Commutator Leavitt Path Algebras

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra LK(E) which lie in the commutator subspace [LK(E), LK(E)]. We then use this result to classify all Leavitt path algebras LK(E) that satisfy LK(E) = [LK(E), LK(E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures