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 algebraic analog of the Cuntz-Krieger algebra C * (E) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings M n (K) and the Leavitt algebras L(1, n) appear as algebras of the form L(E) for various graphs E. In our main result, we give necessary and sufficient conditions on E which imply that L(E) is simple.  2005 Elsevier Inc. All rights reserved.

Let n be a positive integer. For each 0 ≤ j ≤ n − 1 we let C j n denote Cayley graph for the cyclic group Zn with respect to the subset {1, j}. For any such pair (n, j) we compute the size of the Grothendieck group of the Leavitt path algebra L K (C j n ); the analysis is related to a collection of integer sequences described by Haselgrove in the 1940's. When j = 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras L K (C j n ) as the Leavitt path algebras of graphs having at most three vertices. The analysis in the j = 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.

We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L(E) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz-Krieger C An idempotent e in a ring R is called infinite if e R is isomorphic as a right R-module to a proper direct summand of itself. R is called purely infinite in case every nonzero right ideal of R contains an infinite idempotent. Much recent attention has been paid to the structure of purely infinite simple rings, from both an algebraic (see e.g. [3][4][5]) as well as an analytic (see e.g. [7,8,11]) point of view.

Abstract. In this paper, results known about the artinian and noetherian conditions for the Leavitt path algebras of graphs with finitely many vertices are extended to all row-finite graphs. In our first main result, necessary and sufficient conditions on a row-finite graph E are given so that the corresponding (not necessarily unital) Leavitt path K-algebra L(E) is semisimple. These are precisely the algebras L(E) for which every corner is left (equivalently, right) artinian. They are also precisely the algebras L(E) for which every finitely generated left (equivalently, right) L(E)-module is artinian. In our second main result, we give necessary and sufficient conditions for every corner of L(E) to be left (equivalently, right) noetherian. They also turn out to be precisely those algebras L(E) for which every finitely generated left (equivalently, right) L(E)-module is noetherian. In both situations, isomorphisms between these algebras and appropriate direct sums of matrix rings over K or K[x, x −1 ] are provided. Likewise, in both situations, equivalent graph theoretic conditions on E are presented.

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite-dimensional Leavitt path algebras arise.