Yes, UV light kills algae. Fish tank UV purification systems keep water life deader than a zombie.
They do cost in the neighbourhood of $100 USD though. Even a very small system would be more than adequate for your system though. Anyways. The cheapest and fastest way is just to use black everything.
By the way, there's different wavelengths of UV light. Black lights are utterly useless, they're not designed to sterilize things. Water purification UV is specifically designed to kill life, and you shouldn't use any other type of UV to do so. Cause it won't work. =p.