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https://www.amazon.com/exec/obidos/...f=sr_1_1/103-6229518-2985440?v=glance&s=books titledFrom Classical to Quantum Mechanics,

I quote the following: ( [tex]\xi^i [/tex] are coordinate functions)

Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that

[tex]\{\xi^i,\xi^j\} \equiv \omega^i^j[/tex]

is an inversible matrix, we may define the inverse [tex]\omega_i_j[/tex] by requiring

[tex]\omega_i_j \omega^j^k = \delta_i^k[/tex]

We define a tensorial quantity

[tex]\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j[/tex]

which turns out to be a non-degenerate 2-form.

This implies that the dimension of the manifold M is necessarily even.

My questions are the following:

I dont understand the two statement that I have put in red above.

What is a non-degenerate 2-form?

Why does this one above 'turns out' to be non-degenerate?

Why does that imply that M is even?

Additional comments would be welcome. Like concerning the meaning of [tex]\omega [/tex] above.

In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above?

Warm thanks in advance,

Michel

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# Non-degenerate Poisson bracket and even-dimensional manifold

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