Yes. If there was not probable cause to arrest you, then the arrest was unlawful. To preserve your state law causes of action you must file a claim with the city within six months of the incident. You should contact an attorney in your area to do this immediately.
From what you are describint, you are a victim of identity theft. Police can be sued for violating your Constitutional rights, but whether or not they can be sued successfully under the scenario you describe is something you need to bring up with a local lawyer who can fully analyze your individual situation and the particular facts of your case.