We have been discussing maximal isotropic subspaces in Problem Sheet #2 because we proved that they are a way of looking at linear gcs. However, linear gcs are equivalent to maximal isotropic subspaces of \( (V \oplus V^*)_\mathbb{C} \), whereas we have been dealing with maximal isotropic subspaces of \(V\oplus V^*\).
Well, the things we did do not rely on the fact that \( V \) was a real vector space, so they apply for both cases. It is good we did for real vector spaces in case you want to work on Dirac geometry. From now on, we will deal with complex differential forms \( \wedge^\bullet V^*_\mathbb{C} \), where the pairing is defined exactly in the same way.
You can read more about this in the reference I am posting next, where you will find several very important facts that were mentioned in class but not on this blog.
Well, the things we did do not rely on the fact that \( V \) was a real vector space, so they apply for both cases. It is good we did for real vector spaces in case you want to work on Dirac geometry. From now on, we will deal with complex differential forms \( \wedge^\bullet V^*_\mathbb{C} \), where the pairing is defined exactly in the same way.
You can read more about this in the reference I am posting next, where you will find several very important facts that were mentioned in class but not on this blog.
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