A group-graded \(K\)-algebra \(A=\bigoplus_{g\in G}A_g\) is called locally finite in case each graded component \(A_g\) is finite-dimensional over \(K\). For any row-finite graph (i.e., no vertex emits infinitely many edges) and any field \(K\) the Leavitt path algebra \(L_K(E)\) has been defined and vigorously investigated (see e.g. G. Abrams and G. Aranda Pino [J. Algebra 293, No. 2, 319-334 (2005; Zbl 1119.16011)] or P. Ara, M. A. Moreno and E. Pardo [Algebr. Represent. Theory 10, No. 2, 157-178 (2007; Zbl 1123.16006)]).
In the article under review, the authors characterize the graphs \(E\) for which the Leavitt path algebra \(L_K(E)\) is locally finite in the standard \(\mathbb{Z}\)-grading. If \(c=e_1e_2\cdots e_m\) is a cycle in a graph, then an edge \(e\) is called an exit for \(c\) in case there exists \(e_i\) in the cycle which shares the same source vertex as \(e\), but for which \(e\neq e_i\).
The main result is Theorem 3.10, where the following are shown to be equivalent for any graph \(E\) and any field \(K\): (i) \(L_K(E)\) is locally finite; (ii) \(L_K(E)\) is left or right Noetherian; (ii)’ \(L_K(E)\) is left and right Noetherian; (iii) \(E\) has at most finitely many vertices (and edges), and no cycle in \(E\) has an exit. Furthermore, in Theorem 3.8 a description of these algebras is given in terms of direct sums of matrix rings of various sizes over the Laurent polynomial ring \(K[x,x^{-1}]\). In addition, the locally finite just infinite Leavitt path algebras (every nonzero ideal has finite codimension) are described, see Corollary 3.7.