Leavitt Path Algebras

Abstract

The central concept of this thesis is that of Leavitt path algebras, a notion introduced by both Abrams and Aranda Pino in [AA1] and Ara, Moreno and Pardo in [AMP] in 2004. The idea of using a field K and row-finite graph E to generate an algebra LK(E) provides an algebraic analogue to Cuntz and Krieger’s work with C*-algebras of the form C*(E) (which, despite the name, are analytic concepts). At the same time, Leavitt path algebras also generalise the algebras constructed by W. G. Leavitt in [Le1] and [Le2], and it is from this connection that the Leavitt path algebras get their name.

Although the concept of a Leavitt path algebra is relatively new, in the years since the publication of [AA1] there has been a flurry of activity on the subject. Many results were initially shown for row-finite graphs, then extended to countable (but not necessarily row-finite) graphs (as in [AA3]) and then finally shown for completely arbitrary graphs (see, for example, [AR]). Most of the research has focused on the connections between ring-theoretic properties of LK(E) and graphtheoretic properties of E (for example [AA2], [AR] and [ARM2]), the socle and socle series of a Leavitt path algebra ([AMMS1], [AMMS2] and [ARM1]) and analogues between LK(E) and their C*-algebraic equivalents C*(E) (for example [To]). Some papers have classified certain sets of Leavitt path algebras, such as [AAMMS], which classifies the Leavitt path algebras of graphs with up to three vertices (and without parallel edges).

In Chapter 1 we cover the ring-, module- and graph-theoretic background necessary to examine these algebras in depth, as well as taking a brief look at Morita equivalence, a concept that will prove useful at various points in the thesis. We introduce Leavitt path algebras formally in Chapter 2 and look at various results that arise from the definition. We also examine simple and purely infinite simple Leavitt path algebras, as well as the ‘desingularisation’ process, which allows us to construct row-finite graphs from graphs containing infinite emitters in such a way that their corresponding Leavitt path algebras are Morita equivalent. In Chapter 3 we examine the socle and socle series of a Leavitt path algebra, while in Chapter 4 we examine Leavitt path algebras that are von Neumann regular, pi-regular and weakly regular, as well as Leavitt path algebras that are self-injective. Finally, in Appendix A we give a detailed definition of a direct limit, a concept that recurs throughout the thesis.