A Noncommutative Nullstellensatz for Leavitt Path Algebras

Hilbert's Nullstellensatz states that a quotient of the algebra of polynomial functions on an algebraic variety over an algebraically closed field is the algebra of polynomial functions on a (sub)variety if and only if its radical is trivial. We prove a noncommutative analog: A quotient of a Leavitt path algebra over an algebraically closed field is isomorphic to a Leavitt path algebra if and only if its radical is trivial. This suggests that a Leavitt path algebra behaves like a noncommutative algebra of polynomial functions on a directed graph. Along the way we provide a complete answer to the question "When is a quotient of a Leavitt path algebra isomorphic to a Leavitt path algebra?". We also define a Morita invariant stratification and a parametrization of the ideal space of a Leavitt path algebra and show that a generic quotient of a Leavitt path algebra is a Leavitt path algebra. We end this article by pointing out a connection with quantum spaces strengthening the analogy with function algebras and several examples involving algebraic quantum spaces illustrating our results.