No, why would you be able to sue the state? They did nothing wrong, according to what you write: they issued a warrant for the name associated with the crime. Even if they turned out to be wrong, they were neither negligent nor intentionally tried to harm you; rather, they acted in good faith on the information they had. There is no liabilirty for being wrong in good faith.
You can, of course, sue the person who used your name.
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