So I was reading a book on algebraic integrability. In the section about Poisson manifolds they say that the skew-symmetry of the Poisson matrix $X:=(\{x_i,x_j\})_{1\le i,j \le n}$ implies that the Rank of a Poisson structure at a point is always even. Can someone explain why that is? I tried to work it out but couldn't see it.
2026-03-25 07:48:43.1774424923
Rank of a Poisson structure is always even
53 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MANIFOLDS
- a problem related with path lifting property
- Levi-Civita-connection of an embedded submanifold is induced by the orthogonal projection of the Levi-Civita-connection of the original manifold
- Possible condition on locally Euclidean subsets of Euclidean space to be embedded submanifold
- Using the calculus of one forms prove this identity
- "Defining a smooth structure on a topological manifold with boundary"
- On the differentiable manifold definition given by Serge Lang
- Equivalence of different "balls" in Riemannian manifold.
- Hyperboloid is a manifold
- Integration of one-form
- The graph of a smooth map is a manifold
Related Questions in POISSON-GEOMETRY
- How to show that $t \subset g$ is a Lie sub-bialgebra if and only if $t^{\perp} \subset g^*$ is an ideal?
- Poisson bracket and curl - Second equation of MHD
- An analogue of the Poisson bracket in contact geometry?
- poisson and hamiltonian vector fields
- q-quantization of Lie bialgebras
- How exactly do I compute Poisson-Lie brackets?
- proof of the Jacobi Identity for certain poisson brackets
- Proving Jacobi identity for Poisson bracket using antisymmetric matrix
- Casimir Functions of Poisson Structure
- Intuition about Poisson bracket
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
For anyone asking the same question:
A basic fact about skew-symmetric matrices is, that they have even rank. A proof of that can be found here.
The second fact is that the rank of the Poisson matrix is equal to the rank of the poisson structure at a point $m$. I couldn't find an explanation for that so I posted a new question here.
The result follows.