Leavitt Path Algebras

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra L _K (E) for an arbitrary directed graph E and field K. After providing some basic examples, we show how Leavitt path algebras are related to the monoid realization algebras of Bergman, as well as to graph C ^∗-algebras. We then introduce the more general construction of relative Cohn path algebras C _K ^X (E), and show how these are related to Leavitt path algebras. We finish by describing how any Cohn (specifically, Leavitt) path algebra may be constructed as a direct limit of Cohn (specifically, Leavitt) path algebras corresponding to finite graphs. We conclude the chapter with an historical overview of the subject.

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural \(\mathbb{Z}\)-grading on L _K (E). We then present the Reduction Theorem; this result describes how elements of L _K (E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed, including the Uniqueness Theorems. We then establish in the Structure Theorem for Graded Ideals a precise relationship between graded ideals and explicit sets of idempotents (arising from hereditary and saturated subsets of vertices, together with breaking vertices). With this description of the graded ideals having been achieved, we focus in the remainder of the chapter on the structure of all ideals. We achieve in the Structure Theorem for Ideals an explicit description of the entire ideal structure of L _K (E) (including both the graded and non-graded ideals) for an arbitrary graph E and field K. This result utilizes the Structure Theorem for Graded Ideals together with the analysis of the ideal generated by vertices which lie on cycles having no exits. A number of ring-theoretic results follow almost immediately from the Structure Theorem for Ideals, including the Simplicity Theorem. Along the way, we describe the socle of a Leavitt path algebra, and we achieve a description of the finite dimensional Leavitt path algebras.

The richness of the idempotent structure of Leavitt path algebras lies at the heart of the subject; in this chapter we present a number of topics which fall under this umbrella. These include: the purely infinite property (for both simple and non-simple algebras); the structure of the monoid of finitely generated projective modules; the exchange property; von Neumann regularity; and primitive idempotents.

In this chapter we provide descriptions of Leavitt path algebras satisfying various well-studied ring-theoretic properties. These include: primeness and primitivity; chain conditions on one-sided ideals; self-injectivity; and the stable rank.

In this chapter we investigate the connections between Leavitt path algebras (with coefficients in \(\mathbb{C}\)), and their analytic counterparts, the graph C ^∗-algebras. We start by giving a brief overview of graph C ^∗-algebras, and then show how the Leavitt path algebra \(L_{\mathbb{C}}(E)\) naturally embeds as a dense ∗-subalgebra of the graph C ^∗-algebra C ^∗(E). We analyze the structure of the closed ideals in C ^∗(E) for row-finite graphs, and compare this structure to the ideal structure of the corresponding Leavitt path algebra L _K (E). We finish the chapter by considering numerous properties which are simultaneously shared by C ^∗(E) and \(L_{\mathbb{C}}(E)\).

In this chapter we investigate many of the K-theoretic properties of L _K (E). We start by considering the Grothendieck group K ₀(L _K (E)), and then subsequently the Whitehead group K ₁(L _K (E)). Next, we discuss one of the central currently-unresolved questions in the subject (the so-called Algebraic Kirchberg Phillips Question) which asks whether certain K ₀ data is sufficient to classify the purely infinite simple unital Leavitt path algebras up to isomorphism. We conclude with a discussion of tensor products of Leavitt path algebras, and Hochschild homology.

We conclude the book with various observations regarding three important aspects of Leavitt path algebras. First, we describe various generalizations of, and constructions related to, Leavitt path algebras. Next, we present some applications of Leavitt path algebras (specifically, we give some examples of results from outside the subject of Leavitt path algebras per se which have been established using the machinery developed for Leavitt path algebras). Finally, we consider some still-unresolved questions of interest.