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.
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