Amendment XI. Suits Against StatesThis is a recognition of sovereign immunity of sorts enforced in the federal courts. There is nothing in this amendment prohibiting one state from suing another in the federal courts or prohibiting the federal government from suing a state in the federal court. For that matter, this amendment does not expressly prohibit the citizens of a state from suing their own state in a federal court.
The Judicial power of the United States shall not be construed to extend to any suit in law or equity, commenced or prosecuted against one of the United States by Citizens of another State, or by Citizens or Subjects of any Foreign State.
Nothing prevents a state from providing for suits against that state in the state's own courts. Typically states are immune from suit unless they specifically permit the suit. State officers may be subject to suit in federal courts, as long as the suit is not found to be a suit that is really against the state. State officers are not considered to be acting within their lawful state authority if they are acting illegally, e.g. by violating the constitutional rights of a citizen. State officer may be enjoined in the federal court from violating the United States Constitution.
Federal statutes such as 28 USC sec. 1983 authorize suits for damages against state officers acting "under color of law" who violate individual constitutional rights. These are not considered suits against the states, but rather suits against state officer who have forfeited immunity by acting unlawfully. That is the theory anyway.