Broken hand! Are there exercises that can be done to maintain upper body muscle mass while one hand is broken?

can you strap the leg or wrist weights around your cast (I'm not a doc so don't know if this is safe) I know it isn't heavy weights, but it is better than nothing!

You won't lose much muscle mass over only a month. But if you want to keep in the game, try some of the universal machines. Your problem is that you can't hold anything heavy or put any strain on your damaged hand, but there are ways to attach resistance to the muscle groups without picking something up. A physical therapist can suggest some alternative exercises for the arm and chest area. Ask your doctor for a referral.