Gonzalo Aranda Pino

Socle theory for Leavitt path algebras of arbitrary graphs

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavit t path algebra, extending so the results for row-finite graphs in the previous paper (12) ( but with different methods). A concrete description of the socle of a Leavitt path algebra i s obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones.

Kumjian-Pask algebras of higher-rank graphs

We introduce higher-rank analogues of the Leavitt path algebras, which we call the Kumjian-Pask algebras. We prove graded and Cuntz-Krieger uniqueness theorems for these algebras, and analyze their ideal structure.

Chain conditions for Leavitt path algebras

Abstract In this paper we give necessary and sufficient conditions on a row-finite graph E so that the corresponding (not necessarily unital) Leavitt path K-algebra LK (E) is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over K or K[x, x –1].

The center of a Leavitt path algebra

In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a K-vector space is found for Z(L(E)) when L(E) enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra L(E) is central (i.e. the center reduces to the base field K) when L(E) is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.

Decomposable Leavitt path algebras for arbitrary graphs

For any field

The Leavitt path algebra of a graph

K

The Leavitt path algebras of arbitrary graphs

and for a completely arbitrary graph

Purely infinite simple Leavitt path algebras

E

Graph algebras: bridging the gap between analysis and algebra

, we characterize the Leavitt path algebras

Prime Spectrum and Primitive Leavitt Path Algebras

L_K(E)