• DOI:10.1007/S11856-012-0138-5
• Corpus ID: 119249674

The graded structure of Leavitt path algebras

@article{Hazrat2010TheGS,
  title={The graded structure of Leavitt path algebras},
  author={Roozbeh Hazrat},
  journal={Israel Journal of Mathematics},
  year={2010},
  volume={195},
  pages={833-895}
}

• R. Hazrat
• Published 11 May 2010
• Mathematics
• Israel Journal of Mathematics

A Leavitt path algebra associates to a directed graph a ℤ-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ℤ-grading and characterize the (ℤ-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, Cn-comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕ℤ-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.