Applications of normal forms for weighted Leavitt path algebras: simple rings and domains

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Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras Lₖ(n, n + k) constructed by Leavitt. Using Bergman's Diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for 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 path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.