I upgraded my forum to 4.0. Can I delete the old files in the file categories that do not exist in 4.0 like cache, hooks, interface, ips_kernel, lofiversion, public?

Yes you can do that.

You should have deleted before you upgraded.

You could deleted everything minus Uploads Folder, Public Folder and conf_global.php.

After the upgrade you can also delete the Public Folder.

Downloads folder should also be retained if you have the downloads system installed.