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Mathematics Stack Exchange

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.

A Heisenberg group is a Lie group whose Lie algebra is a Heisenberg Lie algebra.

Heisenberg group is also known as the Weyl (or Heisenberg–Weyl) group.

The Heisenberg group historically originates in and still has its strongest ties to quantum physics: there it is a group of unitary operators acting on the space of states induced from those observables on a linear phase space – a symplectic vector space – which are given by linear or by constant functions. So any Heisenberg group is a subgroup of a group of observables in certain simple examples of quantum mechanical systems.

Heisenberg group reveals itself as an important factor in many apparently diverse topics like,

  • Representation Theory of Nilpotent Lie Groups
  • Foundations of Abelian Harmonic Analysis
  • Moduli of Abelian Varieties
  • Structure Theory of Finite Groups
  • Theory of Partial Differential Equations
  • Quantum Mechanics
  • Homological Algebra
  • Ergodic Theory
  • Representation Theory of Reductive Algebraic groups
  • Classical Invariant Theory

This list could easily be lengthened both by adding new topics and making these more specific, for sometimes the applications are multiple.

References:

https://en.wikipedia.org/wiki/Heisenberg_group

https://www.univie.ac.at/nuhag-php/bibtex/open_files/1389_fr750001.pdf

48 questions
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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 $\mathbb{Z}$) Heisenberg $3\times 3$ matrix group. The…
Bartek
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Quotient of the $p$-adic Heisenberg group by certain subgroups

We begin by the following definition. Definition: Let $d \in \mathbb{N}$. The $(2d + 1)$-dimensional Heisenberg group over the ring of $p$-adic integers $\mathbb{Z}_p$, denoted by $\mathbb{H}_d(\mathbb{Z}_p)$, is given…
Surajit
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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 Hilbert space $L^2(\mathbb{R}\to\mathbb{C})$ of…
Jazzmaniac
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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. I have done most of the exercises that come…
user481421
5
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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 reverse engineer this stuff and ask questions.…
cows
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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 homeomorphic to $\mathbb{R}^3$. It suffice to show that…
Infinity
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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 matrices of the form $\begin{pmatrix}1&x&z\\ 0&1&y\\…
Derso
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Heisenberg group modulo prime

According to Wikipedia, the Heisenberg group modulo $p$, where $p$ is an odd prime, has the presentation $$H(\mathbb{F}_p)=\langle x,y,z\mid x^p=y^p=z^p=1, \ xz=zx, \ yz=zy, \ z=xyx^{-1}y^{-1}\rangle.$$ I could even derive it, but the proof seems to…
ashpool
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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 $Z(H_{3}(\mathbb{F}_{p}))$ is well known and…
Nourr Mga
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Classical approximation of a quantum mechanical system

The starting point is the 3-dimensional Heisenberg group $H_3$. The non-degenerate unirreps of $H_3$ are realized on $L^{2}(\mathbb{R})$. Each such representation corresponds to a nonzero value of $\frac{1}{\hbar}$. One knows that each such…
Hasib
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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. Mostly from this Lecture Notes (attached below): …
Tat-iva
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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 of the associated symmetric matrix of $f$ . This…
user1115617
3
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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, which is virtually the discrete Heisenberg group, see…
Bernie
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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 dimensional Heisenberg group. Consider the sequence of…
Juan Pablo Velasquez Rodriguez
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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. So I conjecture that $|G|\leq 27$. As for proving…
Hossmeister
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