Hi!
Divorce is SO hard on kids! My ex was a liar. You don't say how old your daughter is, but it's best she NOT tell you anything about him, AND the other way around. Even if he talks about you, and lies about you, you know the truth, and she will have to find out for herself he is a liar. If your behaviour and words are respectful, he'll make an a$$ out of himself soon enough.
The kids feel SO torn when they have to 'choose' in any way, it isn't fair. It will be hard, but if you are honest, and refuse to listen to anything about him, or contribute to the drama, your daughter will see, on her own, the truth. You won't be able to hide it from her. She may need therapy if she continues to be in the middle. If you refuse to argue, there can't be an argument.
Not trying to tell you what to do. I have 30 year old son, that only knows his parents as divorced. That makes me sorry, but to have stayed married to his father, would have meant a life of yelling and drama! More than there was already.
Good luck, and try not to react when you want to SO bad!!