
Orat's user avatar
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How to compute the unitary dual of a noncommutative profinite group?

Let p $\neq 2$ be a prime number. Let $G=\mathbb{H}(\mathbb{Z}_p)$ be the group of uni-triangular 3x3 matrices wih entries in the ring of p-adic integers, sometimes called the profinite three ...
Juan Pablo Velasquez Rodriguez's user avatar
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An analogue of Beurling's theorem on the Heisenberg group.

Let $f\in L^2(\Bbb R^n)$ and $d\geq0$. Then $$\int_{\Bbb R^n}\int_{\Bbb R^n} \frac{|f(x)||\hat f(y)|}{(1+|x|+|y|)^d} e^{|x||y|} dxdy<\infty$$ implies that $f=0$ when $d\leq n$. The analogue of this ...
zoran's user avatar
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1 answer
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Calculating the Lie bracket on the Heisenberg algebra of $H=Z\times S^1$

I'm working through Mechanics and Symmetry by Marsden and Ratiu. Let $(Z,\Omega)$ be a symplectic vector space and define on $H:=Z\times S^1$ the operation $$(u,\exp i\phi)(v,\exp i\psi)=(u+v,\exp i[\...
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The Heisenberg group/algebra and Symplectic Vector Spaces

I have some questions about the relationship between Heisenberg groups/algebras and symplectic vector spaces. This is my first time properly dealing with many of these topics, so please be patient if ...
leob's user avatar
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Heisenberg group - representation - symplectic manifold

Given a symplectic vector space $(V,\omega)$ of dimension $2n$ ($V$ being a symplectic manifold), a smooth function $\psi \in C^\infty(M)$, a translation operator $T$ acting on $V$ via $(T(Y)\psi)(X) =...
Nibbler's user avatar
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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 ...
Kat's user avatar
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11 votes
1 answer
355 views

Question from J. Milnor paper from 1968 about diffeomorphic manifolds

In the article "A note on curvature and fundamental group"(1968) by J. Milnor the following side question arises: where $G$ and $H$ are continuous (over $\mathbb{R}$) and discrete (over $\...
Bartek's user avatar
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4 votes
1 answer
396 views

Heisenberg group modulo prime

According to Wikipedia, the Heisenberg group modulo $p$, where $p$ is an odd prime, has the presentation $$H(\mathbb{F}_p)=\langle x,y,z\mid x^p=y^p=z^p=1, \ xz=zx, \ yz=zy, \ z=xyx^{-1}y^{-1}\rangle.$...
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Heisenberg Lie group, image of specific element by any representation is nilpotent

consider the Heisenberg group $Heis(\mathbb{R})$ and $\mathfrak{h}$ it's Lie algebra. Consider $C\in \mathfrak{h}$ given by $$C=\begin{bmatrix} 0 && 0 && 1\\ 0 && 0 && ...
roi_saumon's user avatar
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2 votes
1 answer
446 views

Heisenberg group is nilpotent

$F$ is a field and $H(F)$ is the Heisenberg group over $F$. Is it nilpotent? Is it solvable? I did all the math and I found that the commutator subgroup is in the center $Z(H(F))$, so $H(F)/Z(H(F))$ ...
CodingGuy's user avatar
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0 answers
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Compact embedding and uniform convexity fo Sobolev space in Heisenberg group

Let $\Omega\subset\mathbb{H}$ is bounded domain where $\mathbb{H}$ is the Heisenberg group with homogeneous dimension $2N+2$. Then the Sobolev space $W^{1,p}(\Omega)$ is defined as the space of all ...
Mathlover's user avatar
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1 answer
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linear independency of basis elements of the Heisenberg Algebra

I tried one problem from "Introduction to Lie Algebras” by K.Erdmann and M. Wildon and there is one question regarding the Heisenberg Algebra that I am not sure how to do. Assume first that L' is 1-...
mathStudent's user avatar
2 votes
1 answer
272 views

An example for a group so that $ Z(G) ≨ G'$

I am trying to find examples for groups so that their commutator subgroup $G'=[G,G]$ has the following relation to the center $Z(G)$: $Z(G) = G'$ $Z(G) ≨ G'$ $G' ≨ Z(G)$ $G' \nsubseteq Z(G)$ and $Z(G)...
Roy Sht's user avatar
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3 votes
2 answers
90 views

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. ...
Hossmeister's user avatar
4 votes
2 answers
1k views

The center of the group of $n\times n$ upper triangular matrices with a diagonal of ones

Let $\mathbb{F}_{p}$ be a finite field of order $p$ and $H_{n}(\mathbb{F}_{p})$ be the subgroup of $GL_n(\mathbb{F}_{p})$ of upper triangular matrices with a diagonal of ones. Note that the center $Z(...
Nourr Mga's user avatar
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1 vote
1 answer
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Free nilpotent group on 2 generators of class 2.

Classical presentation of Heisenberg group is [x,y,z|[x,y]=z, xz=zx, yz=zy] https://pdfs.semanticscholar.org/276f/aef6c6b534f6058441a4f96d6260c5f32052.pdf Here above on page 12 it is written that we ...
robin3210's user avatar
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3 votes
2 answers
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Commutator subgroup of Heisenberg group.

Dears, Let $H$ be Heisenberg group, a group of $3\times 3$ matrices with $1$ on the main diagonal, zeros below, and elements of $\Bbb R$ above the main diagonal. Its center is the subgroup of all ...
robin3210's user avatar
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0 votes
1 answer
179 views

Lifting representation Heisenberg algebra

I (think) I've found the Heisenberg Lie algebra representation through quantization. Where we have $q \mapsto q$ and $p \mapsto -i \hbar \frac{\partial}{\partial q}$. So this is only a Lie algebra ...
AkatsukiMaliki's user avatar
3 votes
1 answer
368 views

Sources for exploring Nil Geometry

I am first year graduate student and following the Thurston's book on 3-dimensional geometry and topology. Among all the 8 geometries of 3-manifold (from Thruston's classification), I heard from some ...
Infinity's user avatar
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5 votes
1 answer
685 views

Quotient space form by the action of the discrete Heisenberg group on the Heisenberg group

Though I am a beginner to differential topology, pardon me for something very basic. Here is my attempt! H(The set of $3 \times 3$ unipotent matrices over $\mathbb{R}$, Heisenberg group) is ...
Infinity's user avatar
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6 votes
0 answers
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Unitary representation of the Heisenberg group and the universal enveloping algebra

I am studying the Heisenberg group with the Lie algebra generators $\{ U,V,W \}$ and the structure $[U,W]=[V,W]=0$ and $[U,V]=W$. This group has an infinite-dimensional unitary representation on the ...
Jazzmaniac's user avatar
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Can 3 dimensional Heisenberg group be represented irreducibly on L^2(S)?

It is well-known that unitary dual of the 3 dimensional Heisenberg group H represented on $L^2(\mathbb{R})$ is given by a nonzero real number $\lambda\in R^*$(can be interpreted as $1/\hbar$). When $\...
Hasib's user avatar
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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 ...
HK Lee's user avatar
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6 votes
1 answer
477 views

Exponential of operators satisfying Heisenberg Commutation Relation

I'm working through the book "Lie Groups: An Introduction Through Linear Groups", by Wulf Rossmann. In the first section, the author introduces the matrix exponential and derives its basic properties. ...
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