Questions tagged [symplectic-linear-algebra]

Question 1

Let $(B,ω')$ be a symplectic Lie algebra. and the $\delta$ is a symplectic derivation, and $z\in B$. The double extension of $B$ is: $g=ℝe ⊕ B ⊕ ℝd$ as: Central Extension : $I = ℝe ⊕ B$,of $B$ with ...

Question 2

Let $(X,\omega)$ be a symplectic manifold of dimension $2n$ and $J$ be an almost complex structure compatible with $\omega$. In particular, $g(v,w):=\omega(v,Jw)$ defines a Riemannian metric, and ...

Question 3

I was reading this paper which defines $L$ as "symplectic linear transformations, represented by unit-determinant $2 \times 2$" matrixes with entries in $\mathbb{F}_{4}$". For example, ...

Question 4

In research I need to prove a very strange question about the sum of intersections of Lagrangian subspaces being equal to intersection of sum of Lagrangian subspaces. Why I say strange is that it ...

Question 5

A symplectic form on an $n$-dimensional vector space on $GF(2)$ is a symmetric, alternating bilinear form on the space. An orthogonal form $q$ on a space $V$ associated with a symplectic form $f$ is a ...

Question 6

I've skimmed through other posts on this book: Lectures on Symplectic Geometry by da Silva and am wrapping up a question from chapter 3. We have that $T^{*}X=M$ with the projection map $\pi: M\to X$. ...

Question 7

Suppose $E=\{ (x_1, x_2, y_1,y_2): \frac{x_1^2}{a_1^2} + \frac{x_2^2}{a_2^2} + \frac{y_1^2}{b_1^2} + \frac{y_2^2}{b_2^2} \leq 1 \}$. One defines $w_L(E)$ as the supremum of the $\pi r^2$ such that ...

Question 8

The following question is stated as exercise 10.2-2, p. 338 in the book "Introduction to Mechanics and Symmetry - 2nd Edition" by Marsden, Ratiu. Unless otherwise stated, all references ...

Question 9

I am reading the lecture notes of Dipendra Prasad on the Weil representation and theta correspondence. There is quite a bit of notation but I will set it up here. $W$ is a symplectic space over a ...

Question 10

I am reading the following paper where the author says "It is a well-known fact that every symplectic operator on a finite-dimensional complex vector space has a Lagrangian invariant subspace.&...

Question 11

I'm studying the Maslov index in McDuff and Salamon's Introduction to Symplectic Topology, 3rd edition. Theorem 2.2.12 develops the Maslov index for loops of symplectic matrices. In particular, let $r ...

Question 12

If $H$ is a positive definite matrix, it is well known by Williamson's theorem that one can brought $H$ into the block diagonal form $S^T H S = \begin{pmatrix} D & 0 \\ 0 & D \end{pmatrix}$ ...

Question 13

I am trying to prove the following theorem: Let $(V,\omega)$ be a finite-dimensional symplectic and let $W$ be a subspace of $V$. Let $W=W^{\text{rad}}\boxplus U$. Let $e_1,\cdots,e_r$ be a basis of $...

Question 14

Let $V$ be an even dimensional vector space over the finite field $\mathbb{F}_\ell$ where $\ell$ is even, and set $n=\dim_{\mathbb{F}_\ell}(V)$. Let $q:V\to \mathbb{F}_\ell$ be a quadratic form, and ...

Question 15

I am actually interested in understanding the relationship between eigenvalues of matrix $P \in \mathbb{C}^{n \times n}$ and matrix $Q \in \mathbb{R}^{2n \times 2n}$ defined as $$Q=\begin{pmatrix} \Re[...

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Questions tagged [symplectic-linear-algebra]

Understanding the symplectic double extension

Hodge star of a $(0,q)$-form wedged with the symplectic form on a symplectic manifold with a compatible almost complex structure

Are the matrices defined in the question elements of $Sp(1,4)$?

A strange question on sum of intersections of Lagrangian subspaces equal to intersection of sum of them

Counting the totally singular subspaces of a finite dimensional orthogonal vector space on GF(2)

exercise from chapter 3 of Lectures on Symplectic Geometry by da Silva

Gromov width of an elipsoid

Casimir functional for Vlasov-Poisson bracket

Computing the Weil representation

Existence of Invariant Lagrangian subspace

Product Property of Maslov Index for Symplectic Matrices

Symplectic diagonalization of positive semi-definite matrices.

Mutual orthogonality of hyperbolic planes in symplectic space

Order of group of isometries of a quadratic form in characteristic $2$

Interpreting a matrix equation with "matricised" eigenvalues

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