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

TL;DR

This paper develops normal forms for weighted Leavitt path algebras, enabling classification of their algebraic properties and revealing differences from unweighted Leavitt path algebras.

Contribution

It introduces normal forms for weighted Leavitt path algebras and classifies when they are simple, prime, or domains, expanding understanding of their structure.

Findings

Normal forms for elements of weighted Leavitt path algebras.

Classification of wLpas as domains, simple, or graded simple rings.

Establishment of local valuation and properties like primeness and nonsingularity.

Abstract

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (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 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.