# Strongly graded Leavitt path algebras

@article{Nystedt2020StronglyGL,
author={P. Nystedt and Johan Oinert},
journal={arXiv: Rings and Algebras},
year={2020}
}
• Published 2020
• Mathematics
• arXiv: Rings and Algebras
Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.
2 Citations
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#### References

SHOWING 1-10 OF 11 REFERENCES
Group gradations on Leavitt path algebras.
• Mathematics
• 2017
Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewedExpand
The Leavitt path algebra of a graph
• Mathematics
• 2005
Abstract For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K . When K is the field of complex numbers, then L ( E ) is the algebraicExpand
Nonstable K-theory for Graph Algebras
• Mathematics
• 2004
We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property andExpand
Equivalent groupoids have Morita equivalent Steinberg algebras
• Mathematics
• 2013
Let G and H be ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly–Renault–Williams, then the associated Steinberg algebras are MoritaExpand
We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, suchExpand
To every $C^*$ correspondence over a $C^*$-algebra one can associate a Cuntz-Pimsner algebra generalizing crossed product constructions, graph $C^*$-algebras, and a host of other classes of operatorExpand