What is the material of which the transparent part of CD cases are made of? It is nicely rigid and really lightweight.
These are called slimline jewel cases.
From the Wikipedia article on Optical Disc Packaging:
The bulk of slimline cases are made with translucent or transparent polystyrene, and are available in multiple colors. A stronger alternative is made from semi-opaque, semi-flexible polypropylene which is strong enough to protect the disc, but flexible enough not to break easily.
Here is an image of a polystyrene case to compare to yours: