Mixing
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
52239
$endgroup$
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
52239
$endgroup$
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
52239
$endgroup$
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
52239
asked Dec 4 '18 at 0:38
bbwbbw
52239
asked Dec 4 '18 at 0:38
bbwbbw
52239
asked Dec 4 '18 at 0:38
bbwbbw
52239
asked Dec 4 '18 at 0:38
bbwbbw
52239
52239
add a comment |
add a comment |
1 Answer
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$begingroup$
I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(mathbb {R})$ acts transitively on $mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.
answered Jan 24 at 5:56
BenBen
4,283617
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1 Answer
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$begingroup$
I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(mathbb {R})$ acts transitively on $mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.
answered Jan 24 at 5:56
BenBen
4,283617
$endgroup$
add a comment |
$begingroup$
I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(mathbb {R})$ acts transitively on $mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.
answered Jan 24 at 5:56
BenBen
4,283617
$endgroup$
add a comment |
$begingroup$
I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(mathbb {R})$ acts transitively on $mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.
answered Jan 24 at 5:56
BenBen
4,283617
$endgroup$
I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(mathbb {R})$ acts transitively on $mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.
answered Jan 24 at 5:56
BenBen
4,283617
answered Jan 24 at 5:56
BenBen
4,283617
answered Jan 24 at 5:56
BenBen
4,283617
answered Jan 24 at 5:56
BenBen
4,283617
4,283617
add a comment |
add a comment |
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