Is it illegal to share a YouTube video on Facebook of a full album that is not uploaded by the author?

It would be copyright infringement to copy someone's album to Facebook without the permission of the author or owner of the copyright. The fact someone posted the album on YouTube does not change that equation.  Even if it was legally posted on YouTube, you still cannot copy the album unless you have permission. You may be able to post a link on Facebook to a YouTube site that contains the album. At that point you are not copying the work, you are only providing an online address. If YouTube takes down the album, then the link would be useless. Someone might be able to make the case that by posting the link, you are engaging in contributory infringement, but I have not heard of that argument being made successfully. Good luck, Todd