The Symplectic Sum: An Infuriating Construction
Never Let Anybody Chase Your Diagrams for You
The symplectic sum construction upsets me. It is canonical, but not blessed by the categorical dialectic.
Definition 1. A symplectic vector space is a pair where is a vector space and is a non-degenerate anti-symmetric bilinear form.
Example 2. Consider Then,
Exercise 3. Show that a symplectic vector space must have even dimension.
Construction 4. Let be symplectic vector spaces. Then,
makes into a symplectic vector space.
Proof. We show non-degeneracy. Fix Suppose that
for all Take and Then, But, we get
for all So, because is symplectic, we have Thus, for all By non-degeneracy of we have as well. This completes the proof.
Construction 4. is, in an intuitive sense, the canonical symplectic product form. Yet, is not the categorical product.
Definition 5. (The Symplectic Category) Let be symplectic vector spaces. A linear map is symplectic if A symplectomorphism is an invertible symplectic map.
Exercise 6. Check that the composition of symplectic maps is symplectic and that the identity is symplectic. So, there is a category Symp of symplectic spaces and symplectic linear maps.
Exercise 7. Show that every symplectic vector space is symplectomorphic to the one given as Example 2.
Exercise 8. Show that symplectomorphisms preserve volume. (Hint: Use non-degeneracy to show that is a volume form).
“Counter”-example 9. The projection is not symplectic.
Proof. Compute whenever . This completes the proof.