Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [poisson-geometry]

The tag has no usage guidance.

0
votes
0answers
15 views

Recovering the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids

How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in https://arxiv.org/pdf/...
3
votes
1answer
140 views

Jacobi Identity for Poisson Bracket

We define the so-called rigid body Poisson bracket as $\{F,G\}(\Pi) = -\Pi \cdot(\nabla{F} \times \nabla{G}) $. I want to prove Jacobi's identity, which is : $\{F,\{G,H\}\} + \{G,\{H,F\}\}+\{H,\{F,G\}\...
1
vote
1answer
20 views

Left-/right-translate of a two-form

The context is that of coboundary Lie bialgebras discussed in "Lie bialgebras, Poisson Lie groups and dressing transformations" by Y. Kosmann-Schwarzbach. In section 4.2, she defines objects like $r^...
0
votes
1answer
58 views

Darboux coordinates in neighbourhood of a point

Given a bivector field, $(x\partial_x + y\partial_y)\wedge \partial_z$, how does one find a system of Darboux coordinates in a neighbourhood of the point $(x,y,z) = (1,0,0) \in \mathbb{R}^3$?
0
votes
1answer
35 views

Given a bivector, determine the decomposition of $\mathbb{R}^3$ into symplectic leaves

Say I have a bivector, given by $f(z) \partial_x \wedge \partial_y$, where $f(z)$ is a smooth function of $z$. How do I determine the decomposition of $\mathbb{R}^3$ into symplectic leaves from this?
2
votes
1answer
82 views

Relation between Lie bracket and Poisson bracket

For any vector field $X$ on a smooth manifold $Q$, define $f_X : T^* Q \to \mathbb{R}, \omega \mapsto \omega(X_x)$ for $\omega \in T_x^* Q$. We also have that $\{ \cdot ,\cdot\}$ is an arbitrary ...
5
votes
1answer
105 views

Intuition about Poisson bracket

I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket ...
2
votes
0answers
62 views

Casimir Functions of Poisson Structure

I have just started to read Poisson Geometry and was working my way through some problems. This may be basic for many, but I am just trying to understand the way to work through it. The problem says: ...
1
vote
1answer
162 views

Proving Jacobi identity for Poisson bracket using antisymmetric matrix

I want to prove that the Poisson bracket from Hamiltonian mechanics satisfies the Jacobi identity and I want to do so using the matrix $$(J^{ij})=\begin{pmatrix}0 & -I_2 \\ I_2 & 0\end{pmatrix}...
2
votes
1answer
194 views

proof of the Jacobi Identity for certain poisson brackets

I have to prove that these are effectively Poisson bracket. Specifically that the satisfy Jacobi Identity when $a_{ij}=-a_{ji}$. $$ \left\{ f,g\right\} =\stackrel{\scriptscriptstyle i,j=1..3}{\sum}\...
1
vote
1answer
66 views

How exactly do I compute Poisson-Lie brackets?

This question comes from Example 4.2 of the Gekhtman-Shapiro-Vainshtein book Cluster Algebras and Poisson Geometry which I have attached. My goal is to understand how to compute $\{x_1’,x_2’\}=-\...
2
votes
1answer
52 views

q-quantization of Lie bialgebras

I am trying to understand the difference between the "Drinfeld" and the "Lusztig" theory of quantum groups, more specifically with respect to the problem of quantization of Lie bialgebras/Poisson Lie ...
-1
votes
1answer
68 views

poisson and hamiltonian vector fields

for $\pi= dx_1 \wedge dy_1 + x_2^3\, dx_2 \wedge dy_2$ a bivector field on $\mathbb R^4.$ check that $dy_2$ is a poisson vector field, is it a hamiltonian vector field? What should i do to prove it ...
4
votes
0answers
77 views

An analogue of the Poisson bracket in contact geometry?

McDuff and Salamon define an analogue of the Poisson bracket in contact geometry on page 135 in the third version of Introduction to Symplectic Topology. The definition is the following. Let $(M,\...
1
vote
1answer
84 views

Poisson bracket and curl - Second equation of MHD

In the study of Magnetohydrodynamics we have: $$\frac{\partial \textbf{B}}{\partial t} = \nabla \times (\textbf{v} \times B)$$ In Arnold's book "Topological Method in Hydrodynamics" , equation 10.1 , ...
1
vote
1answer
34 views

How to show that $t \subset g$ is a Lie sub-bialgebra if and only if $t^{\perp} \subset g^*$ is an ideal?

Let $g$ be a Lie bialgebra. That is, $g$ is a Lie algebra, there is a linear map $\delta: g \to g \otimes g$ such that $\delta^*: g^* \otimes g^* \to g^*$ is a Lie bracket on $g^*$, and $$ {\...
2
votes
1answer
106 views

Star products and Jacobi Identity

I'm having a "little" problem with one affirmation on Kontsevich's paper. He says that the second order terms $O(\hbar)$ implies, assuming that the associator $A(f,g,h)=0$, that the Jacobi Identity it'...
3
votes
0answers
201 views

Product of vector bundles.

If $\pi_1:E_1\to M$ and $\pi_2:E_2\to M$ are two vector bundles over $M$, then we can construct a new vector bundle $E_1\oplus E_2$ by declaring the fibre at each $x\in M$ to be $(E_1\oplus E_2)_x:=(...
1
vote
1answer
291 views

Lie bracket and Poisson bracket

$\newcommand{\g}{\mathfrak{g}} \newcommand{\Cinf}[1]{\mathcal{C}^\infty (#1)} \renewcommand{\d}{\mathrm{d}} \newcommand{\R}{\mathbb{R}} $ Let $\g$ be a Lie algebra and consider in $\g^*$ the ...
2
votes
0answers
158 views

Simulation of the Second Matern Hard Core point process

Using the second matern point process thinning method, I have obtained a hard core point process from a regular poisson process in $\mathbb{R}^2$. According to the standard text (Stochastic Geometry ...
0
votes
0answers
40 views

Poisson statistics and exponential functions!

Would there be any chance of having a single exponential function outputting poisson data? I mean I have a $y=\exp{\left(\frac{t}{\text{constant}}\right)}$, this is not a probability but can it be ...
1
vote
1answer
283 views

Lie bracket as a tensor

First of all one comment: I know there is a question titled ''Lie bracket is not a vector field'' which proofs Lie bracket is not $C^\infty(M)$-linear. Now, I show my question. In the paper Poisson-...
0
votes
1answer
89 views

property about Schouten (Schouten-Nihenjuis) bracket

I would compute the Schouten bracket of a bivector with itself. I am guessing the bivector is $ P = P^{ij}e_i \wedge e_j$ (repited indices add up), where $P^{ij}=P^{ij}(x)$ are $C^\infty$ functions ...
3
votes
0answers
228 views

Is there a known Hamiltonian for the Lorenz-63 system?

It has been shown that Chua's system has a Hamiltonian-Poisson realization (Arieşanu 2013). That is, there exists a Hamiltonian $H=f(x)$ over $x\in \mathbb{R}^3$ and an skew-symmetric matrix $\Pi\in \...
1
vote
0answers
61 views

Reference request for Poisson Geometry.

I was searching for an introductory text on Poisson geometry.I have tried searching it and end up getting - http://www.math.illinois.edu/~ruiloja/Math595/book.pdf and after looking at a similar ...
0
votes
0answers
68 views

A proposition on Poisson Structures .

I was reading Poisson structure on $\mathbb{R}^{n}$ , here is the Proposition - "Let {.,.} be a Poisson Structure on $\mathbb{R}^{n}$ .then for any $f,g \in C^{\infty} (\mathbb{R}^{n},\mathbb{R})$" ...
2
votes
1answer
101 views

Is a universal enveloping algebra a Poisson algebra?

Let $g$ be a Lie algebra and $U(g)$ its enveloping algebra. I think that $U(g)$ is a Poisson algebra where the Poisson bracket is given by the Lie bracket of $g$ and Leibniz rule. Is this true? Thank ...
0
votes
0answers
104 views

Poisson bivector induced by a Poisson manifold?

A Poisson manifold is a pair $(M, \{\cdot, \cdot\}_M)$ where $M$ is a manifold and $$\{\cdot, \cdot\}: C^\infty(M)\times C^\infty(M)\longrightarrow C^\infty(M)$$ is a Lie bracket such that $$\{f, gh\}=...
1
vote
1answer
33 views

About the definition of $\Pi^\sharp:\Omega^1(M)\longrightarrow \mathfrak{X}^1(M)$?

Let $M$ be a manifold. A $p$-field on $M$ is a section of the bundle $\Lambda^p (TM)$. Let us denote the space of $p$-fields on $M$ by $\mathfrak{X}^p(M)$. Notice, $\mathfrak{X}^1(M)$ is the space of ...
0
votes
1answer
41 views

How to derive the formula $\{t_{ij}, t_{kl}\} = \sum_{a,b} (r^{ajbl} t_{ia}t_{kb} - r^{iakb} t_{aj}t_{bl})$?

I am reading the book. Let $G$ be a Poisson-Lie group and $r = \sum_{s,t} r^{st} X_s \otimes X_t \in g \wedge g$, where $g$ is the Lie algebra of $G$. In the end of page 60, the bracket on $\mathbb{...
0
votes
0answers
98 views

A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
2
votes
1answer
39 views

Does a subalgebra of a Lie algebra $g$ define a Lie subalgebra in the dual $g^*$ if $( g, g^*)$ is a Lie bialgebra?

Question: Let $\mathfrak d = \mathfrak g\bowtie \mathfrak g^*$ be the double of the Lie bialgebra $(\mathfrak g, \mathfrak g^*)$, and let $\mathfrak h$ be a Lie subalgebra of $\mathfrak g$. If $\xi,\...
0
votes
1answer
119 views

Proof involving Poisson bracket

Not being able to understand how each term has been simplified to get from the third step to the fourth step. So how did 1/2m become 1/m and {qj,plpl}pk become {qj,pl}plpk and how did k/4 become k/2 ...
1
vote
0answers
300 views

How to do this Poisson bracket proof

For the proof of the above equation, I understand the first step which has been obtained from the definition but in the second step I don't understand why they are summing over $j$ first (shouldn't ...
0
votes
1answer
176 views

Poisson bracket proof

For this question I understand the first line of the solution which they have obtained from the definition but how have they simplified each term to get to the second line from the first line? The ...
1
vote
1answer
246 views

Poisson bracket proofs

I understand the first sentence you wrote for the need of a different summation index. However, i'm still not able to understand the individual steps. Like how in the first line we have four partial ...
0
votes
1answer
226 views

How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed?

I am reading the book a guide to quantum groups. I have a question on page 18. How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed? Any help will be greatly appreciated!...
2
votes
1answer
78 views

Book Recommendation for Poisson Manifold and Deformation Quantisation

Can someone please recommend a basic introduction to the concept of Poisson Manifolds and Deformation Quantization. I'm new to Theoretical Physics and had to go through a lot of books before I even ...
1
vote
1answer
430 views

Are the symplectic leaves of a Poisson manifold submanifolds?

In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds". In "Lectures on Poisson ...
5
votes
0answers
171 views

Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
1
vote
1answer
112 views

What is a Hamiltonian in a Poisson algebra?

Classical physics on the phase space $T^* M$ (with $M$ a smooth manifold) is done mostly in the following way: one endows $T^*M$ with a Riemannian structure $g^*$ (that will give the kinetic term) and ...
0
votes
0answers
201 views

Does the Poisson bivector give rise to an integrable distribution?

I am reading the book Lectures on the geometry of Poisson manifolds, by Izu Vaisman. To a Poisson structure $\{\cdot,\cdot\}$ on a manifold $M$ we associate the Poisson bivector field $w\in\Gamma(\...
0
votes
1answer
62 views

About poisson manifolds?

A Poisson manifold is a pair $(M, \{\cdot, \cdot\})$ where $M$ is a smooth manifold and $\{\cdot, \cdot\}$ is a Lie bracket on the $\mathbb R$-algebra $C^\infty(M)$ satisfying $$\{f, gh\}=\{f, g\}h+g\{...
1
vote
1answer
342 views

Reference for Poisson geometry

Is there any good reference for Poisson geometry/Poisson manifolds out there? I would like to give a deep look to the subject, but all I seem to be able to find are short chapters or interludes in ...
1
vote
1answer
66 views

Graded Leibniz's Law for Schouten bracket

I'm trying to work on the graded Leibniz's law for Shouten bracket, but I've got the wrong sign whenever how hard I tried. Here's the problem$\newcommand{\vt}{\vartheta} \newcommand{\zt}{\zeta} \...
0
votes
1answer
322 views

Poisson bracket calculation.

A canonical transformation is a transformation from one set of coordinates $q,p$ to a new one $Q(q,p), P(q,p)$. For a function $f(Q,P)$ using the chain rule and using summation notation $$\frac{\...
0
votes
1answer
141 views

Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
1
vote
1answer
32 views

Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
1
vote
1answer
160 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
4
votes
1answer
243 views

What does it mean for a Symplectic Form to be invariant under Group Action?

This should be a very basic question for people familiar with differential manifolds. I'm more or less new to the field so let me apologize in advance for ill-defined questions if arising. I split the ...