For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L _K (E) which lie in the commutator subspace L _K (E), L _K (E). We then use this result to classify all Leavitt path algebras L _K (E) that satisfy L _K (E) = L _K (E),L _K (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.
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