subspaces of a symplectic vector spaces are of special forms.
$begingroup$
Let $(V,omega)$ be a symplectic vector space. Let $F subseteq V$ be a subspace.
Show that
$V$ admits a symplectic basis ${e_1,ldots,e_n,f_1,ldots,f_n}$ with the following properties:
(1) If $F$ is symplectic then $F=span{e_1,ldots,e_k,f_1,ldots,f_k}$ for some $k$
(2) If $F$ is isotropic then $F=span{e_1,ldots,e_k}$ for some $k$
(3) If $F$ is co-isotropic then $F=span{e_1,ldots,e_n,f_1,ldots,f_k}$ for some $k$
(4) If $F$ is Lagrangian then $F=span{e_1,ldots,e_n}$
The converse of these statements are easy to prove by applying definitions. But I have no idea how to prove these.
linear-algebra differential-geometry symplectic-geometry symplectic-linear-algebra
asked Dec 4 '18 at 0:38
bbwbbw
47038
$endgroup$
add a comment |
$begingroup$
Let $(V,omega)$ be a symplectic vector space. Let $F subseteq V$ be a subspace.
Show that
$V$ admits a symplectic basis ${e_1,ldots,e_n,f_1,ldots,f_n}$ with the following properties:
(1) If $F$ is symplectic then $F=span{e_1,ldots,e_k,f_1,ldots,f_k}$ for some $k$
(2) If $F$ is isotropic then $F=span{e_1,ldots,e_k}$ for some $k$
(3) If $F$ is co-isotropic then $F=span{e_1,ldots,e_n,f_1,ldots,f_k}$ for some $k$
(4) If $F$ is Lagrangian then $F=span{e_1,ldots,e_n}$
The converse of these statements are easy to prove by applying definitions. But I have no idea how to prove these.
linear-algebra differential-geometry symplectic-geometry symplectic-linear-algebra
asked Dec 4 '18 at 0:38
bbwbbw
47038
$endgroup$
add a comment |
$begingroup$
Let $(V,omega)$ be a symplectic vector space. Let $F subseteq V$ be a subspace.
Show that
$V$ admits a symplectic basis ${e_1,ldots,e_n,f_1,ldots,f_n}$ with the following properties:
(1) If $F$ is symplectic then $F=span{e_1,ldots,e_k,f_1,ldots,f_k}$ for some $k$
(2) If $F$ is isotropic then $F=span{e_1,ldots,e_k}$ for some $k$
(3) If $F$ is co-isotropic then $F=span{e_1,ldots,e_n,f_1,ldots,f_k}$ for some $k$
(4) If $F$ is Lagrangian then $F=span{e_1,ldots,e_n}$
The converse of these statements are easy to prove by applying definitions. But I have no idea how to prove these.
linear-algebra differential-geometry symplectic-geometry symplectic-linear-algebra
asked Dec 4 '18 at 0:38
bbwbbw
47038
$endgroup$
Let $(V,omega)$ be a symplectic vector space. Let $F subseteq V$ be a subspace.
Show that
$V$ admits a symplectic basis ${e_1,ldots,e_n,f_1,ldots,f_n}$ with the following properties:
(1) If $F$ is symplectic then $F=span{e_1,ldots,e_k,f_1,ldots,f_k}$ for some $k$
(2) If $F$ is isotropic then $F=span{e_1,ldots,e_k}$ for some $k$
(3) If $F$ is co-isotropic then $F=span{e_1,ldots,e_n,f_1,ldots,f_k}$ for some $k$
(4) If $F$ is Lagrangian then $F=span{e_1,ldots,e_n}$
The converse of these statements are easy to prove by applying definitions. But I have no idea how to prove these.
linear-algebra differential-geometry symplectic-geometry symplectic-linear-algebra
linear-algebra differential-geometry symplectic-geometry symplectic-linear-algebra
asked Dec 4 '18 at 0:38
bbwbbw
47038
asked Dec 4 '18 at 0:38
bbwbbw
47038
asked Dec 4 '18 at 0:38
bbwbbw
47038
asked Dec 4 '18 at 0:38
bbwbbw
47038
asked Dec 4 '18 at 0:38
bbwbbw
47038
47038
add a comment |
add a comment |
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