Symplectic Spaces
Published (updated: ) in linear-algebra. Tags: linear-algebra, symplectic, vector-spaces.
We come across vector spaces equipped with an inner-product frequently. But instead of an inner product, we could have a skew-symmetric and non-degenerate bilinear form on it. Such spaces are called as symplectic vector spaces. More formally,
Definition: Let be an -dimensional real vector space and be a bilinear map such that
1. . (skew-symmetry)
2. for all implies that . (non-degeneracy)
Then the pair is called a symplectic vector space where is called the symplectic form.
The biggest difference when working with a symplectic space as compared to an inner-product space is that the subspaces need not be well-behaved. That is, you can have subspaces of which are not symplectic with respect to the restricted form . This makes the theory of symplectic spaces much more interesting. Let’s first construct the analog of an orthonormal basis for symplectic spaces.
Let be a symplectic space. We chose two vectors and from such that . This is possible because of non-degeneracy (condition 2). Normalize one of these vectors so that . Now we argue that can be decomposed as the direct sum of and , where and .
If then . If also , then this would imply that .
If and we have , , then we can write .
So . Now we can perform the same set of steps on the space to get vectors . Iterating this many times, we can write , where . Hence the set forms a special basis for our symplectic space . This basis has the following properties
for .
for .
This set is called as a symplectic basis for . This basis is not unique but the number is invariant, as should be clear from the proof. If we look carefully, we have also shown an important feature of symplectic spaces, that it has to have an even dimension always. In fact given any even number , we can easily construct a symplectic space of this dimension: , where we declare the standard basis to be the symplectic basis. This fully defines the form because of bilinearity. is the model symplectic space to keep in mind. We call a set of vectors to be cross-orthonormal if it is of the above form (need not necessarily form a basis). We will also need to define what an isomorphism is in this category.
An isomorphism between symplectic spaces is a linear isomorphism that also preserves the symplectic form, in the following senseis then called a symplectomorphism between and .
Now we investigate the different types of subspaces that a symplectic space could have.
- Let be a vector subspace of such that is non-degenerate. Then we call it a symplectic subspace of .
- Let be a subspace such that is zero, that is for all . Then we call an isotropic subspace of .
- If is an isotropic space, then is called as a co-isotropic subspace of .
Given a symplectic basis , its easy to construct examples of such subspaces.
- is symplectic.
- is isotropic.
- is co-isotropic.
We can see from these examples that there is a borderline case that looks like . This is indeed special and is called as a Lagrangian subspace. Its a subspace that is both isotropic and co-isotropic. We conclude this note with an interesting result involving Lagrangian subspaces.
Let be a Lagrangian subspace of and let denote its dual. Then is naturally symplectomorphic to equipped with the following symplectic form