Free path groupoid grading on Leavitt path algebras (Q2828347)

The notion of partial action of a group on an algebra, introduced in previous works by Exel, Dokuchaev and Ferrero, appears also in a purely algebraic context in a work by \textit{M. Ferrero} and \textit{J. Lazzarin} [J. Algebra 319, No. 12, 5247--5264 (2008; Zbl 1148.16022)]. Associated to this notion, we have that of a partial skew group rings which is also considered in the previous reference. Leavitt path algebras have been realized as partial skew group rings by the first author and \textit{D. Royer} [Commun. Algebra 42, No. 8, 3578--3592 (2014; Zbl 1300.16024); J. Algebra 333, No. 1, 258--272 (2011; Zbl 1235.16014); Funct. Anal. Appl. 45, No. 2, 117--127 (2011; Zbl 1271.46043); translation from Funkts. Anal. Prilozh. 45, No. 2, 45--59 (2011)]. In this approach, the group acting is the free group on the edges. In the paper under review, Leavitt path algebras are realized as partial skew groupoid rings. The acting groupoid is the free path groupoid which, in words of the authors, results on a more natural definition of the partial action. The Steinberg model of Leavitt path algebras also involves a graph groupoid which, although related, is not the same as the free path groupoid. By using this description of Leavitt path algebras as partial skew groupoid rings, it is obtained a characterization of (free path groupoid) graded isomorphisms of Leavitt path algebras that preserves generators.