From what you've described, it appears that your girlfriend has somewhat of a codependent relationship with her ex. Her boundaries are not clearly defined.
Q "she has lied again, a trivial lie but why?"
People lie for a purpose. It's an opportunistic behavior. There may be different reasons as to why she's doing it. 1) does not want to hurt you 2) is not sure what to say 3) has an agenda i.e. to keep some sort of a relationship w/ her ex even if it is a platonic one and be with you as well 4) she may have not introduced you to her family because they know something you don't i.e. that she's still involved with her ex... 5) she's embarrassed for some reason from her family, etc/they don't get along...
If you're not happy, you should not make a whole lot of compromises on your end that your girlfriend can't match on hers. From what you've shared, she seems to be more involved with her ex from the work, the house they share, etc. and the time she spends with him is time she could be spending with you. This is something that has to be addressed seriously. She may or may not realize that this is a codependent relationship b/w the two of them. Either way, codependency is never healthy.
You said "she's lied again." Past behavior is a good indicator of future behavior.