Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt pathalgebras (with graphs of weight 1) and cover the algebras $L_K(n, n + k)$constructed by Leavitt. Using Bergman's Diamond lemma, we give normal forms forelements of a weighted Leavitt path algebra. This allows us to produce a basisfor a wLpa. Using the normal form we classify the wLpas which are domains,simple and graded simple rings. For a large class of weighted Leavitt pathalgebras we establish a local valuation and as a consequence we prove thatthese algebras are prime, semiprimitive and nonsingular but contrary to Leavittpath algebras, they are not graded von Neumann regular.
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