Can I sue the woman who invited my husband to live with her knowing he was married?

No, not in this day and age. Once upon a time adultery was a crime and you might have been able to sue in tort, but the courts really don't deal much with philandering spouses any more.

You do have rights with respect to your husband, however. I presume that you will be filing for divorce, and you may be entitled to temporary maintenance during the divorce and perhaps maintenance for a period of years thereafter (there's lots of facts that go into that determination). you should get an attorney to help you with the divorce and with the motion for temporary orders that might include maintenance, health insurance etc.
If you have children by this man, he has an legal obligation to contribute to their support today and until they have reached majority.

Two things to remember:
1. get an attorney to make sure you understand your rights so you can assert them
2. living well is the best revenge. Go have a great life!